Breaking Universality in the Lower Order Terms in the 1-level and 2-level Density of Holomorphic Cusp Newforms

This page follows the manuscript closely. It is written as mathematical exposition rather than as a summary page.

1 Introduction

Since the 1970s, the study of zeros of $L$-functions has revealed striking connections with random matrix theory. In particular, the local statistics of zeros high on the critical line are conjecturally governed by the eigenvalue statistics of the Gaussian Unitary Ensemble, a phenomenon often referred to as universality. The behavior of zeros near the central point $s=1/2$, however, is more sensitive to the arithmetic of the underlying family. This distinction is particularly important for questions such as the Birch and Swinnerton–Dyer conjecture, which are governed by the order of vanishing at the central point [Hej, Mon, Od1, Od2, RS].

The Katz–Sarnak philosophy predicts that, for a family of $L$-functions ordered by analytic conductor, the statistics of zeros near the central point are governed by the eigenvalue statistics of a classical compact group determined by the family [katz1999random, katz1999zeros]. This prediction has been supported by a large body of work, including results on moments [CF2000, CFKRS2005, KeSn1, KeSn2, KeSn3] and on $n$-level densities for suitable test functions [DM1, FI, Gu, HR, HM, Mil2, OS, RR1, Ro, Rub, Yo2].

Given an $L$-function $L(s,f)$ associated with a holomorphic cusp newform $f \in H^*_k(N)$ of level $N$ and weight $k$, we assume the Generalized Riemann Hypothesis (GRH).¹ Thus, all non-trivial zeros of $L(s,f)$ are of the form $\rho_f=1/2+i\gamma_f$. We then enumerate zeros by their imaginary part $0 \leq \gamma_f^{(1)}\leq \gamma_f^{(2)}\leq \cdots$ with $\gamma_f^{(-j)}=-\gamma_f^{(j)}$ by symmetry. We define the $n$-level density by \[D_n(f;\Phi)\ :=\ \sum_{\substack{j_1,\ldots,j_n\\j_i \neq \pm j_k}}\phi_1\left( \gamma^{(j_1)}_f \frac{\log (R)}{2\pi}\right)\cdots \phi_n\left( \gamma^{(j_n)}_f \frac{\log (R)}{2\pi}\right),\] where² $\Phi(x_1, \ldots, x_n)=\phi_1(x_1)\cdot \phi_2(x_2)\cdots\phi_n(x_n)$ is a Schwartz test function and $R$ is the analytic conductor of $f$. We rescale the zeros near the central point by $\log (R)/2\pi.$ Note that for $f \in H^{*}_k(N)$, its analytic conductor $R$ is $(64\pi^2)^{-1}N(k+1)(k+3)$, and so $\log(R) \sim\log(N)$.

The Katz–Sarnak prediction concerns the leading term of the density. Lower-order terms, however, retain finer arithmetic information that is invisible in the limiting random-matrix model, and they dictate the rate of the convergence to the conjectural limit. Indeed, families with the same symmetry type can exhibit different lower-order behavior. This phenomenon has been observed in several families of $L$-functions; for example, [young2005lower] showed that lower-order terms in families of elliptic curves can depend strongly on the arithmetic of the family (see also [GAO_ZHAO_2023, Fiorilli_2015, Fourvy_2003], for investigations of the lower order terms for other families of $L$-functions).

We study the lower-order behavior of the 1- and 2-level density of zeros for families of $L$-functions derived from holomorphic cusp newforms. In particular, by taking $L$-functions associated to level-$N$ weight-$k$ newforms, we demonstrate that the lower-order terms depend explicitly on the prime factorization of $N$ as $N \to \infty$.

As emphasized in [Miller2009_LOTerms_1LevelDensity], the Central Limit Theorem provides a useful analogy. After normalization, the leading Gaussian limit is universal, while the Berry–Esseen theorem shows that the rate of convergence retains information about higher moments [Berry1941, Esseen1942]. The same distinction appears here: the Katz–Sarnak main term is universal at the level of symmetry type, whereas the lower-order terms retain arithmetic information specific to the factorization of the level.

Such lower-order differences can encode arithmetic phenomena that are invisible in the limiting symmetry type. In particular, work on elliptic-curve families shows that family-dependent lower-order terms influence the distribution of zeros near the central point and can account for effects such as excess rank [Mil2, Miller2005, Miller2009_LOTerms_1LevelDensity].

1.1 Main results

Definition 2. Let $H_k^*(N)$ be the set of cuspidal newforms of level $N$ and weight $k$. We fix $k$, and let $\mathcal{F}_{N}$ be the associated set of $L$-functions defined through [eq:lfndefn]. The family of $L$-functions is $\mathcal{F}:= \bigcup_N\mathcal{F}_{N}$, where we leave the union ambiguous as we ultimately study what happens as $N \to \infty$ through different factorizations. To each $f \in \mathcal{F}_{N}$ we assign a weight $w_R(f)$ defined in [eq:wR_defn], where $R$ is the analytic conductor of $f$ (note that $R$ is fixed across all $f \in \mathcal{F}_{N}$). Let \[\begin{aligned} W_R(\mathcal{F}_{N})\ :=\ \sum_{f\in \mathcal{F}_{N}}w_R(f), \end{aligned}\] and let $\phi,\phi_1,\phi_2$ be our test functions which are Schwartz, whose Fourier transform defined by \[\begin{aligned} \widehat\phi(y)\ :=\ \int_{-\infty}^\infty \phi(x) e^{-2\pi i xy}dx \end{aligned}\] have compact support. Then, we investigate the lower-order terms of the weighted first-level and second-level densities defined by \[\label{eq:2ndlvl-density-def} D_1(\mathcal{F},\phi)\ :=\ \lim_{N\to \infty}\frac{1}{W_R(\mathcal{F}_{N})}\sum_{f \in \mathcal{F}_{N}}w_R(f)D_1(f ; \phi),\] \[\label{eq:2ndlvl-density-def-2} D_2(\mathcal{F},\phi_1,\phi_2)\ :=\ \lim_{N\to \infty}\frac{1}{W_R(\mathcal{F}_{N})}\sum_{f \in \mathcal{F}_{N}}w_R(f)D_2(f ; \phi_1,\phi_2),\]where \[D_1(f, \phi) \ :=\ \displaystyle\sum_{i}\phi\left(\frac{\log (R)}{2\pi}\gamma_f^{(i)}\right),\] \[D_2(f; \phi_1,\phi_2)\ :=\ \sum_{\substack{i,j\\i \neq \pm j}}\phi_1 \left( \frac{\log (R)}{2\pi} \gamma_f^{(i)} \right)\phi_2 \left( \frac{\log (R)}{2\pi} \gamma_f^{(j)} \right).\] We occasionally write $D_1 = D_1(\mathcal{F})=D_1(\mathcal{F}, \phi)$, and similarly for $D_2$. We also abuse notation slightly by writing \[\begin{aligned} D_1(\mathcal{F}_N, \phi)\ =\ \frac{1}{W_R(\mathcal{F}_N)}\sum_{f \in \mathcal{F}_N}w_R(f)D_1(f;\phi) \end{aligned}\] so $D_1(\mathcal{F},\phi) = \lim_{N\to \infty}D_1(\mathcal{F}_N,\phi)$, and similarly for $D_2$.

Let $N = q_1^{a_1}\cdots q_n^{a_n}$, where $q_1,\dots,q_n$ are distinct primes, $a_i \in \mathbb{Z}^+$, and $q_i \asymp N^{\delta_i}$ for $\delta_i \geq 0$. We study how the behavior of the lower order terms is affected by different $\delta_1, \dots, \delta_n$. Adapting the strategy of [Miller2009_LOTerms_1LevelDensity] to these more general levels, we decompose the densities $D_1(\mathcal{F})$ and $D_2(\mathcal{F})$ into explicit formulas which may be broken down and evaluated. This decomposition is also what allows us to identify precisely where the universality of the lower-order terms can break.

For the one-level density, the explicit formula [eq:1lvl density breakdown] separates $D_1(\mathcal{F})$ into expressions involving $U_{k,N}(\phi)$ and $S_1(\mathcal{F}_n,\phi)$, defined in [eq:Ukn_def] and [eq:S1_def] respectively. The former is evaluated independently of the factorization of $N$ in Theorem 12. The latter is decomposed in Theorem 15 as \[\begin{aligned} S_1(\mathcal{F}_N,\phi) \ =\ S_{A'}(\mathcal{F}_N)+S_A(\mathcal{F}_N) +O\left( \log^{-4}(R) \right). \label{eq:S1_firstmention} \end{aligned}\] The expressions $S_{A'}(\mathcal{F}_N)$ and $S_A(\mathcal{F}_N)$ are defined in [eq:SA'_defn] and [eq:SA_defn]. These definitions decompose further into cases dependent on whether a prime $p$ divides the level $N$; we define these in [def:A'_r] and [def:A_r] as $A'_{r,N}(p)$ and $A_{r,N}(p)$.

The two-level density is treated by first applying the inclusion-exclusion identity [eq:2lvl_break_down]. Consequently, the two-level density is determined by the one-level densities together with the product term $S_2(\mathcal{F}_N,\phi_1,\phi_2)$ defined in [def:S2]. The term $S_2$ is then decomposed in Theorem 17 as \[\begin{aligned} S_2(\mathcal{F}_N,\phi_1,\phi_2) = S_{B''}(\mathcal{F}_N)+S_{B'}(\mathcal{F}_N)+S_{B_f}(\mathcal{F}_N) +S_{B_\infty}(\mathcal{F}_N) +O\left ( \log^{-4}(R) \right ) . \label{eq:S2_firstmention} \end{aligned}\] The corresponding quantities $B''_{r_1,r_2,N}$, $B'_{r_1,r_2,N}$, and $B_{r_1,r_2,N}$ are defined in [not:B'' i,j]–[not:B i,j]. The four terms above arise from the four possibilities for whether the primes $p_1$ and $p_2$ divide the level; the explicit splitting is displayed in [eq:S_2 formula].

This gives the following hierarchy of expressions: \[\boxed{ \begin{array}{c} D_1,\ D_2 \\[2mm] \downarrow \\ U_{k,N},\ S_1,\ S_2 \\[2mm] \downarrow \\ S_{A'},\ S_A,\ S_{B''},\ S_{B'},\ S_{B_f},\ S_{B_\infty} \\[2mm] \downarrow \\ A'_{r,N},\ A_{r,N},\ B''_{r_1,r_2,N},\ B'_{r_1,r_2,N},\ B_{r_1,r_2,N}. \end{array}}\]

Once we sufficiently decompose our expressions of interest, we evaluate the components for different factorization profiles of $N$. This is the content of Section 6, where we obtain the relevant uniform estimates for $A'_{r,N}(p)$, $A_{r,N}(p)$, $B''_{r_1,r_2,N}(p_1,p_2)$, $B'_{r_1,r_2,N}(p_1,p_2)$, and $B_{r_1,r_2,N}(p_1,p_2).$

In Section 7, we evaluate $S_{A'},S_{A},S_{B''},S_{B'},S_{B_f}$, and $S_{B_{\infty}}$ based on the estimates from Section 6. However, because of the $O\left ( \log^{-4}(R) \right )$ terms present in [eq:S1_firstmention] and [eq:S2_firstmention], we primarily focus on whether or not a given term exceeds this error. Section 6 shows that $A$, $A'$, $B$, $B'$, and $B''$ can be decomposed into a main term and an error term. For some of these expression when $p_1$ and/or $p_2$ are non-constant, the main term vanishes. The main (resp. error) term from $A$, $A'$, $B$, $B'$, and $B''$ will contribute only towards the main (resp. error) term in $S_{A'},S_{A},S_{B''},S_{B'},S_{B_f}$, and $S_{B_{\infty}}$. Thus, depending on how the level $N$ factors, certain expressions like $S_{A'}$ (Theorem 32), $S_{B''}$ (Theorem 40), and $S_{B'}$ (Theorem 41) sometimes fall below $O\left ( \log^{-4}(R) \right )$, and sometimes fall above it. This dependence on the factorization of $N$ is how the universality is broken for lower order terms.

Theorem 3. Let $N = q_1^{a_1}\cdots q_n^{a_n}$ for fixed $a_i \in \mathbb{Z}^+$, fixed distinct primes $q_1,\dots,q_n$, and fixed $\delta_i \geq 0$ with $q_i \asymp N^{\delta_i}$. Let $\mathcal{F}:= \bigcup \mathcal{F}_{N}$ be the family of $L$-functions corresponding to these levels $N$ as $N \to \infty$, and let $Q := \left \{ q_i \mid a_i = 1,\ \delta_i = 0 \right \}$. Let $\phi$ be an even Schwartz function with $\widehat\phi$ supported in $[-\sigma,\sigma]$ for $\sigma < 0.22$ and $\widehat\phi(0) \neq 0$. If $\mathcal{F}$ and $\widetilde\mathcal{F}$ are two such families, with associated sets $Q$ and $\widetilde Q$, whose levels run through sequences with $N \asymp \widetilde N$, then \[\begin{aligned} \label{eq:D1-comparison} \lim_{N \to \infty} \log(R)\left [ D_1(\mathcal{F}_{N},\phi)-D_1(\widetilde\mathcal{F}_{\widetilde N},\phi) \right ] \ =\ -2\widehat\phi(0)\left ( \beta_{Q,1}-\beta_{\widetilde Q,1} \right ) , \end{aligned}\] with $\beta_{Q,1}$ as in [eq:gammaQ-def]. This limit is nonzero whenever exactly one of $Q,\widetilde Q$ is empty. In other words, the universality of the lower-order terms in $D_1(\mathcal{F},\phi)$ breaks when $N$ has a constant prime factor $q \Vert N$ as $N \to \infty$.

A similar statement emerges at the second level. For the test functions $\phi_1,\phi_2$, let \[\begin{aligned} \label{eq:Xi-def} \Xi(\phi_1,\phi_2)\ :=\ \int_{-\infty}^\infty \phi_1(x)dx\int_{-\infty}^\infty \phi_2(x)dx-\int_{-\infty}^\infty \phi_1(x)\phi_2(x)dx. \end{aligned}\]

Theorem 4. Let $N = q_1^{a_1}\cdots q_n^{a_n}$ for fixed $a_i \in \mathbb{Z}^+$, fixed distinct primes $q_1,\dots,q_n$, and fixed $\delta_i \geq 0$ with $q_i \asymp N^{\delta_i}$. Let $\mathcal{F}:= \bigcup \mathcal{F}_{N}$ be the family of $L$-functions corresponding to these levels $N$ as $N \to \infty$, and let $Q := \left \{ q_i \mid a_i = 1,\ \delta_i = 0 \right \}$. Let $\phi_1,\phi_2$ be even Schwartz functions with $\widehat\phi_1,\widehat\phi_2$ supported in $[-\sigma,\sigma]$ for $\sigma < 0.11$ and $\Xi(\phi_1,\phi_2) \neq 0$. If $\mathcal{F}$ and $\widetilde\mathcal{F}$ are two such families of squarefree levels $N \asymp \widetilde N$, with associated sets $Q$ and $\widetilde Q$, then \[\begin{aligned} \label{eq:D2-comparison} \lim_{N \to \infty} \log(R)\left [ D_2(\mathcal{F}_{N},\phi_1,\phi_2)-D_2(\widetilde\mathcal{F}_{\widetilde N},\phi_1,\phi_2) \right ] \ =\ -4\,\Xi(\phi_1,\phi_2)\left ( \beta_{Q,1}-\beta_{\widetilde Q,1} \right ) , \end{aligned}\] with $\beta_{Q,1}$ as in [eq:gammaQ-def]. This limit is nonzero whenever exactly one of $Q,\widetilde Q$ is empty. In other words, the universality of the lower-order terms in $D_2(\mathcal{F},\phi_1,\phi_2)$ breaks when $N$ has a constant prime factor $q \Vert N$ as $N \to \infty$.

1.2 The explicit formula

The explicit formula described above is introduced here. We write $D_2(f,\phi_1,\phi_2)$ in terms of $D_1(f,\phi_i)$ using inclusion-exclusion. \[\label{eq:2lvl_break_down} D_2(f; \phi_1, \phi_2)\ =\ D_1(f, \phi_1)D_1(f, \phi_2)-2D_1(f, \phi_1\phi_2)+\frac{1-\varepsilon_f}{2}\phi_1(0)\phi_2(0),\] where $\varepsilon_f = \pm1$ depending on whether the functional equation associated with the completed $L$-function $\Lambda(f,s)$ is even or odd. Then, we obtain \[\label{eq:2lvl_from_1_lvl} D_2\left(\mathcal{F}\right) \ =\ \lim_{N\to \infty } \frac{1}{W_R(\mathcal{F}_{N})}\sum_{f\in \mathcal{F}_{N}} w_R(f)\left(D_1(f, \phi_1)D_1(f, \phi_2)-2D_1(f, \phi_1\phi_2)+\frac{1-\varepsilon_f}{2}\phi_1(0)\phi_2(0)\right).\] We assume that the reader is familiar with standard properties of $L$-functions (see for example [iwaniec-kwalski2004analytic]), and in particular with Satake parameters. We denote the Satake parameters of $f$ at $n$ by $\alpha_f(n)$ and $\beta_f(n)$. The Satake parameters at $n$ are related to the corresponding Hecke eigenvalue, $\lambda(n)$. For $p \nmid N$ we have \[\lambda_f(p) \ = \ \alpha_f(p) + \beta_f(p),\]\[\alpha_f(p) \beta_f(p) \ =\ 1\] and $|\alpha_f(p)|\ =\ 1$. For $p \mid N,$ we take $\alpha_f(p) \ =\ \lambda_f(p)$ and $\beta_f(p) \ =\ 0$. Our main tool is the explicit formula (see [ILS] (4.11), for example), \[\label{eq:1lvl density breakdown} D_1(f, \phi) \ = \ \frac{U_{k,N}(\phi)}{\log(R)} - 2\sum_{p}\sum_{m=1}^\infty \frac{\alpha_f(p)^{m}+\beta_f(p)^{m}}{p^{m/2}} \frac{\log(p)}{\log(R) } \widehat\phi\left(m \frac{\log(p)}{\log(R) }\right)\] where³ \[\label{eq:Ukn_def} U_{k,N} (\phi)\ :=\ 2\widehat\phi(0)\log\left(\frac{\sqrt{N}}{\pi} \right) + \int_{-\infty}^\infty \psi\left(\frac{k}{4}+ \frac{2\pi i x}{\log(R) }\right)\phi(x)dx + \int_{-\infty}^\infty \psi\left(\frac{k}{4}+ \frac{1}{2} +\frac{2\pi i x}{\log(R)}\right)\phi(x)dx,\] and $\psi$ is the digamma function. In addition, define \[\label{eq:S1_def} S_1(\mathcal{F}_{N},\phi)\ := \ -2\sum_{p}\sum_{m=1}^\infty\frac{1}{W_R(\mathcal{F}_{N})}\sum_{f\in\mathcal{F}_{N}} w_R(f) \frac{\alpha_f(p)^m+\beta_f(p)^m}{p^{m/2}}\frac{\log(p)}{\log(R)}\widehat\phi\left(m\frac{\log(p)}{\log(R)}\right)\] and

Then we can rewrite $D_1(\mathcal{F},\phi)$ as \[\label{eq:D1explicitformula} D_1(\mathcal{F},\phi) \ = \ \lim_{N\to \infty}\frac{U_{k,N}(\phi)}{\log(R)} + S_1(\mathcal{F}_{N},\phi),\] and using [eq:2lvl_from_1_lvl] we can rewrite $D_2(\mathcal{F}, \phi_1,\phi_2)$ as \[\begin{aligned} \label{eq:D2explicitformula} D_2\left(\mathcal{F}, \phi_1,\phi_2\right) \ =& \ \lim_{N\to \infty}\frac{U_{k,N}(\phi_1)U_{k,N}(\phi_2)}{\log^2(R)} +\left(\frac{U_{k,N}(\phi_{2})}{\log(R)}\right)S_1(\mathcal{F}_{N},\phi_1) \notag \\ &+\left(\frac{U_{k,N}(\phi_{1})}{\log(R)}\right)S_1(\mathcal{F}_{N},\phi_2)+S_2(\mathcal{F}_{N},\phi_1,\phi_2) \notag- 2\left(\frac{U_{k,N}(\phi_1\phi_2)}{\log(R)} + S_1(\mathcal{F}_{N},\phi_1\phi_2)\right) \\ \ &+\phi_1(0)\phi_2(0)\frac{\sum_{f\in\mathcal{F}_{N}}w_R(f)(1-\varepsilon_f)}{2W_R(\mathcal{F}_{N})}. \end{aligned}\]

Therefore, by computing $U_{k,N}(\phi), ~ S_1(\mathcal{F}_{N},\phi),$ and $S_2(\mathcal{F}_{N},\phi_1,\phi_2)$ up to $O\left ( \log^{-4}(R) \right )$ error for arbitrary $\phi,\phi_1,$ and $\phi_2$ (even) Schwartz functions with compactly supported Fourier transforms, we can calculate $D_1(\mathcal{F},\phi)$ and $D_2\left(\mathcal{F},\phi_1,\phi_2\right)$ up to $O\left ( \log^{-4}(R) \right )$ error.

2 Preliminaries

We define the $N^{\text{th}}$ congruence subgroup of $\mathrm{SL}_2(\mathbb{Z})$ by \[\label{eq:cong-subgroup} \Gamma_0(N) \ := \ \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}_2(\mathbb{Z}) \ \Bigg| \ c \equiv 0 \pmod{N} \right\}.\] We are interested in how the $N^{\text{th}}$ congruence subgroup acts on certain holomorphic functions $f: \mathbb{H}\to \mathbb{C}$, where $\mathbb{H}$ denotes the upper half complex plane. Specifically, we are interested in functions $f$ that admit a symmetry when acted on by $\Gamma_0(N)$ by the following relation: \[f\left( \frac{az + b}{cz + d} \right)\ = \ (cz + d)^{k} f(z).\] If these functions satisfy the extra stipulation that they vanish at their cusps, we call these functions $f$ a holomorphic cusp form of weight $k$ and level $N$. Such functions admit a Fourier expansion \[\label{eq:Fourier} f(z)\ =\ \sum_{n=1}^{\infty}a_f(n)e^{2\pi inz},\] where the coefficients $a_f(n)$ are normalized so that $a_f(1)=1.$⁴ We denote the space of all holomorphic cusp forms of weight $k$ and level $N$ by $S_k(N)$. This space is a finite-dimensional Hilbert space with inner product given by \[\label{eq:Petersson} \langle f,g\rangle \ := \ \frac{1}{\nu(N)}\int_{\Gamma_0(N) \backslash\mathbb{H}}f(z)\overline{g(z)}\,y^k\,\frac{dx\,dy}{y^2},\] where $\nu(N):=[\mathrm{SL}_2(\mathbb{Z}):\Gamma_0(N)]$. This inner product is called the Petersson inner product. Given a modular form $f$, we define the Hecke Operator as \[\label{eq:Hecke} T_nf(z) \ := \ n^{k-1}\sum_{\substack{ad=n \$a,N)=1}}\sum_{b=0}^{d-1}d^{-k}f\left(\frac{az+b}{d}\right) .\] Importantly, for \(f \in S_k(N)$, $T_n(f(z)) \in S_k(N)$ for all $n.$ In 1970, Atkin and Lehner [AL] built a theory on newforms of $S_k(N)$. Given a form $f \in S_k(d)$ where $d\mid N$, we may also consider this to be an element of $S_k(N)$. The forms of $S_k(N)$ obtained this way are known as old forms and span a vector subspace $S_k^{\text{old}}(N)$. We denote its orthogonal complement under the Petersson inner product as $S_k^{\text{new}}(N)$, giving us the decomposition \[S_{k}(N) \ = \ S_k^{\text{new}}(N) \oplus S_k^{\text{old}}(N).\] A Hecke eigenform $f\in S_k^{\text{new}}(N)$ is known as a newform and the set of equivalence classes for such a function is one dimensional [MurtyMurty1997]. Moreover, we can find an orthogonal basis $\mathcal{B}_k(N)$ for $S_k(N)$ comprised of newforms. See [ILS] for construction of this basis for square-free level and [BarrettEtAl2016arXiv] for non-square-free level. Since $S_k(N)$ is finite-dimensional, we know $|\mathcal{B}_k(N)|$ is finite [ILS]. Given a Hecke eigenform $f\in\mathcal{B}_k(N)$, we refer to its eigenvalue under $T_n$ as the $n^{\text{th}}$ Hecke eigenvalue of $f$, $\lambda_f(n)$. By the work of Deligna [Deligne1974], it is known that $\lambda_f(p)\in [-2,2]$. Moreover, the Hecke eigenvalues of $f$ are closely related to its Fourier coefficients, satisfying the relation \[\label{eq:Fourier-Hecke} a_f(n) \ = \ \lambda_f(n)n^{(k-1)/2}.\] Furthermore, the Hecke eigenvalues of $f$ possess the useful multiplicative property \[\begin{aligned} \label{eq:mult-prop} \lambda_f(m)\lambda_f(n)\ =\sum_{\substack{d\mid(m,n)\\ (d,N)=1}}\lambda_f\left(\frac{mn}{d^2}\right), \end{aligned}\] so that if $(m,n)=1$, then \[\begin{aligned} \lambda_f(mn) \ = \ \lambda_f(m)\lambda_f(n). \end{aligned}\] We thus define the $L$-function associated to $f$ as: \[\label{eq:lfndefn} L(s,f) \ := \ \sum_{n=1}^{\infty}\frac{\lambda_f(n)}{n^s}, \quad \operatorname{Re}(s)>1.\]

For a cusp form $f$, we define the normalized Fourier coefficients by \[\Psi_f(n) \ := \ \left( \frac{\Gamma(k-1)}{(4\pi)^{k-1}}\right)^{1/2}\|f\|^{-1}\lambda_f(n),\] where $\|f\|^2:=\langle f,f\rangle$, and $\Gamma$ is the gamma function. We wish to consider the sum \[\Delta_{k,N}(m,n) \ = \ \sum_{f \in \mathcal{B}_k(N)}\overline{\Psi_f(m)}\Psi_f(n)\] where $\mathcal{B}_k(N)$ is an orthonormal basis of $S_k(N).$ This sum is computed using the Petersson trace formula [Petersson1932, iwaniec-kwalski2004analytic].

Moreover, we define \[\begin{aligned} \label{def:Z_func} Z(s, f)\ & := \ \sum_{n=1}^{\infty} \frac{\lambda_f\left(n^2\right)}{n^s}\ =\ \frac{\zeta_N(s) L(s, f \otimes f)}{\zeta(s)}, \\\label{def:Z_N_func} Z_N(s,f)\ & :=\ \sum_{n\mid N^\infty} \frac{\lambda_f(n^2)}{n^s}. \end{aligned}\] We wish to evaluate the arithmetically weighted sum \[\begin{aligned} \label{def:Deltastar} \Delta^*_{k,N}(m,n)\ =\ \sum_{f \in H^*_k(N)}\frac{Z_N(1,f)}{Z(1,f)}\lambda_f(m)\lambda_f(n). \end{aligned}\] Iwaniec, Luo, and Sarnak [ILS] derived the following expression in the case of $N$ squarefree.

As Proposition 10 requires $N$ to be squarefree, for general $N$, we use Proposition 4.1 in [BarrettEtAl2016arXiv] which removes the squarefree restriction on $N$.

3 Computing ${U_{k,N}(\phi)}$

We start the computation of the 1st and 2nd level density by computing ${U_{k,N}(\phi)}$ up to $O\left ( \log^{-4}(R) \right )$ error.

Proof. Using (8.363.3) of [GradshteynRyzhik1965] we obtain that the leading term is $\psi(\frac{k}{4})\widehat\phi(0)$. Therefore, we subtract the leading term and compute the error term more accurately. Using the fact that $\phi$ is even \[\begin{aligned} &\int_{-\infty}^\infty \phi(x)\psi\left(\frac{k}{4}+ \frac{2\pi i x}{\log(R)}\right) dx - \psi\left(\frac{k}{4}\right)\widehat\phi(0)\notag\\ & \ = \ \frac{1}{2}\int_{-\infty}^\infty \phi(x)\left[\psi\left(\frac{k}{4}+ \frac{2\pi i x}{\log(R)}\right) + \psi\left(\frac{k}{4}- \frac{2\pi i x}{\log(R)}\right) - 2\psi\left(\frac{k}{4}\right)\right]dx. \label{eq:goal 1 what we want di gamma expand} \end{aligned}\]

Suppose $f:\Omega\to \mathbb{C},$ where $\Omega$ is an open subset of $\mathbb{C}$. Suppose further that $z_0 \in \Omega$. Taylor expanding about $z_0 \in \Omega$ yields \[\label{eq:symm Taylor expansion up to x^4 term} f(z_0+z) + f(z_0-z) - 2f(z_0) \ =\ f''(z_0)z^2 + O(z^4)\] for all $z$ such that $|z|<K$ where $K$ is the radius of convergence. Moreover, we know from Cauchy’s Inequality that $K \geq \text{dist}(z_0,\partial\Omega)$. We apply the formula in [eq:symm Taylor expansion up to x^4 term] with $\Omega = \{z\in\mathbb{C}: \text{Re}(z) > 0\}$, $f(z) = \psi(z)$, $z_0 = \frac{k}{4}$, and $z = 2\pi ix/\log(R)$. We remark that $\text{dist}(\frac{k}{4},\partial(\{z\in\mathbb{C}: \text{Re}(z) > 0\})) = \frac{k}{4}$ and hence $K\geq \frac{k}{4}$. Then, we obtain \[\label{eq:digamma taylor expand} \psi\left(\frac{k}{4} + z\right) + \psi\left(\frac{k}{4} - z\right) - 2 \psi\left(\frac{k}{4}\right) \ =\ \psi^{''}\left(\frac{k}{4}\right) z^2 + O(z^4)\] for all $x$ such that $|x|<k\log(R)/(8\pi)$.

we integrate [eq:digamma taylor expand] against the test function $\phi$ for $|x| < k\log(R)/(8\pi)$ and [eq:goal 1 what we want di gamma expand]. Then \[\begin{aligned} \label{eq: digamma substution} &\int_{-k\log(R)/(8\pi)}^{k\log(R)/(8\pi)}\phi(x)\psi\left(\frac{k}{4}+ \frac{2\pi i x}{\log(R)}\right) dx - \psi\left(\frac{k}{4}\right)\widehat\phi(0) \notag \\ & \ = \ \frac{1}{2}\int_{-k\log(R)/(8\pi)}^{k\log(R)/(8\pi)} \phi(x)\left[-\psi''\left(\frac{k}{4}\right)\left(\frac{2\pi x}{\log(R)}\right)^2 + O\left(\frac{(2\pi x)^4}{\log^4(R)}\right)\right]dx \notag \\ & \ = \ \frac{1}{2}\int_{-k\log(R)/(8\pi)}^{k\log(R)/(8\pi)} \phi(x)\left[-\psi''\left(\frac{k}{4}\right)\left(\frac{2\pi x}{\log(R)}\right)^2\right]dx + O\left ( \log^{-4}(R) \right ) \left(\int_{-k\log(R)/(8\pi)}^{k\log(R)/(8\pi)} \phi(x)x^4dx\right) \notag \\ & \ = \ \frac{1}{2}\int_{-k\log(R)/(8\pi)}^{k\log(R)/(8\pi)} \phi(x)\left[-\psi''\left(\frac{k}{4}\right)\left(\frac{2\pi x}{\log(R)}\right)^2\right]dx + O\left ( \log^{-4}(R) \right ) . \end{aligned}\]

We show the tail part of the integral is negligible. Consider the series expansion of $\psi(z)$ from [abramowitz1972handbook], which holds for all $z \notin \mathbb{Z}^{\leq 0}$: \[\psi(z) \ = \ -\gamma + \sum_{n=0}^\infty \frac{z-1}{(n+1)(n+z)}\] where $\gamma$ is the Euler–Mascheroni constant. For $z = \frac{k}{4}+2\pi i x/ \log(R),$ we then have that for all $n \geq 1,$: \[\begin{aligned} \label{ineq:helping convergence of tail} \left| \frac{\left(\frac{k}{4}+2\pi i x/ \log(R)\right)-1}{(n+1)(n+\left(\frac{k}{4}+2\pi i x/ \log(R)\right))}\right| & \ = \ \left|\frac{8 i \pi x + (-4 + k) \log(R)}{(1 + n)\left(8 i \pi x + (k + 4n) \log(R)\right)}\right| \notag \\ & \ \leq \ \frac{8\pi x + |k-4| \log(R)}{(1 + n)\left((k + 4n) \log(R)\right)} \notag \\ & \ \leq \ \frac{8\pi x+|k-4|\log(R)}{n^2}. \end{aligned}\] We also know that because $\phi(x)$ is Schwartz, $\int_{k\log(R)/(8\pi)}^\infty \phi(x) P(x)dx = O\left ( \log^{-B}(R) \right )$ for any polynomial $P(x)$. Therefore, using this fact along with [ineq:helping convergence of tail], we find \[\begin{aligned} \label{eq: remainder part 1/R} &\left|\int_{k\log(R)/(8\pi)}^\infty \phi(x)\left(-\gamma + \sum_{n=0}^\infty \frac{z-1}{(n+1)(n+z)}\right) dx \right|\notag\\ & \ \leq \ (-\gamma +1)\left|\int_{k\log(R)/(8\pi)}^\infty \phi(x)dx \right|+ \sum_{n=1}^\infty\left|\int_{k\log(R)/(8\pi)}^\infty\phi(x)\left(\frac{8\pi x+|k-4|\log(R)}{n^2}\right)dx \right| \notag\\ & \ = \ \left(-\gamma +1+ \left(\sum_{n=1}^\infty \frac{|k-4|\log(R)}{n^2}\right)\right)\left|\int_{k\log(R)/(8\pi)}^\infty \phi(x)dx\right| + 8\pi \left(\sum_{n=1}^\infty \frac{1}{n^2}\right)\left|\int_{k\log(R)/(8\pi)}^\infty x\phi(x)dx\right| \notag \\ & \ = \ O\left ( \log^{-4}(R) \right ) . \end{aligned}\] Similarly, we know that \[\left|\int_{-\infty}^{-k\log(R)/(8\pi)} \phi(x)\left(-\gamma + \sum_{n=0}^\infty \frac{z-1}{(n+1)(n+z)}\right) dx \right| \ = \ O\left ( \log^{-4}(R) \right ) .\] In addition, we know that \[\label{eq: other piece of remainder} \int_{k\log(R)/(8\pi)}^{\infty} \phi(x)\left[-\psi''\left(\frac{k}{4}\right)\left(\frac{2\pi x}{\log(R)}\right)^2\right]dx + \int_{-\infty}^{-k\log(R)/(8\pi)} \phi(x)\left[-\psi''\left(\frac{k}{4}\right)\left(\frac{2\pi x}{\log(R)}\right)^2\right]dx\] is $O\left ( \log^{-4}(R) \right )$ because $\phi$ is Schwartz. Therefore, combining [eq: digamma substution], [eq: remainder part 1/R], and [eq: other piece of remainder], and noticing that \[\begin{aligned} \int_{-\infty}^{\infty} \phi(x)x^2dx = -\widehat\phi''(0) \end{aligned}\] we have \[\int_{-\infty}^\infty \phi(x)\psi\left(\frac{k}{4}+ \frac{2\pi i x}{\log(R)}\right) dx - \psi\left(\frac{k}{4}\right)\widehat\phi(0) \ = \ \frac{2\pi^2\psi''\left(\frac{k}{4}\right)}{\log^2(R)}\widehat\phi''(0) + O\left ( \log^{-4}(R) \right )\] as claimed. ◻

By the same argument, but for $z_0 = \frac{k}{4}+ \frac{1}{2}$, we obtain the following lemma.

4 Computing ${S_1((\mathcal{F}_{N},\phi)}$ up to $O\left ( \log^{-4}(R) \right )$ error

Similar to [Miller2009_LOTerms_1LevelDensity], for a level $N$ define⁵ \[\begin{aligned} A'_{r,N}(p)\ &:=\ \frac{1}{W_R(\mathcal{F}_N)}\sum_{\substack{f \in \mathcal{F}_N \\ p\, \mid N}}w_R(f)\lambda_f(p)^r, \label{def:A'_r} \\ A_{r,N}(p)\ &:=\ \frac{1}{W_R(\mathcal{F}_N)} \sum_{\substack{f \in \mathcal{F}_N \\ p\, \nmid N}}w_R(f)\lambda_f(p)^r, \label{def:A_r} \end{aligned}\] where $W_R(\mathcal{F}_{N}):=\sum_{f \in \mathcal{F}_{N}}w_R(f)$. This brings us to the following theorem.

Theorem 15. Let \[\mathcal{S}_1( \mathcal{F}_{N},\phi)\ :=\ -2\sum_p\sum_{m=1}^{\infty}\frac{1}{W_{R}(\mathcal{F}_{N})}\sum_{f \in \mathcal{F}_{N}}w_R(f)\frac{\alpha_f(p)^m+\beta_f(p)^m}{p^{m/2}}\frac{\log(p)}{\log(R)}\widehat{\phi}\left(m \frac{\log(p)}{\log(R)} \right),\] where $\log(R)$ is the average log conductor, then \[\begin{aligned} \mathcal{S}_1( \mathcal{F}_{N},\phi)\ =& \ \ S_{A'}(\mathcal{F}_{N}) + S_{A}(\mathcal{F}_{N}) +O\left ( \log^{-4}(R) \right ) , \end{aligned}\] where \[\begin{aligned} S_{A'}(\mathcal{F}_{N})\ :=\ & -2\sum_p\sum_{m=1}^{\infty}\frac{A'_{m, N}(p)}{p^{m/2}}\frac{\log(p)}{\log(R)}\widehat{\phi}\left(m \frac{\log(p)}{\log(R)} \right) \label{eq:SA'_defn} \\ S_{A}(\mathcal{F}_{N})\ :=\ &-2\widehat{\phi}\left(0 \right)\sum_p\frac{2A_{0, N}(p)\log(p)}{p(p+1)\log(R)}+2\sum_p \frac{2A_{0, N}(p)\log(p)}{p\log(R)}\widehat{\phi}\left(2 \frac{\log(p)}{\log(R)} \right) \label{eq:SA_defn} \notag \\ &-2\sum_p\frac{A_{1, N}(p)\log(p)}{p^{1/2}\log(R)}\widehat{\phi}\left(\frac{\log(p)}{\log(R)} \right)+2\widehat{\phi}\left(0 \right)\sum_p\frac{A_{1,N}(p)(3p+1)\log(p)}{p^{1/2}(p+1)^2\log(R)} \notag \\ &-2\sum_p\frac{A_{2,N}(p)\log(p)}{p\log(R)}\widehat{\phi}\left(2 \frac{\log(p)}{\log(R)} \right)+2\widehat{\phi}\left(0 \right) \sum_p\frac{A_{2, N}(p)(p^2+3p+1)\log(p)}{p(p+1)^3 \log(R)} \notag \\ &+\frac{\widehat{\phi}''(0)}{4\pi^2}\sum_p\frac{A_{0, N}(p)(32p^2+24p+8)\log^3(p)}{p(p+1)^3\log(R)} \notag\\ &-\frac{\widehat{\phi}''(0)}{4\pi^2}\sum_p\frac{A_{1, N}(p)(27p^3-17p^2+5p+1)\log^3(p)}{p^{1/2}(p+1)^4} \notag\\ &-\frac{\widehat{\phi}''(0)}{4\pi^2}\sum_p\frac{A_{2, N}(p)(64p^4-4p^3+44p^2+20p+4)\log^3(p)}{p(p+1)^5\log^3(R)}\\&-2\widehat{\phi}\left(0 \right)\sum_{p}\sum_{r=3}^{\infty}\frac{A_{r, N}(p)p^{r/2}(p-1)\log(p)}{(p+1)^{r+1}\log(R)} \notag \\ &+\frac{\widehat{\phi}''(0)}{4\pi^2}\sum_p\sum_{r=3}^{\infty}\frac{A_{r, N}(p-1)(r^2(p-1)^2-12rp-8p)p^{r/2}\log^3(p)}{(p+1)^{r+3}\log(R)}. \end{aligned}\]

Identical to Miller’s analysis in [Miller2009_LOTerms_1LevelDensity], we break the explicit formula from [ILS] into the case when $p\mid N$ and $p\nmid N$ and use properties of the Satake parameters to obtain \[\begin{aligned} \mathcal{S}_1({\mathcal{F}_{N}},\phi)\ =&\ -2\sum_p\sum_{m=1}^{\infty}\frac{1}{W_R(\mathcal{F}_{N})}\sum_{\substack{f \in \mathcal{F}_{N} \notag \\p\mid N}}w_R(f)\frac{\lambda_f(p)^m}{p^{m/2}}\frac{\log(p)}{\log(R)}\widehat{\phi}\left(m \frac{\log(p)}{\log(R)} \right) \notag \\ &-2\sum_p\frac{1}{W_R(\mathcal{F}_{N})}\sum_{\substack{f \in \mathcal{F}_{N} \notag \\p \nmid N}}w_R(f)\frac{\lambda_f(p)}{p^{1/2}}\frac{\log(p)}{\log(R)}\widehat{\phi}\left(\frac{\log(p)}{\log(R)} \right) \notag \\ &-2\sum_p\frac{1}{W_R(\mathcal{F}_{N})}\sum_{\substack{f \in \mathcal{F}_{N}\\p \nmid N}}w_R(f)\frac{\lambda_f(p)^2-2}{p}\frac{\log(p)}{\log(R)}\widehat{\phi}\left(2\frac{\log(p)}{\log(R)} \right) \notag \\ &-2\sum_p\sum_{m=3}^{\infty}\frac{1}{W_R(\mathcal{F}_{N})}\sum_{\substack{f \in \mathcal{F}_{N}\\p \nmid N}}w_R(f)\frac{\alpha_f(p)^m+\beta_f(p)^m}{p^{m/2}}\frac{\log(p)}{\log(R)}\widehat{\phi}\left(m \frac{\log(p)}{\log(R)} \right)\label{eq:yes3}. \end{aligned}\] The first three sums are already in the desired form, as they can easily be expressed as weighted averages of Hecke eigenvalues. We look to evaluate the last sum. Through Taylor expansion, we have \[\widehat{\phi}(mx)-\widehat{\phi}(x)\ =\ \frac{m^2-1}{2}\widehat{\phi}''(0)x^2+O\left(m^4x^4 \right).\] Thus we have \[\label{eq:taylorexpansionphi} \widehat{\phi}\left(m\frac{\log(p)}{\log(R)}\right)-\widehat{\phi}\left(\frac{\log(p)}{\log(R)}\right)\ =\ \frac{m^2-1}{2}\widehat{\phi}''(0)\left( \frac{\log(p)}{\log(R)}\right)^2+O\left(m^4\left(\frac{\log(p)}{\log(R)} \right)^4 \right).\] Substituting ([eq:taylorexpansionphi]) into the last sum in ([eq:yes3]), we obtain \[\begin{aligned} \label{eq:S1 first} \sum_p\sum_{m=3}^{\infty}\frac{1}{W_R(\mathcal{F}_{N})}\sum_{\substack{f \in \mathcal{F}_{N}\\p \nmid N}}w_R(f)\frac{\alpha_f(p)^{m}+\beta_f(p)^{m}}{p^{m/2}} \frac{\log(p)}{\log(R)}\Bigg(\widehat{\phi} \Big( & \left. \frac{\log(p)}{\log(R)} \Big) +\frac{m^2-1}{2}\widehat{\phi}''(0)\left( \frac{\log(p)}{\log(R)}\right)^2 \right. \notag \\ &+ O\left(m^4\left(\frac{\log(p)}{\log(R)} \right)^4 \right) \Bigg). \end{aligned}\]

Define \[\begin{aligned} \label{def: M_(c,k)} M_{c,k}(p)\ :=\ \sum_{m=c}^{\infty}m^k\frac{\alpha_f(p)^{m}+\beta_f(p)^{m}}{p^{m/2}}. \end{aligned}\] We write equation [eq:S1 first] as \[\begin{aligned} \sum_p\frac{1}{W_R(\mathcal{F}_{N})}\sum_{\substack{f \in \mathcal{F}_{N}\\p \nmid N}}w_R(f)M_{3,0}(p) &\frac{\log(p)}{\log(R)}\widehat{\phi} \left(\frac{\log(p)}{\log(R)} \right)\notag\\ &+\frac{\widehat{\phi}''(0)}{2}\sum_p\frac{1}{W_R(\mathcal{F}_{N})}\sum_{\substack{f \in \mathcal{F}_{N}\\p \nmid N}}w_R(f)(M_{3,2}(p)-M_{3,0}(p))\frac{\log^3(p)}{\log^3 (R)} \notag \\ &+O\left(\sum_p\frac{1}{W_R(\mathcal{F}_{N})}\sum_{\substack{f \in \mathcal{F}_{N}\\p \nmid N}}w_R(f)M_{3,4}(p)\frac{\log^5 (p)}{\log^5(R)} \right) . \end{aligned}\] Substituting $\widehat{\phi}\left(\frac{\log(p)}{\log(R)} \right)\ =\ \widehat{\phi}(0)+\frac{\widehat{\phi}''(0)}{2}\frac{\log^2(p)}{\log^2(R)}+O\left(\frac{\log^4(p)}{\log^4(R)}\right)$ gives \[\begin{aligned} \label{eq: S1 fourth} \widehat{\phi}(0)\sum_p\frac{1}{W_R(\mathcal{F}_{N})}\sum_{\substack{f \in \mathcal{F}_{N}\\p \nmid N}} w_R(f)M_{3,0}(p)\frac{\log(p)}{\log(R)}+\frac{\widehat{\phi}''(0)}{2}\sum_p\frac{1}{W_R(\mathcal{F}_{N})}\sum_{\substack{f \in \mathcal{F}_{N}\\p \nmid N}} w_R(f)M_{3,2}(p)\frac{\log^3(p)}{\log^3(R)}& \notag \\ +O\left(\sum_p\frac{1}{W_R(\mathcal{F}_{N})}\sum_{\substack{f \in \mathcal{F}_{N}\\p \nmid N}}w_R(f)M_{3,4}(p)\frac{\log^5 (p)}{\log^5(R)} \right)& \end{aligned}\] In Miller’s proof of Theorem 1.1, he shows \[M_{3,0}(p)\ =\ \frac{2}{p(p+1)}-\frac{p^{1/2}(3p+1)}{p(p+1)^2}\lambda_f(p)-\frac{(p^2+3p+1)}{p(p+1)^3}\lambda_f(p)^2+\sum_{m=3}^{\infty}\frac{p^{m/2}(p-1)\lambda_f(p)^m}{(p+1)^{m+1}}.\] We compute $M_{3,2}(p).$ See Appendix 8 for the proof.

Substituting equation [eq: M_3,2 lemma compute] into equation [eq: S1 fourth] yields the main term in Theorem 15. we look at the error term \[\begin{aligned} \sum_p\frac{1}{W_R(\mathcal{F}_{N})}\sum_{\substack{f \in \mathcal{F}_{N}\\p \nmid N}}w_R(f)M_{3,4}(p)\frac{\log^5 (p)}{\log^5(R)}.\label{eq:MOh} \end{aligned}\] As $\mid \alpha_f(p)^m+\beta_f(p)^m\mid \ \leq \ 2$ for all $m \ \in \ \mathbb{N}$, we have \[\begin{aligned} \eqref{eq:MOh}\ &=\ O \left( \sum_p\sum_{m=3}^{\infty}\frac{1}{W_R(\mathcal{F}_{N})}\sum_{\substack{f \in \mathcal{F}_{N}\\p \nmid N}}\frac{\alpha_f(p)^{m}+\beta_f(p)^{m}}{p^{m/2}} \frac{\log(p)}{\log(R)}\left(m^4\left(\frac{\log(p)}{\log(R)} \right)^4 \right) \right) \notag \\ \ &=\ O \left( \sum_p \left( \frac{\log(p)}{\log(R)} \right)^5 \sum_{m=3}^{\infty} \sum_{\substack{f \in \mathcal{F}_{N}\\p \nmid N}} \frac{2 w_R(f)}{W_R(\mathcal{F}_{N})}\frac{m^4}{p^{m/2}} \right) \notag \\ \ &=\ O \left( \sum_p \left( \frac{\log(p)}{\log(R)} \right)^5 \sum_{m=3}^{\infty} \frac{m^4}{p^{m/2}} \right). \end{aligned}\]

Thus we have \[\begin{aligned} \sum_p \left( \frac{\log(p)}{\log(R)} \right)^5 \sum_{m=3}^{\infty} \frac{m^4}{p^{m/2}} \ \lesssim \ \frac{1}{\log^4(R)}\sum_p\frac{\log^5 (p)}{p^{3/2}} \ &\lesssim \ \frac{1}{\log^5(R)}. \end{aligned}\] As the sum $\sum_p\frac{\log^5 (p)}{p^{3/2}}$ converges by the comparison test, our error term is sufficient. 0◻

5 Computing ${S_2(\mathcal{F}_{N},\phi_1,\phi_2)}$ up to $O\left ( \log^{-4}(R) \right )$ error

We are interested in finding a formula for $S_2(\mathcal{F}_{N},\phi_1,\phi_2)\:=\frac{1}{W_R(\mathcal{F}_{N})}\sum_{f\in \mathcal{F}_{N}}w_R(f)S(\phi_1)S(\phi_2)$ where \[\begin{aligned} \label{eq: S_1S_2 formula} S(\phi_1)S(\phi_2) \ = \ &\left(\sum_{p_1}\sum_{m_1=1}^\infty \frac{\alpha_f(p_1)^{m}+\beta_f(p_1)^{m_1}}{p_1^{m_1/2}} \frac{\log(p_1)}{\log(R)} \widehat\phi_1\left(m_1 \frac{\log(p_1)}{\log(R)}\right)\right)\notag\\&\cdot \left(\sum_{p_2}\sum_{m_2=1}^\infty \frac{\alpha_f(p_2)^{m_2}+\beta_f(p_2)^{m_2}}{p^{m_2/2}} \frac{\log(p_2)}{\log(R)} \widehat\phi_2\left(m_2 \frac{\log(p_2)}{\log(R)}\right)\right) \end{aligned}\] up to $O\left ( \log^{-4}(R) \right )$ error term.

For fixed $f \in \mathcal{F}_{N},$ we break this sum given by [eq: S_1S_2 formula] into cases depending on whether each $p_1$ and $p_2$ divide $N$ or not. We freely change the order of summation because convergence is guaranteed by $\widehat\phi_1$ and $\widehat\phi_2$ being Schwartz. Using the fact that if $p\mid N$, then $\alpha_f(p)^m+\beta_f(p)^m\ = \ \lambda_f(p)^m$, we obtain: where $W_R(\mathcal{F}_{N})\:= \ \sum_{f\in \mathcal{F}_{N}} w_R(f).$ Further, we introduce the notations: \[\begin{aligned} \label{not:B'' i,j} B''_{r_1,r_2,N}(p_1,p_2)&\ = \ \frac{1}{W_R(\mathcal{F}_{N})}\sum_{\substack{f \in \mathcal{F}_{N} \\p_1 \mid N \\ p_2 \mid N}} w_R(f) \lambda_f(p_1)^{r_1}\lambda_f(p_2)^{r_2} \\ \label{not:B' i,j} B'_{r_1,r_2,N}(p_1,p_2)&\ = \ \frac{1}{W_R(\mathcal{F}_{N})}\sum_{\substack{f \in \mathcal{F}_{N} \\p_1 \mid N \\ p_2 \nmid N}} w_R(f) \lambda_f(p_1)^{r_1}\lambda_f(p_2)^{r_2}\\\label{not:B i,j} B_{r_1,r_2,N}(p_1,p_2)&\ = \ \frac{1}{W_R(\mathcal{F}_{N})}\sum_{\substack{f \in \mathcal{F}_{N} \\p_1 \nmid N \\ p_2 \nmid N}} w_R(f) \lambda_f(p_1)^{r_1}\lambda_f(p_2)^{r_2}. \end{aligned}\] Whenever there is no confusion, we drop the indication of the level $N$, and write $B''_{r_1,r_2}(p_1,p_2)$, $B'_{r_1,r_2}(p_1,p_2)$, or $B_{r_1,r_2}(p_1,p_2)$. We compute each of the sums explicitly, using the notations above.

To prove the theorem, we show that the first line of [eq:S_2 formula] is $S_{B''}(\mathcal{F}_{N}),$ the sum of the second and third line of [eq:S_2 formula] equals $S_{B'}(\mathcal{F}_{N})$, and the fourth line of [eq:S_2 formula] equals $S_{B_f}(\mathcal{F}_{N}) + S_{B_\infty}(\mathcal{F}_{N})$. Below, we break the proof into three parts, where each part shows one of the equalities mentioned above.

5.1 First line of [eq:S_2 formula]: $p_1 \mid N$, $p_2 \mid N$

This case is simple. Using the notation established above, the first line of [eq:S_2 formula] is equal to \[\begin{aligned} &\frac{1}{W_R(\mathcal{F}_{N})}\sum_{p_1,p_2}\sum_{\substack{m_1\in N\\m_2\in N}}\sum_{\substack{f\in \mathcal{F}_{N}\\p_1\mid N\\p_2\mid N }}w_R(f)\frac{\lambda_f(p_1)^{m_1}}{p_1^{m_1/2}} \frac{\lambda_f(p_2)^{m_2}}{p_2^{m_2/2}} \frac{\log(p_1)\log (p_2)}{\log^2(R)}\widehat\phi_1 \left(m_1 \frac{\log(p_1)}{\log(R)}\right)\widehat\phi_2 \left(m_2 \frac{\log(p_2)}{\log(R)}\right) \notag \\ &\ = \ \sum_{p_1,p_2}\sum_{\substack{m_1\in N\\m_2\in N}} B''_{m_1,m_2}(p_1,p_2) \frac{\log(p_1)\log(p_2)}{p_1^{m_1/2}p_2^{m_2/2}\log^2(R)}\widehat\phi_1 \left(m_1 \frac{\log(p_1)}{\log(R)}\right)\widehat\phi_2 \left(m_2 \frac{\log(p_2)}{\log(R)}\right) \ = \ S_{B''}(\mathcal{F}_{N}). \end{aligned}\]

5.2 Sum of second and third lines of [eq:S_2 formula]: $p_1 \mid N$, $p_2 \nmid N$ and $p_2 \mid N$, $p_1 \nmid N$

We begin by noting that the sums for $p_1 \mid N, p_2 \nmid N$ and $p_2 \mid N, p_1 \nmid N$ are symmetric with the exception of the test function. Therefore, for $A=\{(1,2), (2,1)\},$ the second and third line of [eq:S_2 formula] sum to For $(i,j)\in A$ fixed, we denote by $(\bigstar)$ the expression inside of the brackets in [eq:case 2 sum]. By symmetry, it suffices to compute ($\bigstar$) for $(i,j) = (1,2)$.

We start by breaking the $(\bigstar)$ into three parts, depending on whether $m_2 =1, m_2=2,$ or $m_2 \geq 3$. \[\begin{aligned} (\bigstar)\,= \, \nonumber\,&\frac{1}{W_R(\mathcal{F}_{N})}\sum_{p_1,p_2}\sum_{m_1\in \mathbb{N}}\sum_{\substack{f\in \mathcal{F}_{N}\\p_1\mid N\\p_2\nmid N }}w_R(f)\frac{\lambda_f(p_1)^{m_1}}{p_1^{m_1/2}}\frac{\lambda_f(p_2)}{\sqrt{p_2}} \frac{\log(p_1)\log (p_2)}{\log^2(R)}\widehat\phi_1 \left(m_1 \frac{\log(p_1)}{\log(R)}\right)\widehat\phi_2 \left(\frac{\log(p_2)}{\log(R)}\right) \\\nonumber&+ \frac{1}{W_R(\mathcal{F}_{N})}\sum_{p_1,p_2}\sum_{m_1\in \mathbb{N}}\sum_{\substack{f\in \mathcal{F}_{N}\\p_1\mid N\\p_2\nmid N }}w_R(f)\frac{\lambda_f(p_1)^{m_1}}{p_1^{m_1/2}}\frac{(\lambda_f(p_2)^2-2)}{p_2} \frac{\log(p_1)\log (p_2)}{\log^2(R)}\\ &\cdot\widehat\phi_1 \left(m_1 \frac{\log(p_1)}{\log(R)}\right)\widehat\phi_2 \left(\frac{2\log(p_2)}{\log(R)}\right)\\\label{eq: case 2 sum m2 geq 3} &+\frac{1}{W_R(\mathcal{F}_{N})}\sum_{p_1,p_2}\sum_{\substack{m_1\in \mathbb{N}\\m_2\geq 3}}\sum_{\substack{f\in \mathcal{F}_{N}\\p_1\mid N\\p_2\nmid N }}w_R(f)\frac{\lambda_f(p_1)^{m_1}}{p_1^{m_1/2}}\frac{\alpha_f(p_2)^{m_2}+\beta_f(p_2)^{m_2}}{p_2^{m_2/2}} \frac{\log(p_1)\log (p_2)}{\log^2(R)} \notag \\ &\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \cdot \widehat\phi_1 \left( \frac{m_1\log(p_1)}{\log(R)}\right)\widehat\phi_2 \left(\frac{m_2\log(p_2)}{\log(R)}\right). \end{aligned}\]

We write $(\bigstar)=(\bigstar')+(\bigstar'')+(\bigstar''')$ where, for example, $(\bigstar')$ is the first line in [eq: case 2 sum m2 geq 3]. Using the definition of $B'_{r_1,r_2}(p_1,p_2)$ in [not:B' i,j] and $A_{r}(p)$ in [def:A_r], as well as the convention that $A'_{r}(p) = B'_{r,0}(p,q)$ (for any prime $q\nmid N$), we can easily deal with the case when $m_2=1$ and $m_2=2$. We have \[\label{eq for them case 2, m1 =1} (\bigstar') \ =\ \sum_{p_1,p_2}\sum_{m_1\in \mathbb{N}} B'_{m_1,1}(p_1,p_2) \frac{\log(p_1)\log(p_2)}{p_1^{m_1/2}\sqrt{p_2}\log^2(R)} \widehat\phi_1\left(\frac{m_1\log(p_1)}{\log(R)}\right)\widehat\phi_2\left(\frac{\log(p_2)}{\log(R)}\right)\] and We aim to simplify $(\bigstar''')$, which is the case when $m_2 \geq 3$. The purpose of the following lemma is to remove the dependency of $m_2$ in the argument of $\widehat\phi_2$ by arguing that we can replace $\widehat\phi_2\left(m_2\frac{\log(p_2)}{\log(R)}\right)$ with $\widehat\phi_2(0)$ at the cost of $O\left ( \log^{-4}(R) \right )$.

Proof. By assumption, $\widehat\phi_2$ is even and therefore $\widehat\phi'_2(0)=0$. Thus, using the Taylor expansion around the origin, we notice that \[\label{eq.lemma, case 2:taylor expansion} \widehat\phi_2 \left(m_2 \frac{\log(p_2)}{\log(R)}\right) - \widehat\phi_2(0)\ =\ O\left(m_2^2 \frac{\log(p_2)^2}{\log(R)^2}\right).\] Moreover, we know that for every $m \in \mathbb{N}$ and prime $p,$ \[\label{eq.lemma, case 2: alpha beta bound} |\alpha_f(p)^m + \beta_f(p)^m| \ \leq \ 2 .\] Thus, because for $p\mid N$, $\lambda_f(p)^{m} = \alpha_f(p)^m + \beta_f(p)^m,$ we have if $p\mid N$, \[|\lambda_f(p)^{m}|\ \leq \ 2.\] Therefore, we obtain the following estimate: \[\begin{aligned} \label{case2, lemma first eq} &\frac{1}{W_R(\mathcal{F}_{N})}\sum_{\substack{p_1\\ p_2 < R^{\sigma}}}\sum_{\substack{m_1\in \mathbb{N}\\m_2\geq 3}}\sum_{\substack{f\in \mathcal{F}_{N}\\p_1\mid N\\p_2\nmid N }}w_R(f)\frac{\lambda_f(p_1)^{m_1}}{p_1^{m_1/2}}\frac{\alpha_f(p_2)^{m_2}+\beta_f(p_2)^{m_2}}{p_2^{m_2/2}} \frac{\log(p_1)\log (p_2)}{\log^2(R)} \notag \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \cdot \widehat\phi_1 \left(m_1 \frac{\log(p_1)}{\log(R)}\right)\left(\widehat\phi_2 \left(m_2\frac{\log(p_2)}{\log(R)}\right)- \phi_2 (0)\right) \notag \\ &\lesssim \frac{1}{\log^4(R)}\left(\sum_{p_1} \sum_{m_1 \in \mathbb{N}}\frac{\log(p_1)}{p_1^{m_1/2}} \widehat\phi_1\left(m_1\frac{\log(p_1)}{\log(R)}\right)\right) \left(\sum_{p_2}\log(p_2)^3 \sum_{m_2 \geq 3}\frac{m_2^2}{p_2^{m_2/2}}\right)\notag \\ & \lesssim \frac{1}{\log^4(R)}\left(\sum_{p_1} \sum_{m_1 \in \mathbb{N}}\frac{\log(p_1)}{p_1^{m_1/2}} \widehat\phi_1\left(m_1\frac{\log(p_1)}{\log(R)}\right)\right) \left(\sum_{p_2}\frac{\log(p_2)^3 }{p_2^{3/2}}\right), \end{aligned}\] where the last line follows using the fact that $m_2^2 \lesssim \left(\frac{3}{2}\right)^{m_2/2}$ and the geometric series formula. In fact, the proof is complete because $\left(\sum_{p_1} \sum_{m_1 \in \mathbb{N}}\frac{\log(p_1)}{p_1^{m_1/2}} \widehat\phi_1\left(m_1\frac{\log(p_1)}{\log(R)}\right)\right)$ and $\left(\sum_{p_2}\frac{\log(p_2)^3 }{p_2^{3/2}}\right)$ are convergent sums. ◻

With Lemma 18 and using the notation of $M_{3,0}(p)$ from [def: M_(c,k)], we haveWe know from [Miller2009_LOTerms_1LevelDensity] that \[\begin{aligned} M_{3,0}(p)\ = \ \frac{2}{p(p+1)}-\frac{p^{1/2}(3p+1)}{p(p+1)^2}\lambda_f(p)-\frac{(p^2+3p+1)}{p(p+1)^3}\lambda_f(p)^2+\sum_{m=3}^{\infty}\frac{p^{m/2}(p-1)\lambda_f(p)^m}{(p+1)^{m+1}}. \end{aligned}\] Moreover, by observing that $M_{3,0}(p_2)=\sum_{m_2=0}^\infty P_m(p_2)$, we have \[\begin{aligned} \label{eq for case 2 m1 geq 2} (\bigstar''')\ = \ \sum_{p_1,p_2}\sum_{m_1=1,m_2=0}^{\infty}B'_{m_1,m_2}(p_1,p_2)\frac{P_{m_2}(p_2)}{p_1^{m_1/2}}\frac{\log(p_1)\log (p_2)}{\log^2(R)}\widehat\phi_1\left(\frac{m_1\log(p_1)}{\log(R)}\right)\widehat\phi_2\left(0\right). \end{aligned}\] substituting [eq for them case 2, m1 =1], [eq for them case 2, m1 =2], and [eq for case 2 m1 geq 2] for $(\bigstar'),(\bigstar'')$, and $(\bigstar''')$ respectively, we obtain that the second line of [eq:S_2 formula] equals $S_{B'}(\mathcal{F}_{N})$.

5.3 Fourth line of [eq:S_2 formula]: $p_1 \nmid N$, $p_2 \nmid N$

Lastly, we show that up to $O(\log^{-4} R)$ error, the fourth line of [eq:S_2 formula] equals $S_{B_f}(\mathcal{F}_{N}) +S_{B_\infty}(\mathcal{F}_{N})$ .

For fixed primes $p_1,p_2$, let The fourth line of [eq:S_2 formula] is equal to \[\begin{aligned} \nonumber & \frac{1}{W_R(\mathcal{F}_{N})}\sum_{p_1,p_2}\notag\\ &\quad\sum_{\substack{m_1\in N\\m_2\in N}}\sum_{\substack{f\in \mathcal{F}_{N}\\p_1\nmid N\\p_2\nmid N }}w_R(f)\frac{\alpha_f(p_1)^{m_1}+\beta_f(p_1)^{m_1}}{p_1^{m_1/2}} \frac{\alpha_f(p_2)^{m_2}+\beta_f(p_2)^{m_2}}{p_2^{m_2/2}}\frac{\log(p_1)\log (p_2)}{\log^2(R)}\widehat\phi_1 \left(m_1 \frac{\log(p_1)}{\log(R)}\right)\widehat\phi_2 \left(m_2 \frac{\log(p_2)}{\log(R)}\right)\notag\\ &\ = \ \frac{1}{W_R(\mathcal{F}_{N})}\sum_{p_1,p_2}\big(C(1,1)+C(1,2)+C(2,1)+C(2,2)\big) \label{eq:rewrite fourth line finite} \\ & ~~~~~\ +\frac{1}{W_R(\mathcal{F}_{N})}\sum_{p_1,p_2}\sum_{m_1,m_2\geq 3} \big(C(m_1,1)+C(m_1,2)+C(1,m_2)+C(2,m_2)+C(m_1,m_2)\big).\label{eq:rewrite fourth line infinite} \end{aligned}\] We show that the finite part, line [eq:rewrite fourth line finite], equals $S_{B_f}(\mathcal{F}_{N})$ while the infinite part, line [eq:rewrite fourth line infinite], equals $S_{B_\infty}(\mathcal{F}_{N}).$

The former immediately follows from the multiplicativity of Hecke eigenvalues. We know that for all prime $p$, $\alpha_f(p) + \beta_f(p) = \lambda_f(p)$ and $\alpha_f(p)^2 + \beta_f(p)^2 = \lambda_f(p)^2-2$. For the latter, we need a lemma. See Appendix 9 for the proof.

using lemmas 19 and 20, as well as using formulas for $M_{3,0}(p)$ from [Miller2009_LOTerms_1LevelDensity], we obtain \[\begin{aligned} &\frac{1}{W_R(\mathcal{F}_{N})}\sum_{p_1,p_2}\sum_{m_2\geq 3} C(1,m_2) \notag \\ &\ = \ \frac{1}{W_R(\mathcal{F}_{N})} \sum_{p_1,p_2} \sum_{\substack{f\in \mathcal{F}_{N}\\p_1\nmid N\\p_2\nmid N }}w_R(f) \frac{\lambda_f(p_1)}{\sqrt{p_1}}M_{3,0}(p_2)\frac{\log(p_1)\log(p_2)}{\log^2(R)} \widehat\phi_1\left(\frac{\log(p_1)}{\log(R)}\right)\widehat\phi_2(0) + O\left ( \log^{-4}(R) \right ) \notag \\ &\ = \ \widehat{\phi}_1(0)\sum_{p_1,p_2}\sum_{m_1=0}^{\infty}\widehat{\phi}_2\left(\frac{\log(p_2)}{\log(R)} \right)\frac{\log(p_1) \log(p_2)}{\log^2(R)}B_{m_1,1}(p_1,p_2)\frac{P_{m_1}(p_1)}{\sqrt{p_2}} + O\left ( \log^{-4}(R) \right ) . \end{aligned}\] Performing a similar substitution, we have \[\begin{aligned} &\sum_{p_1,p_2}\sum_{m_2\geq 3} \frac{C(2,m_2)}{W_R(\mathcal{F}_{N})}\nonumber\\ &= \ \widehat{\phi}_1(0)\sum_{p_1,p_2<R^\sigma}\sum_{m_1=0}^{\infty}\frac{\log(p_1) \log(p_2)}{\log^2(R)}\widehat{\phi}_2\left( \frac{2\log(p_2)}{\log(R)}\right)(B_{m_1,2}(p_1,p_2)-2B_{m_1,0}(p_1,p_2))\frac{P_{m_1}(p_1)}{p_2} + O\left ( \log^{-4}(R) \right ) , \notag \\ &\sum_{p_1,p_2}\sum_{m_1\geq 3} \frac{C(m_1,1)}{W_R(\mathcal{F}_{N})}\nonumber\\ &= \ \widehat{\phi}_2(0)\sum_{p_1,p_2<R^\sigma}\sum_{m_1=0}^{\infty}\widehat{\phi}_1\left(\frac{\log(p_1)}{\log(R)} \right)\frac{\log(p_1) \log(p_2)}{\log^2(R)}B_{m_1,1}(p_1,p_2)\frac{P_{m_1}(p_2)}{\sqrt{p_2}}+ O\left ( \log^{-4}(R) \right ) ,\nonumber\\ &\sum_{p_1,p_2}\sum_{m_1\geq 3} \frac{C(m_1,2)}{W_R(\mathcal{F}_{N})} \nonumber\\ & =\ \widehat{\phi}_2(0)\sum_{p_1,p_2<R^\sigma}\sum_{m_1=0}^{\infty}\frac{\log(p_1) \log(p_2)}{\log^2(R)}\widehat{\phi}_1\left( \frac{2\log(p_1)}{\log(R)}\right)(B_{m_1,2}(p_1,p_2)-2B_{m_1,0}(p_1,p_2))\frac{P_{m_1}(p_2)}{p_2} + O\left ( \log^{-4}(R) \right ) ,\nonumber\\ &\sum_{p_1,p_2}\sum_{m_1,m_2\geq 3} \frac{C(m_1,m_2)}{W_R(\mathcal{F}_{N})} \nonumber\\ & = \ \widehat{\phi}_1(0)\widehat{\phi}_2(0)\sum_{p_1,p_2<R^\sigma}\sum_{m_1,m_2=0}^\infty\frac{\log(p_1) \log(p_2)}{\log^2(R)}B_{m_1,m_2}(p_1,p_2)P_{m_1}(p_1)P_{m_2}(p_2) + O\left ( \log^{-4}(R) \right ) . \end{aligned}\] With this, we have $S_{B\infty}(\mathcal{F}_{N})$ is equivalent to [eq:rewrite fourth line infinite], finishing the proof of Theorem 17.

6 Formulas for family specific terms

6.1 Computing terms

We use the harmonic weights⁶ to facilitate computing explicitly the asymptotics. Recall we are working with the weights \[\begin{aligned} w_R(f)\ =\ \frac{Z_N(1,f)}{Z(1,f)} \label{eq:wR_defn} \end{aligned}\] where $Z_N(s,f)$ and $Z(s,f)$ are defined in [def:Z_N_func] and [def:Z_func] respectively. We consider $\mathcal{F}_{N} = H_k^*(N)$ the family of holomorphic cusp newforms with weight $k$ and level $N$.

Recall that our extended explicit formula is expressed in terms of weighted average moments of Hecke eigenvalues, namely \[\sum_{f \in H^{*}_k(N)}\lambda(p_1)^{r_1}\lambda(p_2)^{r_2},\] wherein restrictions are placed on $p_1,p_2$ relative to the level $N$. We wish to compute these average weighted moments for all possible factorizations of $N$. We summarize our results in Lemma 25 through Corollary 29.

For an integer $r \ \geq \ 1$ define \[\label{eq:nu-c-N1-def} \nu(r) \ = \ r\prod_{p\mid r}\left(1+\frac1p\right), \qquad c(r) \ = \ \prod_{q^2\mid r}\frac{q^2}{q^2-1}, \qquad N_1 \ = \ \prod_{p\parallel N}p.\] We use $\xi_d(\ell)$ as the standard local coefficients in the orthonormal oldform basis of [BarrettEtAl2016arXiv].

Proof. The coefficients, $\xi_d(\ell)$, here are multiplicative in the prime-power components of both $d$ and $\ell$, so the left-hand side of [eq:modified-oldclass-sum] factors into simply a product of local terms. Thus at every prime which does not divide $D_L$ the local calculation is exactly the same as the one in Lemma 3.2 of [BarrettEtAl2016arXiv]. Hence the only new calculation occurs at a prime $p\mid D_L$ and since $D\mid N_1$; the condition $p\mid D_L$ tells us that $p\parallel N$, $p\parallel L$, and $p\nmid M$. In particular, the local divisor $d$ can only be 1 or $p$, having that $p$ divides $D$ exactly once. Consequently, the local contribution is \[\begin{aligned} \label{eq:local-D-factor-start} T_p \ = \ \lambda_f(p)+\xi_p(1)^2\lambda_f(p)+\sqrt p\,\xi_p(1)\xi_p(p). \end{aligned}\] By [BarrettEtAl2016arXiv] and using the fact that $p\nmid M$ we have \[\xi_p(p) \ = \ \rho_f(p)^{-1/2}\] and \[\xi_p(1) \ = \ -\frac{\lambda_f(p)}{\sqrt p(1+1/p)}\rho_f(p)^{-1/2} \ = \ -\frac{\sqrt p\,\lambda_f(p)}{p+1}\rho_f(p)^{-1/2}.\] Consequently, it follows that \[\xi_p(1)^2 \ = \ \frac{p\lambda_f(p)^2}{(p+1)^2\rho_f(p)} \ = \ \frac{1-\rho_f(p)}{\rho_f(p)},\] so \[1+\xi_p(1)^2 \ = \ \rho_f(p)^{-1}.\] At the same time \[\sqrt p\,\xi_p(1)\xi_p(p) \ = \ -\frac{p\lambda_f(p)}{(p+1)\rho_f(p)},\] and substituting these identities into [eq:local-D-factor-start] yields \[\begin{aligned} T_p & \ = \ \frac{\lambda_f(p)}{\rho_f(p)}-\frac{p\lambda_f(p)}{(p+1)\rho_f(p)}\notag \ = \ \frac{\lambda_f(p)}{(p+1)\rho_f(p)}. \end{aligned}\] Thus, the usual local factor $\rho_f(p)^{-1}$ from Barrett et al. is replaced at every prime $p\mid D_L$ by \[\frac{\lambda_f(p)}{p+1}\rho_f(p)^{-1}.\] Multiplying these modified local factors over the primes that divide $D_L$, retaining the factors from [barrett2017] at every remaining prime, and observing that the primes of $D$ contained in $M$ contribute the factor $\lambda_f(D_M)$, which comes from $\lambda_f\left(\frac{Db}{\ell}\right)$ leaves us with [eq:modified-oldclass-sum]. ◻

Proof. Beginning with equation (3.7) of [BarrettEtAl2016arXiv] we have since $(a, N) = 1$ the divisor in the first Fourier coefficient factor must be equal to 1, so \[\begin{aligned} \label{eq:barrett-before-specialization} \Delta_{k,N}(a,Db) & \ = \ \frac{12}{(k-1)\nu(N)}\sum_{LM=N}\frac{M}{\varphi(M)}\sum_{f\in H_k^*(M)}\frac{\lambda_f(a)}{Z(1,f)} \notag \\ &\qquad\times\sum_{d\mid L}\xi_d(1)\sum_{\ell\mid(d,D)}\xi_d(\ell)\ell^{1/2}\lambda_f\left(\frac{Db}{\ell}\right). \end{aligned}\] we substitute Lemma 21 into [eq:barrett-before-specialization]. After doing this it is clear to see that in equation (2.35) of [BarrettEtAl2016arXiv], we can rewrite the remaining arithmetic factors to obtain \[\begin{aligned} \label{eq:forward-with-ZN} \Delta_{k,N}(a,Db) \ = \ \frac{12}{(k-1)N}c(N)\sum_{LM=N}\sum_{f\in H_k^*(M)}\frac{Z_N(1,f)}{Z(1,f)}\lambda_f(a)\lambda_f(D_Mb)\prod_{p\mid D_L}\frac{\lambda_f(p)}{p+1}. \end{aligned}\] Continuing forward, all that remains is to rewrite the local factors in [eq:forward-with-ZN]. So, let $p\mid D_L$ then $p\nmid M$, which now has that \[\lambda_f(p)\lambda_f(p^{2j}) \ = \ \lambda_f(p^{2j+1})+\lambda_f(p^{2j-1}),\] where the second term is omitted for $j \ = \ 0$. Therefore \[\begin{aligned} \lambda_f(p)Z_p(1,f) \ &= \ \sum_{j\geq0}\frac{\lambda_f(p^{2j+1})}{p^j}+\sum_{j\geq1}\frac{\lambda_f(p^{2j-1})}{p^j} = \ \left(1+\frac1p\right)\sum_{j\geq0}\frac{\lambda_f(p^{2j+1})}{p^j}, \end{aligned}\] giving the local identity \[\label{eq:odd-local-series} \frac{\lambda_f(p)}{p+1}Z_p(1,f) \ = \ \frac1p\sum_{j\geq0}\frac{\lambda_f(p^{2j+1})}{p^j}.\] Further, at every prime $p\mid L$ such that $p\nmid M$ and $p\nmid D_L$, we can keep the usual expansion \[Z_p(1,f) \ = \ \sum_{j\geq0}\frac{\lambda_f(p^{2j})}{p^j}.\] since $D_L$ is squarefree and $(D_L,M) \ = \ 1$, the multiplication of these local identities gives \[\label{eq:local-series-product} Z_N(1,f)\lambda_f(a)\lambda_f(D_Mb)\prod_{p\mid D_L}\frac{\lambda_f(p)}{p+1} \notag \ = \ \frac{Z_M(1,f)}{D_L}\sum_{\substack{\ell\mid L^\infty \$\ell,M)=1}}\frac{\lambda_f(aD_L\ell^2)\lambda_f(D_Mb)}{\ell}.\] Thus the series in [eq:local-series-product] are absolutely convergent given that only a finite number of primes are involved. Hence substituting [eq:local-series-product] into [eq:forward-with-ZN] and collecting the sum over \(f\in H_k^*(M)$ gives [eq:generalized-forward-trace]. ◻

Proof. Let $\mathcal R$ denote the right-hand side of [eq:generalized-trace-shifted], then apply Lemma 22 at level M to each term \[\Delta_{k,M}(aD_L\ell^2,D_Mb).\] While possibly surprising, this operation is valid because \[D_M\mid\prod_{p\parallel M}p\] and \[(aD_L\ell^2b, M)=1.\] for a given factorization $QW \ = \ M$ define \[D_Q \ = \ (D,Q), \qquad D_W \ = \ (D,W).\] Thus, since every prime that divides $D$ occurs exactly once in $N$, the integers $D_L$, $D_Q$, and $D_W$ must be pairwise coprime and satisfy \[D \ = \ D_LD_QD_W.\] So substituting [eq:generalized-forward-trace] into the definition of $\mathcal R$ yields \[\begin{aligned} \label{eq:inversion-expanded} \mathcal R \ = \ \sum_{LQW=N}\frac{\mu(L)}{D_LD_Q}\sum_{\substack{\ell\mid L^\infty \$\ell,QW)=1}}\frac1\ell\sum_{\substack{t\mid Q^\infty \\(t,W)=1}}\frac1t\Delta^*_{k,W}(aD_LD_Q\ell^2t^2,D_Wb). \end{aligned}\] Fixing \(W\mid N$ and $f\in H_k^*(W)$ plus combining the local factors of the two inner series prime by prime says that if $p\mid W$ then the corresponding Euler factor is already included in $Z_W(1,f)$. However, if $p\nmid W$ and $p\nmid D$ then exactly one of the $\ell$ or $t$ sums contributes the factor \[Z_p(1,f) \ = \ \sum_{j\geq0}\frac{\lambda_f(p^{2j})}{p^j}.\] If $p\mid D$ and $p\nmid W$, then $p$ belongs to exactly one of $L$ or $Q$, and the corresponding local factor by [eq:odd-local-series] is \[\frac1p\sum_{j\geq0}\frac{\lambda_f(p^{2j+1})}{p^j} \ = \ \frac{\lambda_f(p)}{p+1}Z_p(1,f).\] Therefore, for a fixed divisor $W$ of $N$, the entire inner contribution is \[\begin{aligned} \label{eq:ZN-DW-definition} \mathcal Z_{N,D}(W) \ := \ \sum_{f\in H_k^*(W)}\frac{Z_N(1,f)}{Z(1,f)}\lambda_f(a)\lambda_f(D_Wb)\prod_{\substack{p\mid D \\p\nmid W}}\frac{\lambda_f(p)}{p+1}. \end{aligned}\] Importantly, this expression is independent of the factorization $LQ=N/W$ and therefore we can write \[\begin{aligned} \mathcal R & \ = \ \sum_{LQW=N}\mu(L)\mathcal Z_{N,D}(W) \ = \ \sum_{W\mid N}\mathcal Z_{N,D}(W)\sum_{L\mid N/W}\mu(L). \end{aligned}\] Notably, the inner sum is zero unless $W \ = \ N$, in which case it is equal to 1, but when $W \ = \ N$, we have $D_W \ = \ D$ and the product in [eq:ZN-DW-definition] is empty. Thus \[\mathcal R \ = \ \mathcal Z_{N,D}(N) \ = \ \Delta^*_{k,N}(a,Db),\] which proves [eq:generalized-trace-shifted]. Finally, it suffices to derive the equivalent form [eq:generalized-trace-ils-form]. So let $p\mid D_L$. Then since $p\nmid M$, after summing over an orthonormal basis of $S_k(M)$, equation [eq:odd-local-series] gives \[\begin{aligned} \frac1p\sum_{j\geq0}\frac{\Delta_{k,M}(ap^{2j+1}u,v)}{p^j} \ = \ \frac1{p+1}\sum_{j\geq0}\frac{\Delta_{k,M}(ap^{2j}u,pv)}{p^j}, \end{aligned}\] given that all displayed factors other than the indicated powers of $p$ are to be coprime to $p$. Applying this identity independently at each prime that divides $D_L$ replaces the factor $D_L^{-1}$ and the shifted index $aD_L\ell^2$ in [eq:generalized-trace-shifted] by $\nu(D_L)^{-1}$ and the index $Db$, respectively. This gives [eq:generalized-trace-ils-form] and completes the proof. ◻

Proof. Taking $D = a = b = 1$ in [eq:generalized-trace-shifted] yields \[\begin{aligned} \label{eq:weight-trace-expansion} W_R(\mathcal{F}_{N}) \ = \ \frac{k-1}{12}\sum_{LM=N}\mu(L)Mc(M)^{-1}\sum_{\substack{\ell\mid L^\infty \$\ell,M)=1}}\frac{\Delta_{k,M}(\ell^2,1)}{\ell}. \end{aligned}\] By Proposition 8 and the fact that \((\ell,M) = 1$, we have \[\label{eq:weight-petersson-bound} \Delta_{k,M}(\ell^2,1) \ = \ \delta_{\ell,1}+O\left(\frac{\tau(M)\ell^{1/2+\varepsilon}}{M}\right).\] The diagonal term in [eq:weight-trace-expansion] is $\frac{(k-1)\mathcal A(N)}{12}$. For the off-diagonal contribution, choosing $\varepsilon> 0$ sufficiently small gives \[\sum_{\ell\mid L^\infty}\ell^{-1/2+\varepsilon} \ = \ \prod_{p\mid L}\left(1-p^{-1/2+\varepsilon}\right)^{-1} \ll_\varepsilon L^\varepsilon.\] Since $\sum_{LM=N}|\mu(L)|\tau(M) \ \ll_\varepsilon\ N^\varepsilon$, combining these bounds shows the total off-diagonal contribution is $O(N^\varepsilon)$, proving [eq:generalized-weight-asymptotic].

Because $\mathcal A$ is a Dirichlet convolution of multiplicative functions, it is multiplicative. Direct calculation on prime powers yields the formulas for $\mathcal A(p^e)$. In all cases, \[\frac{\mathcal A(p^e)}{p^e} \ \geq \ \left(1-\frac1p\right)^2.\] Multiplying over all primes dividing $N$ and using the standard estimate $\prod_{p\mid N}\left(1-\frac1p\right) \gg_\varepsilon N^{-\varepsilon}$ yields [eq:A-lower-bound]. Equations [eq:generalized-weight-asymptotic] and [eq:A-lower-bound] imply [eq:weight-lower-bound] for all sufficiently large $N$.

For [eq:normalized-general-trace-bound], define $x = aD_L\ell^2$ and $y = D_Mb$ in [eq:generalized-trace-shifted]. The hypotheses imply $(x,y) = 1$. If $x = y$, then $x = y = 1$, which contradicts $Dab > 1$. Thus, the diagonal term vanishes. Proposition 8 yields \[\begin{aligned} \label{eq:trace-petersson-bound} \Delta_{k,M}(aD_L\ell^2,D_Mb) \ \ll_{k,\varepsilon} \ \frac{\tau(M)}{M}(Dab)^{1/4+\varepsilon}\ell^{1/2+2\varepsilon}. \end{aligned}\] Substituting [eq:trace-petersson-bound] into [eq:generalized-trace-shifted], using $D_L^{-1}\leq 1$, and applying the divisor bounds gives \[\Delta^*_{k,N}(a,Db) \ \ll_{k,\varepsilon} \ (Dab)^{1/4+\varepsilon}N^\varepsilon.\] Dividing by the lower bound in [eq:weight-lower-bound] and renaming $\varepsilon$ proves [eq:normalized-general-trace-bound]. ◻

Proof. In [eq:weight-trace-expansion], the condition $\mu(L) \neq 0$ forces $L$ to be squarefree. For $\ell\mid L^\infty$ with $(\ell,M) = 1$ and $LM = N$, every prime dividing $\ell$ divides $L$ but not $M$. Thus, $\ell$ is supported only on primes dividing $N_1$. By assumption, there are at most $t$ such primes. Applying the sharper Petersson estimate from [BarrettEtAl2016arXiv] yields \[\label{eq:sharper-weight-petersson} \Delta_{k,M}(\ell^2,1) \ = \ \delta_{\ell,1}+O\left(\frac{\ell\log(2\ell^2)}{M(\ell+kM)^{1/2}}\right).\] Multiplying by $M/\ell$, the off-diagonal contribution for a fixed factorization $LM = N$ is bounded by \[\sum_{\substack{\ell\mid L^\infty\$\ell,M)=1}}\frac{\log(2\ell^2)}{(\ell+kM)^{1/2}}.\] For fixed \(0 < \eta < 1/2$, we have $(\ell+kM)^{-1/2} \ll_{k,\eta} M^{-1/2+\eta}\ell^{-\eta}$. Since $\ell$ is supported on at most $t$ primes, the Euler product gives \[\sum_{\substack{\ell\mid L^\infty\$\ell,M)=1}}\frac{\log(2\ell^2)}{(\ell+kM)^{1/2}} \ \ll_{k,t,\eta} \ M^{-1/2+\eta}.\] Whenever \(\mu(L) \neq 0$, we have $L\mid\operatorname{rad}(N)$, so $M = \frac{N}{L} \geq \frac{N}{\operatorname{rad}(N)} = Q(N)$. Since $c(M)^{-1} \leq 1$, summing over $L$, absorbing the divisor sum into $N^\eta$, and renaming $\eta$ gives \[W_R(H_k^*(N))-\frac{k-1}{12}\mathcal A(N) \ \ll_{k,t,\varepsilon} \ N^\varepsilon Q(N)^{-1/2},\] proving [eq:factorization-sensitive-weight]. If $N$ is squareful, every prime dividing $N$ occurs to at least the second power, so $\operatorname{rad}(N)^2 \leq N$. Hence $Q(N) \geq N^{1/2}$, yielding [eq:squarefull-weight-quarter]. ◻

Proof. For $N = p^e$, $\mu(L) \neq 0$ implies $L \in \{1, p\}$. The corresponding levels are $M = p^e$ and $M = p^{e-1}$, with the latter giving the larger error. Therefore, \[W_R(H_k^*(p^e))-\frac{k-1}{12}\mathcal A(p^e) \ \ll_{k,e} \ p^{-(e-1)/2}.\] ◻

Lemma 28. Let $\mathcal{F}_{N} \ := \ H_k^*(N)$ for level $N \geq 2$, and let $r, r_1,r_2 \geq 1$. Define \[\varepsilon_r \ = \ r-2\left\lfloor\frac r2\right\rfloor \in \{0,1\}, \qquad b_{r,r-2j} \ = \ \binom rj-\binom r{j-1}\] (where $\binom r{-1} \ := \ 0$), and let $C_j \ = \ \frac{1}{j+1}\binom{2j}{j}$ be the $j$-th Catalan number. The following estimates hold uniformly in the prime factorization of $N$.

If $p\mid N$, then \[A'_r(p) \ = \ \begin{cases} p^{-r/2}, & p\parallel N \text{ and } r\equiv0\pmod2, \\[5pt] O_{k,\varepsilon}\left(\dfrac{p^{-\lfloor r/2\rfloor+1/4+\varepsilon}}{N^{1-\varepsilon}}\right), & p\parallel N \text{ and } r\equiv1\pmod2, \\[9pt] 0, & p^2\mid N. \end{cases}\] If $p\nmid N$, then \[A_r(p) \ = \ \begin{cases} C_{r/2}+O\left(\dfrac{2^r(p^r)^{1/4+\varepsilon}}{N^{1-\varepsilon}}\right), & r\equiv0\pmod2, \\[9pt] O\left(\dfrac{2^r(p^r)^{1/4+\varepsilon}}{N^{1-\varepsilon}}\right), & r\equiv1\pmod2. \end{cases}\] For $p_1 = p_2 = p$, \[B''_{r_1,r_2}(p,p) \ = \ A'_{r_1+r_2}(p), \qquad B_{r_1,r_2}(p,p) \ = \ A_{r_1+r_2}(p).\] For $p_1\neq p_2$ with $p_1,p_2\mid N$: if $p_1^2\mid N$ or $p_2^2\mid N$, then $B''_{r_1,r_2}(p_1,p_2) = 0$. If $p_1\parallel N$ and $p_2\parallel N$, define $d = p_1^{\varepsilon_{r_1}}p_2^{\varepsilon_{r_2}}$. Then \[B''_{r_1,r_2}(p_1,p_2) \ = \ \begin{cases} p_1^{-r_1/2}p_2^{-r_2/2}, & r_1\equiv r_2\equiv0\pmod2, \\[6pt] O\left(\dfrac{p_1^{-\lfloor r_1/2\rfloor}p_2^{-\lfloor r_2/2\rfloor}d^{1/4+\varepsilon}}{N^{1-\varepsilon}}\right), & \text{otherwise}. \end{cases}\] For $p_1\mid N$ and $p_2\nmid N$: if $p_1^2\mid N$, then $B'_{r_1,r_2}(p_1,p_2) = 0$. If $p_1\parallel N$, then \[B'_{r_1,r_2}(p_1,p_2) \ = \ \begin{cases} p_1^{-r_1/2}C_{r_2/2}+O\left(\dfrac{2^{r_2}p_1^{-\lfloor r_1/2\rfloor}(p_1^{\varepsilon_{r_1}}p_2^{r_2})^{1/4+\varepsilon}}{N^{1-\varepsilon}}\right), & r_1\equiv r_2\equiv0\pmod2, \\[12pt] O\left(\dfrac{2^{r_2}p_1^{-\lfloor r_1/2\rfloor}(p_1^{\varepsilon_{r_1}}p_2^{r_2})^{1/4+\varepsilon}}{N^{1-\varepsilon}}\right), & \text{otherwise}. \end{cases}\] For $p_1 \neq p_2$ and $p_1,p_2\nmid N$: \[B_{r_1,r_2}(p_1,p_2) \ = \ \begin{cases} C_{r_1/2}C_{r_2/2}+O\left(\dfrac{2^{r_1+r_2}(p_1^{r_1}p_2^{r_2})^{1/4+\varepsilon}}{N^{1-\varepsilon}}\right), & r_1\equiv r_2\equiv0\pmod2, \\[12pt] O\left(\dfrac{2^{r_1+r_2}(p_1^{r_1}p_2^{r_2})^{1/4+\varepsilon}}{N^{1-\varepsilon}}\right), & \text{otherwise}. \end{cases}\]

Proof. For $p\mid N$, Equation (2.11) from [BarrettEtAl2016arXiv] yields \[\lambda_f(p)^2 \ = \ \begin{cases} p^{-1}, & p\parallel N, \\ 0, & p^2\mid N. \end{cases}\] Thus, when $p\parallel N$, positive powers of $\lambda_f(p)$ reduce as \[\label{eq:level-power-reduction} \lambda_f(p)^r \ = \ p^{-\lfloor r/2\rfloor}\lambda_f(p)^{\varepsilon_r},\] and vanish when $p^2\mid N$. For even $r$, averaging [eq:level-power-reduction] yields $A'_r(p) = p^{-r/2}$. For odd $r$, $A'_r(p) = p^{-\lfloor r/2\rfloor}\frac{\Delta^*_{k,N}(1,p)}{W_R(\mathcal{F}_{N})}$, and the estimate follows from [eq:normalized-general-trace-bound] with $D = p$ and $a = b = 1$.

For $p\nmid N$, repeated application of the Hecke relation yields \[\label{eq:hecke-power-expansion} \lambda_f(p)^r \ = \ \sum_{j=0}^{\lfloor r/2\rfloor}b_{r,r-2j}\lambda_f(p^{r-2j}).\] This identity follows by induction. We note that $\sum_{j=0}^{\lfloor r/2\rfloor}b_{r,r-2j} = \binom r{\lfloor r/2\rfloor} \leq 2^r$ and $b_{2j,0} = C_j$. Averaging [eq:hecke-power-expansion] produces a diagonal term only for even $r$ at $j = r/2$, with coefficient $C_{r/2}$. Remaining terms are bounded using [eq:normalized-general-trace-bound] with $D = a = 1$ and $b = p^{r-2j}$. Summing the errors establishes $A_r(p)$.

The identical prime identities for $B''$ and $B$ are immediate. Assume $p_1 \neq p_2$ and $p_1, p_2 \mid N$. If either $p_1^2\mid N$ or $p_2^2\mid N$, the corresponding eigenvalue power vanishes, giving $B''_{r_1,r_2}(p_1,p_2) = 0$. For $p_1, p_2 \parallel N$, applying [eq:level-power-reduction] yields \[\lambda_f(p_1)^{r_1}\lambda_f(p_2)^{r_2} \ = \ p_1^{-\lfloor r_1/2\rfloor}p_2^{-\lfloor r_2/2\rfloor}\lambda_f\left(p_1^{\varepsilon_{r_1}}p_2^{\varepsilon_{r_2}}\right).\] If $r_1$ and $r_2$ are both even, the remaining eigenvalue is 1, providing the main term $p_1^{-r_1/2}p_2^{-r_2/2}$. Otherwise, applying [eq:normalized-general-trace-bound] with $D = p_1^{\varepsilon_{r_1}}p_2^{\varepsilon_{r_2}}$ and $a=b=1$ yields the error term.

For the mixed moment with $p_1\parallel N$ and $p_2\nmid N$, combining [eq:level-power-reduction] and [eq:hecke-power-expansion] gives \[\lambda_f(p_1)^{r_1}\lambda_f(p_2)^{r_2} \ = \ p_1^{-\lfloor r_1/2\rfloor}\sum_{j=0}^{\lfloor r_2/2\rfloor}b_{r_2,r_2-2j}\lambda_f\left(p_1^{\varepsilon_{r_1}}p_2^{r_2-2j}\right).\] A diagonal term appears only when both $r_1$ and $r_2$ are even, contributing $p_1^{-r_1/2}C_{r_2/2}$. Remaining terms are bounded via [eq:normalized-general-trace-bound] with $D = p_1^{\varepsilon_{r_1}}$, $a=1$, and $b=p_2^{r_2-2j}$. Using $\sum_jb_{r_2,r_2-2j} \leq 2^{r_2}$ yields the stated error for $B'_{r_1,r_2}(p_1,p_2)$.

Finally, for $p_1 \neq p_2$ and $p_1,p_2\nmid N$, applying [eq:hecke-power-expansion] yields \[\begin{aligned} \lambda_f(p_1)^{r_1}\lambda_f(p_2)^{r_2} \ = \ \sum_{j_1=0}^{\lfloor r_1/2\rfloor}\sum_{j_2=0}^{\lfloor r_2/2\rfloor} &b_{r_1,r_1-2j_1}b_{r_2,r_2-2j_2}\lambda_f(p_1^{r_1-2j_1})\lambda_f(p_2^{r_2-2j_2}). \end{aligned}\] Since the trace indices are coprime, they are equal only when both are 1. This occurs when $r_1$ and $r_2$ are even, yielding the main term $C_{r_1/2}C_{r_2/2}$. Remaining terms are bounded using [eq:normalized-general-trace-bound] with $D = 1$, $a = p_1^{r_1-2j_1}$, and $b = p_2^{r_2-2j_2}$, completing the proof via $\left(\sum_{j_1}b_{r_1,r_1-2j_1}\right)\left(\sum_{j_2}b_{r_2,r_2-2j_2}\right) \leq 2^{r_1+r_2}$. ◻

Corollary 29. Let $N = \prod_{i=1}^t q_i^{e_i}$ vary through a sequence of levels with fixed $t$ and fixed exponents $e_i\geq1$, and suppose \[q_i \ \asymp \ N^{\delta_i}, \qquad \delta_i \ \geq \ 0, \qquad \sum_{i=1}^t e_i\delta_i \ = \ 1.\] Then the moments at primes dividing the level from Lemma 28 separate according to the individual factorization exponents. If $e_i \geq 2$, every positive moment at $q_i$ vanishes: \[\label{eq:profile-repeated-prime-vanishing} A'_r(q_i) \ = \ 0,\] and any $B''$ or $B'$ moment containing $q_i$ as a level-dividing prime vanishes. If $e_i = 1$, then for $m \geq 1$, \[\label{eq:profile-even-level} A'_{2m}(q_i) \ = \ q_i^{-m} \asymp N^{-m\delta_i},\] while \[\label{eq:profile-odd-level} A'_{2m+1}(q_i) \ \ll_{k,\varepsilon} \ N^{-1-\delta_i(m-1/4)+\varepsilon}.\] If $i \neq j$, $e_i = e_j = 1$, and $r,s \geq 1$, then \[\label{eq:profile-two-level} B''_{r,s}(q_i,q_j) \ = \ \begin{cases} q_i^{-r/2}q_j^{-s/2}, & r \equiv s \equiv 0 \pmod2,\\[5pt] O\left(N^{-1-\delta_i\lfloor r/2\rfloor-\delta_j\lfloor s/2\rfloor+\frac14(\delta_i\varepsilon_r+\delta_j\varepsilon_s)+\varepsilon}\right), & \text{otherwise}. \end{cases}\] Consequently, the only primes dividing the level capable of producing nondecaying even-moment main terms along such a sequence are those with $e_i = 1$ and $\delta_i = 0$. Several fixed primes may simultaneously satisfy $q_i\parallel N$, and their even-even $B''$ interactions continue to appear as products of their local factors. If $q_i\parallel N$ is fixed, $q\nmid N$, and $r,s$ are both even, then \[\label{eq:profile-mixed-fixed-main} B'_{r,s}(q_i,q) \ = \ q_i^{-r/2}C_{s/2} +O\left( \frac{2^s q_i^{-r/2}q^{s(1/4+\varepsilon)}}{N^{1-\varepsilon}}\right).\]

Proof. The vanishing in [eq:profile-repeated-prime-vanishing] follows from $\lambda_f(q_i)^2=0$ for $q_i^2\mid N$. If $e_i = 1$, $\lambda_f(q_i)^2 = q_i^{-1}$, yielding [eq:profile-even-level]. For odd moments, Lemma 28 yields \[A'_{2m+1}(q_i) \ \ll_{k,\varepsilon} \ \frac{q_i^{-m+1/4+\varepsilon}}{N^{1-\varepsilon}}.\] Substituting $q_i \asymp N^{\delta_i}$ and renaming $\varepsilon$ gives [eq:profile-odd-level]. The same substitution in the $B''$ estimate of Lemma 28 yields [eq:profile-two-level], and [eq:profile-mixed-fixed-main] is its mixed even-even specialization. ◻

7 Lower order terms

We compute the various expressions up to $O(\log^{-4}(R))$ error. In particular, we prove that universality of the terms breaks when there is a factor of bounded size in the factorization of $N$ as $N\to\infty$.

Theorem 32. Let $\phi$ be an even Schwartz function such that $\widehat\phi$ has compact support. Let $\mathcal{F}= \bigcup \mathcal{F}_{N}$ be a family of newforms defined by a sequence $N \to \infty$, where $N = q_1^{a_1}\cdots q_n^{a_n}$ for fixed distinct primes $q_1,\dots,q_n$, with $q_i \asymp N^{\delta_i}$ for fixed $\delta_i \geq 0$, and $a_i \in \mathbb{Z}^+$. Let $Q := \left \{ q_i \mid a_i = 1,\ \delta_i = 0 \right \}$ and set \[\begin{aligned} \label{eq:gammaQ-def} \beta_{Q,1}\ :=\ \sum_{q \in Q}\frac{\log(q)}{q^2-1}, \qquad \beta_{Q,3}\ :=\ \sum_{q \in Q}\frac{q^2\left ( q^2+1 \right ) \log^3(q)}{\left ( q^2-1 \right ) ^3}, \end{aligned}\] with the convention that an empty sum is $0$. Then \[\begin{aligned} \label{eq:SA'-explicit} S_{A'}(\mathcal{F}_{N})\ =\ -\frac{2\widehat\phi(0)\beta_{Q,1}}{\log(R)}-\frac{4\widehat\phi''(0)\beta_{Q,3}}{\log^3(R)}+O\left ( \log^{-5}(R) \right ) . \end{aligned}\] In particular, $S_{A'}(\mathcal{F}_{N}) = O\left ( \log^{-5}(R) \right )$ if $Q = \emptyset$, while $S_{A'}(\mathcal{F}_{N}) \asymp \log^{-1}(R)$ if $Q \neq \emptyset$ and $\widehat\phi(0) \neq 0$.

Proof. Recall the definition \[\begin{aligned} S_{A'}(\mathcal{F}_{N}) \ := \ -2\sum_p\sum_{r=1}^\infty \frac{A'_{r,N}(p)}{p^{r/2}}\frac{\log (p)}{\log (R)}\widehat\phi\left ( r\frac{\log(p)}{\log (R)} \right ) . \end{aligned}\] By the results in Section 6, $p^2 \mid N \implies A'_{r,N}(p) = 0$, so we need only sum over $p \Vert N$, which are the primes $q_i$ with $a_i =1$. Moreover, one can see that \[\begin{aligned} \sum_{r=1}^\infty \left\lvert \frac{A'_{r,N}(p)}{p^{r/2}}\frac{\log p}{\log R}\widehat\phi\left ( r \frac{\log p}{\log R} \right ) \right\rvert\ \lesssim\ \sum_{r=1}^\infty\frac{1}{p^{r/2}}\ <\ \infty, \end{aligned}\] so the sum converges absolutely, hence we may break into even and odd $r$.

When $r \equiv 1 \pmod 2$, Section 6 gives $A'_{r,N}(p) \ll_{k,\varepsilon} p^{-\left \lfloor r/2 \right \rfloor +1/4+\varepsilon}N^{-1+\varepsilon}$, so the odd terms contribute (taking $\varepsilon$ small enough) \[\begin{aligned} \ \lesssim\ \frac{1}{N^{1-\varepsilon}}\sum_{p \Vert N}\frac{\log(p)}{\log(R)}\sum_{\substack{r \geq 1 \\ r \equiv 1 (2)}}p^{-\left \lfloor r/2 \right \rfloor -r/2+1/4+\varepsilon} \ \lesssim\ \frac{n\log(N)}{N^{1-\varepsilon}\log(R)}\ \lesssim\ \frac{1}{N^{1/2}}, \end{aligned}\] which vanishes in the sense of Remark 31. When $r = 2m$ is even, Section 6 gives $A'_{2m,N}(p) = p^{-m}$ exactly, and we are left with \[\begin{aligned} \label{eq:SA'-even-part} S_{A'}(\mathcal{F}_{N})\ =\ -2\sum_{p \Vert N}\frac{\log(p)}{\log(R)}\sum_{m = 1}^\infty \frac{1}{p^{2m}}\widehat\phi\left ( 2m\frac{\log(p)}{\log(R)} \right ) +O\left ( N^{-1/2} \right ) . \end{aligned}\] If $p \Vert N$ has $\delta_p > 0$, then $p \asymp N^{\delta_p}$, and bounding $\widehat\phi$ and $\log(p)/\log(R)$ trivially shows that the corresponding summand is $\lesssim p^{-2} \lesssim N^{-2\delta_p}$, again negligible by Remark 31. Only the fixed primes $q \in Q$ survive.

Fix such a $q \in Q$. As $\widehat\phi$ is even and Schwartz we have $\widehat\phi'(0) = 0$, and Taylor’s theorem gives, for every $y \in \mathbb R$, \[\begin{aligned} \label{eq:phihat-taylor-4} \widehat\phi(y)\ =\ \widehat\phi(0)+\frac{\widehat\phi''(0)}{2}y^2+E(y), \qquad \left\lvert E(y) \right\rvert\ \leq\ \frac{\lVert \widehat\phi^{(4)} \rVert_\infty}{24}y^4 . \end{aligned}\] Setting $y = 2m\log(q)/\log(R)$, so that the quadratic term carries a factor $4m^2$, and using \[\begin{aligned} \sum_{m=1}^\infty \frac{1}{q^{2m}}\ =\ \frac{1}{q^2-1},\qquad \sum_{m=1}^\infty \frac{m^2}{q^{2m}}\ =\ \frac{q^2\left ( q^2+1 \right ) }{\left ( q^2-1 \right ) ^3}, \qquad \sum_{m=1}^\infty \frac{m^4}{q^{2m}}\ <\ \infty, \end{aligned}\] we obtain \[\begin{aligned} \sum_{m=1}^\infty \frac{1}{q^{2m}}\widehat\phi\left ( 2m\frac{\log(q)}{\log(R)} \right ) \ =\ \frac{\widehat\phi(0)}{q^2-1}+\frac{2\widehat\phi''(0)\log^2(q)}{\log^2(R)}\cdot\frac{q^2\left ( q^2+1 \right ) }{\left ( q^2-1 \right ) ^3}+O\left ( \frac{\log^4(q)}{\log^4(R)} \right ) . \end{aligned}\] Multiplying by $-2\log(q)/\log(R)$ and summing over the finitely many $q \in Q$ yields [eq:SA'-explicit]. Finally, every summand of $\beta_{Q,1}$ is positive, so $\beta_{Q,1} > 0$ whenever $Q \neq \emptyset$, and the last assertion follows. ◻

Theorem 33. Suppose the test function $\phi$ is an even Schwartz function with $\widehat\phi$ supported in $[-\sigma,\sigma]$ for $\sigma<0.22$. For all our cases, we have where \[\begin{aligned} \gamma_{PNT3 } \ &\coloneqq \ 1+ \int_1^\infty \frac{E(t)}{t^2}dt \approx -1.33258\nonumber \\ \nonumber \gamma_{A,1} \ &\coloneqq \ \int_{1}^{\infty}\frac{E(t)}{t^{2}}\Bigl((\log(t))^{2}-2\log(t)\Bigr)dt \approx -10.0881\\\nonumber \gamma_{A,2} \ &\coloneqq \ \sum_p\frac{4\log(p)}{p(p+1)} \approx 1.5382\\\nonumber \gamma_{A,3} \ &\coloneqq \ \sum_p\frac{2(p^2+3p+1)\log(p)}{p(p+1)^3 }\approx 0.8852\\ \nonumber \gamma_{A,4} \ &\coloneqq \ \sum_p\frac{(32p^2+24p+8)\log^3(p)}{p(p+1)^3} \approx 43.6045 \\\nonumber \gamma_{A,5} \ &\coloneqq \ \sum_p\frac{(-64p^4+4p^3-44p^2-20p-4)\log^3(p)}{p(p+1)^5}\&\approx -72.6540\\ \nonumber \gamma_{A,6} \ &\coloneqq \ \sum_p\frac{2(p-1)\log(p)}{(p+1)}\sum_{r=2}^{\infty}\frac{C_rp^{r}}{(p+1)^{2r}} \approx 0.8321\\ \gamma_{A,7} \ &\coloneqq \ \sum_p\frac{(p-1)\log^3(p)}{(p+1)^3}\sum_{r=3}^{\infty}\frac{C_r p^{r}(4r^2(p-1)^2-24rp-8p)}{(p+1)^{2r}} \approx 5.8746529 \end{aligned}\] and where we let $\theta(t)=\sum_{p\leq t}\log(p)$ and define $E(t):=\theta(t)-t$ to be the error.

Proof of Theorem 33, We note that, regardless of factorization, we have \[\begin{aligned} A_{r,N}(p) \ = \ & \begin{cases} C_{r/2} + O\left(\frac{r2^rp^{r/4}\log(p)\log^2(N)}{N}\right) &\text{ if $r \equiv 0 \bmod 2$ }\\ O\left(\frac{r2^rp^{r/4}\log(p)\log^2(N)}{N}\right)&\text{ if $r \equiv 1 \bmod 2$}. \end{cases} \end{aligned}\] Thus we have \[\begin{aligned} A_{0,N}(p)\ =& \ 1+O\left(\frac{\log(p)\log^2(N)}{N}\right)\nonumber\\ A_{1,N}(p)\ =& \ O\left(\frac{p^{1/4}\log(p)\log^2(N)}{N}\right)\nonumber\\ A_{2,N}(p)\ =& \ 1 +O\left(\frac{p^{1/2}\log(p)\log^2(N)}{N}\right). \end{aligned}\] Since \[\begin{aligned} \frac{\log^2(N)}{N \log(R)}\sum_{p\leq R^{\sigma}} \frac{\log^2(p)}{p^2 } &\ \lesssim\ \frac{\log^2(N)}{R^{\sigma}N}\\ \frac{\log^2(N)}{N \log(R)}\sum_{p\leq R^{\sigma}} \frac{\log^2(p)}{p^{5/4} }&\ \lesssim\ \frac{\log^2(N)}{R^{\sigma/4}N} \\ \frac{\log^2(N)}{N \log(R)}\sum_{p\leq R^{\sigma}} \frac{\log^2(p)}{p^{3/2} } &\ \lesssim\ \frac{\log^2(N)}{R^{\sigma/2}N}, \end{aligned}\] the error terms for all the finite sums in Theorem [theorem: S_1] that are multiplied by a Schwartz function are sufficiently small and we have the sums equal \[\begin{aligned} &-2\widehat{\phi}\left(0 \right)\sum_p\frac{2\log(p)}{p(p+1)\log(R)}+2\widehat{\phi}\left(0 \right) \sum_p\frac{(p^2+3p+1)\log(p)}{p(p+1)^3 \log(R)} \notag \\ &+\frac{\widehat{\phi}''(0)}{4\pi^2}\sum_p\frac{(32p^2+24p+8)\log^3(p)}{p(p+1)^3\log^3(R)} -\frac{\widehat{\phi}''(0)}{4\pi^2}\sum_p\frac{(64p^4-4p^3+44p^2+20p+4)\log^3(p)}{p(p+1)^5\log^3(R)}. \end{aligned}\] We deal with the terms \[\begin{aligned} &2\sum_p \frac{2A_{0, \mathcal{F}}(p)\log(p)}{p\log(R)}\widehat{\phi}\left(2 \frac{\log(p)}{\log(R)} \right) \notag \\ -&2\sum_p\frac{A_{1, \mathcal{F}}(p)\log(p)}{p^{1/2}\log(R)}\widehat{\phi}\left(\frac{\log(p)}{\log(R)} \right)\notag \\ -&2\sum_p\frac{A_{2,\mathcal{F}}(p)\log(p)}{p\log(R)}\widehat{\phi}\left(2 \frac{\log(p)}{\log(R)} \right). \end{aligned}\] In Appendix 10, we estimate many sums up to our required error, which we employ to compute these expressions. In many cases, the error terms are left with convergent but non-elementary integrals that must be estimated numerically. Since the main terms for $A_2(\mathcal{F}_{N})(p)$ and $A_0(\mathcal{F}_{N})(p)$ agree and $A_1(\mathcal{F}_{N})(p)$ is an error, using Lemma 46, we have \[\begin{aligned} 2\sum_p&\frac{2A_{0,\mathcal{F}}(p)\log(p)}{p\log(R)}\widehat{\phi}\left(2 \frac{\log(p)}{\log(R)} \right)-2\sum_p\frac{A_{2,\mathcal{F}}(p)\log(p)}{p\log(R)}\widehat{\phi}\left(2 \frac{\log(p)}{\log(R)} \right)\\ &= \ \frac{\phi(0)}{2} +\frac{2\widehat\phi(0)}{\log(R)}\Bigl(1+\int_{1}^{\infty}\frac{E(t)}{t^{2}}dt\Bigr) +\frac{\widehat\phi''(0)}{\pi^2(\log(R))^{3}} \int_{1}^{\infty}\frac{E(t)}{t^{2}}\Bigl((\log(t))^{2}-2\log(t)\Bigr)dt +O\left ( \log^{-4}(R) \right ) \end{aligned}\] as the main term for these two differences. We show the error terms in these cases are sufficiently small. Using compact support, our error terms are \[\begin{aligned} &\frac{\log(R)}{N}\sum_{p \leq R^{\sigma}} \frac{\log^2(p)}{p}\ \lesssim\ \frac{\log^3(R)}{N},\nonumber\\ &\frac{\log(R)}{N}\sum_{p \leq R^{\sigma}}\frac{\log^2(p)}{p^{1/4}}\ \lesssim\ \frac{R^{3\sigma/4}\log(N)}{N}\nonumber,\\ &\frac{\log(R)}{N}\sum_{p \leq R^{\sigma}}\frac{\log^2(p)}{p^{1/2}}\ \lesssim\ \frac{R^{\sigma/2}\log(N)}{N},\ \end{aligned}\] which for restricted support are all sufficient. We evaluate the last two sums. Using [lem: Computation of small m] and [lem: Computation of tail m], we have \[\sum_{p}\sum_{r=3}^{\infty}\frac{A_{r, \mathcal{F}}(p)p^{r/2}(p-1)\log(p)}{(p+1)^{r+1}\log(R)}\ =\ \sum_p\frac{(p-1)\log(p)}{(p+1)\log(R)}\sum_{r=2}^{\infty}\frac{C_rp^{r}}{(p+1)^{2r}}+O\left ( \log^{-4}(R) \right ) .\] We aim to evaluate \[\sum_p\sum_{r=3}^{\infty}\frac{A_{r, \mathcal{F}}(p)(p-1)(r^2(p-1)^2-12rp-8p)p^{r/2}\log^3(p)}{(p+1)^{r+3}\log^3(R)}.\] We first show the tail is negligible. We see \[\begin{aligned} \sum_p\sum_{r=1+2\log(R)}^{\infty}\frac{A_{r, \mathcal{F}}(p)((p-1)(r^2(p-1)^2-12rp-8p)p^{r/2}\log^3(p)}{(p+1)^{r+3}\log^3(R)} \ \lesssim \ \frac{1}{\log^3}\sum_p\log^3(p)\sum_{r=1+2\log(R)}^{\infty} r^2\left(\frac{2p^{1/2}}{p+1} \right)^r. \end{aligned}\] Further, \[\begin{aligned} \frac{1}{\log^3(R)}\sum_p\log^3(p)\sum_{r=1+2\log(R)}^{\infty} r^2\left(\frac{2p^{1/2}}{p+1} \right)^r &\lesssim\ \frac{1}{\log^3(R)}\sum_p\log^3(p)\left(\frac{p^{1/2}}{p+1} \right)^{2\log(R)+1}\sum_{r=1}^{\infty} r^2\left(\frac{2p^{1/2}}{p+1} \right)^r\nonumber\\ &\lesssim\ \frac{1}{\log^3(R)}\sum_p\log^3(p)\left(\frac{p^{1/2}}{p+1} \right)^{2\log(R)+1}\sum_{r=0}^{\infty} r^2\left(\frac{2p^{1/2}}{p+1} \right)^r\nonumber\\ &\lesssim\ \frac{1}{\log^3(R)}\sum_p\log^3(p)\left(\frac{2p^{1/2}}{p+1} \right)^{2\log(R)+1}\nonumber\\ &\lesssim\ \frac{1}{R^{-.11}\log^3(R)}. \end{aligned}\] Thus the tail is negligible. We show the error term that arises from $A_r(\mathcal{F}_{N})$ in the truncated sum is also negligible. Using the same asymptotic for the rational function in the sum, we have the error is \[\begin{aligned} &\frac{1}{N\log^3(R)}\sum_{p\leq R^{\sigma}}\log^4(p)\sum_{r=3}^{2\log(R)}r^3\left(\frac{2p^{3/4}}{(p+1)}\right)^r \notag \\ &\ \lesssim\ \frac{1}{N\log^3(R)}\left[\sum_{p< 2027}\log^4(p)\sum_{r=3}^{2\log(R)}r^3\left(\frac{2p^{3/4}}{(p+1)}\right)^r \sum_{2027<p< R^{\sigma}}\log^4(p)\sum_{r=3}^{2\log(R)}r^3\left(\frac{2p^{3/4}}{(p+1)}\right)^r \right]\notag \\ &\ \lesssim\ \frac{1}{N}\left[2027\cdot \left(\frac{2\cdot3^{3/4}}{4}\right)^{2\log(R)} + \sum_{p\in[2027,R^{\sigma}]}\left(\frac{2p^{3/4}}{p+1} \right) \right] \notag \\ &\ \lesssim\ \frac{R^{\max\{.11,3\sigma/4\}}}{N}, \end{aligned}\] which is negligible for our support. Thus \[\sum_p\frac{(p-1)\log^3(p)}{p+1}\sum_{r=2}^{\log(R)}\frac{C_r p^{r}(1 + 2r)\bigl(p^{2}(2r - 1) - p (10 + 4 r) + (2r - 1)\bigr)}{(p+1)^{2r}\log^3(R)}\] as our main term. This sum can be extended to infinity at the cost of $R^{-0.11}$, which is negligible. Thus shown \[\begin{aligned} S_A&(\mathcal{F}_{N})\ =\ \notag\\ &\frac{\phi(0)}{2} +\frac{2\widehat\phi(0)}{\log(R)}\Bigl(1+\int_{1}^{\infty}\frac{E(t)}{t^{2}}dt\Bigr)+\frac{\widehat\phi''(0)}{\pi^2(\log(R))^{3}} \int_{1}^{\infty}\frac{E(t)}{t^{2}}\Bigl((\log(t))^{2}-2\log(t)\Bigr)dt\nonumber\\ &-2\widehat{\phi}\left(0 \right)\sum_p\frac{2\log(p)}{p(p+1)\log(R)}+2\widehat{\phi}\left(0 \right) \sum_p\frac{(p^2+3p+1)\log(p)}{p(p+1)^3 \log(R)}+\frac{\widehat{\phi}''(0)}{4\pi^2}\sum_p\frac{(32p^2+24p+8)\log^3(p)}{p(p+1)^3\log^3(R)}\nonumber\\ &-\frac{\widehat{\phi}''(0)}{4\pi^2}\sum_p\frac{(64p^4-4p^3+44p^2+20p+4)\log^3(p)}{p(p+1)^5\log^3(R)}+2\widehat\phi(0)\sum_p\frac{(p-1)\log(p)}{(p+1)\log(R)}\sum_{r=2}^{\infty}\frac{C_rp^{r}}{(p+1)^{2r}}\nonumber\\ &+\frac{\widehat{\phi}''(0)}{4\pi^2}\sum_p\frac{(p-1)\log^3(p)}{(p+1)\log^3(R)}\sum_{r=3}^{\infty}\frac{C_r (4r^2(p-1)^2-24rp-8p)p^{r}}{(p+1)^{2r}}+ O\left ( \log^{-4}(R) \right ) . \end{aligned}\]Substituting our constants in yields the theorem. 0◻

For $S_{B_\infty }(\mathcal{F}_{N})$, we start by noticing that the expressions for $A_r(p)$ and $B_{r_1,r_2}(p_1,p_2)$ are the same up to error terms for the families with prime level or families with level being two distinct primes. The error term only differs by an exponent on $1/N$, making the computations more or less the same.

First, we compute the main terms; the computation is exactly the same for all factorizations.

Theorem 37. Suppose the test functions $\phi_1,\phi_2$ are even Schwartz functions with $\widehat\phi_1, \widehat\phi_2$ supported in $[-\sigma,\sigma]$ for $\sigma<0.22$, and with $\widehat\phi_i''$ the second derivative of $\widehat\phi_i$. Then for all of our cases, \[\begin{aligned} S_{B_\infty}(\mathcal{F}_{N}) \ = \ & \frac{\widehat\phi_1(0)\widehat\phi_2(0)}{\log(R)} \left(\frac{3}{4}-\log(2)-\gamma_3+\tfrac{\gamma_4-\gamma_5}{2} +3\gamma_2-4\gamma_2\log 2+2\gamma_2(\gamma_4-\gamma_5)-4\gamma_2\gamma_3\right) \\ &+\frac{\widehat\phi_1(0)\widehat\phi_2(0)}{\log^2(R)} \left({2\gamma_1-\frac{\gamma_6}{2}+2\gamma_7-2\gamma_8-\frac{7}{18}+\gamma_9+\gamma_{10}+\gamma_{11}}\right)+\frac{\widehat\phi_1(0)\widehat\phi_2''(0)+\widehat\phi_1''(0)\widehat\phi_2(0) }{4\pi^2\log^2(R)}\gamma_1 \end{aligned}\] where \[\begin{aligned} \gamma_{1} \ &:= \ \int_{1}^\infty \frac{E(t)(2-5\log(t))}{t^{7/2}}dt &&\approx 0.2953\\ \gamma_2 \ &:= \ \int_1^\infty \frac{E(t)(2t+1)}{(t^2+t)^2}dt &&\approx 0.1914\\ \gamma_3 \ &:= \ \int_1^\infty \frac{E(t)(2t+1)}{(t^2+t)^2}dt &&\approx 0.1914\\ \gamma_4 \ &:= \ \int_1^\infty E(t)\frac{2t^3+8t^2+4t+1}{t^2(t+1)^4}dt &&\approx -0.000501\\ \gamma_5 \ &:= \ \int_1^\infty E(t)\frac{4t^{3} - 5t^{2} + 4t + 1}{t^{2} \left(t + 1\right)^{4}}dt &&\approx -0.4854\\ \gamma_6 \ &:= \ \int_0^\infty t\frac{e^t+3+e^{-t}}{(e^t+1)^3}dt&&\approx 0.210279 \\ \gamma_7 \ &:= \ \int_1^\infty \frac{E(t)((-3t^3-12t^2-8t-2)\log(t)+t^3+4t^2+4t+1)}{t^3(t+1)^4} &&\approx 0.07780\\ \gamma_8 \ &:= \ \int_1^\infty \frac{E(t)((-9t^4+10t^2+10t+3)\log(t)+3t^4+3t^3-2t^2-3t-1)}{t^4(t+1)^4}dt &&\approx 0.06586\\ \gamma_9 \ &:= \ \sum_{p_1, p_2} \log \left(p_1\right) \log(p_2)\left[P_0\left(p_1\right) P_0\left(p_2\right)+\sum_{\ell=1}^{\infty} C_\ell\left(P_0\left(p_1\right) P_{2 \ell}\left(p_2\right)+P_{2 \ell}\left(p_1\right) P_0\left(p_2\right)\right)\right]&&\approx 0.5135 \\ \gamma_{10} \ &:= \ \sum_{p_1 p_2} {\log (p_1) \log(p_2)} \sum_{\ell_1, \ell_2=1}^{\infty} C_{\ell_1} C_{\ell_2} P_{2\ell_1}(p_1) P_{2\ell_2}(p_2) &&\approx 0.4014\\ \gamma_{11} \ &:= \ \sum_p \log^2(p)\sum_{\ell_1, \ell_2=1}^{\infty}\left[\left(C_{\ell_1+\ell_2}-C_{\ell_1} C_{\ell_2}\right) P_{2\ell_1}\left(p\right) P_{2\ell_2}(p)+C_{\ell_1+\ell_2-1} P_{2\ell_1-1}(p) P_{2 \ell_2-1}(p)\right] &&\approx 1.9648. \end{aligned}\]

The above theorems follow from substituting the terms $A', A, B'', B,$ and $B$ obtained from Section 6 for each case into the formulas for $S_{B''}(\mathcal{F}_{N}), S_{B'}(\mathcal{F}_{N}), S_{B_f}(\mathcal{F}_{N}),$ and $S_{B_\infty}$ given in Theorem 17. The main idea is to divide the proof into calculation of main terms and error terms. Factorizations that do not contribute a main term in the formulas for $A', A, B'', B,$ and $B$ will not contribute a main term towards $S_{B''}(\mathcal{F}_{N}), S_{B'}(\mathcal{F}_{N}), S_{B_f}(\mathcal{F}_{N}),$ and $S_{B_\infty}$. Thus, by examining which factorization yield main terms, we can determine when universality breaks. we explain how the main term and error terms are computed.

There are two parts to the proof of the theorem. The first is evaluating the main terms from merely plugging in what we have already proved. For example, one of the main terms that come up in $S_{B_\infty}$ is, after plugging in our results, \[\begin{aligned} \label{eq:SumExample} \widehat{\phi}_1(0)\sum_{p}\sum_{m_1=0}^{\infty}\widehat{\phi}_2\left(\frac{\log(p)}{\log(R)} \right)\frac{\log^2 (p)}{\sqrt{p}\log^2(R)}\left(\frac{-(3p+1)}{p(p+1)^2)}+\sum_{\ell=2}^\infty C_\ell \frac{p^{\ell-1}(p-1)}{(p+1)^{2\ell}}\right). \end{aligned}\] Notice that the substitution still has a sum over $p$ of $\widehat\phi_2$ evaluated at $\log(p)/\log(R)$. The general procedure is as follows: to move the dependence on $\widehat\phi_i$ outside of the sum, we substitute its power series expansion and apply the Prime Number Theorem to compute the sum associated to the constant term. The other sums are generally error terms, since they involve larger powers of $\log(R)$ in the denominator. Often, we use techniques involving generating functions to simplify our expressions.

In the example [eq:SumExample], we find that the expression is equal to, up to $O\left ( \log^{-4}(R) \right )$ error, \[\widehat{\phi}_1(0)\left(-\frac{4\widehat\phi_2(0)}{9\log^2(R)}+\frac{4\pi^2\widehat\phi_2(0)+\widehat\phi_2''(0)}{4\pi^2\log^2(R)}\int_1^\infty \frac{E(t)(2-5\log(t))}{t^{7/2}}dt \right).\] As mentioned above, the remaining lemmas we use to get rid of the dependency on the test function as well as to simplify evaluating the main terms can be found in Appendix 10.

The second part of the proof is to argue that the error terms of terms $A', A, B'', B,$ and $B$ obtained in Section 6 become error terms in computing $S_{B''}(\mathcal{F}_{N}), S_{B'}(\mathcal{F}_{N}), S_{B_f}(\mathcal{F}_{N}),$ and $S_{B_\infty}$ as well. Below are the two main lemmas we use to bound the errors.

Proof of Lemma 38. Note that \[\begin{aligned} \sum_{p <R^{\sigma}}\log^n(p)\sum_{m=0}^{2\log(R)} &\frac{2^m m p^{3m/4}(p-1)}{(p+1)^{m+1}} \notag\\ & \ \leq\ \sum_{p <R^{\sigma}}\log^n(p)\sum_{m=0}^{2\log(R)} m\left(\frac{2p^{3/4}}{p+1}\right)^m \notag \\ &\ \lesssim\ \sum_{p<2027} \log^n(p)\sum_{m=0}^{2\log(R)} m\left(\frac{2p^{3/4}}{p+1}\right)^m+ \sum_{p\in[2027,R^{\sigma}]}\log^n(p)\sum_{m=0}^{2\log(R)} m\left(\frac{2p^{3/4}}{p+1}\right)^m \notag \\ &\ \lesssim\ 2027\cdot \left(\frac{2\cdot3^{3/4}}{4}\right)^{2\log(R)} + \sum_{p\in[2027,R^{\sigma}]}\left(\frac{2p^{3/4}}{p+1}\right) \notag\\ &\ \lesssim\ R^{0.12} +\sum_{p\in[2027,R^{\sigma}]} p^{-1/4} \notag \\ & \ \lesssim \ R^{0.12} + R^{3\sigma/4}. \end{aligned}\] In the second to last line above, we can bound $2027\cdot \left(\frac{2\cdot3^{3/4}}{4}\right)^{2\log(R)} \lesssim R^{0.12}$ because
$\log((2\cdot3^{3/4}/{4})^2) = \log(3\sqrt{3}/4) \approx 0.1136 < 0.12$. 0◻

Proof. Observe that: \[\begin{aligned} \sum_{p} \log^n(p) \sum_{m=2\log(R)}^{\infty} &\frac{2^{m}p^{m/2}(p-1)}{(p+1)^{m+1}} \notag\\ &\ \leq\ \sum_{p} \log^n(p) \sum_{m=2\log(R)}\left(\frac{2\sqrt{p}}{p+1}\right)^m \ \lesssim\ \sum_{p}\log^n(p) \left(\frac{2\sqrt{p}}{p+1}\right)^{2\log(R)} \notag \\ &\ \lesssim\ \left(2027 \cdot \left(\frac{2\sqrt{2}}{3}\right)^{2\log(R)} +\sum_{p \geq 2027} \frac{\log^n(p)}{p^{2\log(R)/3}}\right) \ \lesssim\ {R^{-0.11}}. \end{aligned}\] Again, it is crucial that $2\sqrt{2}/3<1$. The restriction for the support ($\sigma < 0.22$) comes from the fact that $\log(({2\sqrt{2}}/{3})^2) = \log(8/9) \approx -0.1178$ and that $-2\log(({2\sqrt{2}}/{3})^2) \geq0.22$. ◻

Every error term from those of $A', A, B'', B,$ and $B$ obtained from Section 6 is in big-O notation where the implied constant is absolute in $k.$ Therefore, we can bring in all the sum into the big-O notation. Moreover, every error term has some infinite sum over $m_1$ or $m_2$ over $\mathbb{N}.$ Whenever we have this sum in $m_i$, we break the into two parts, into cases when $m_i < 2\log(R)$ and when $m_i \geq 2\log(R).$ In the first case, when $m_i < 2\log(R)$, we bound the error using the terms given from Section 6 then apply Lemma 38. On the other hand, in the case where $m_i \geq 2\log(R),$ we use the bound $|B^*_{r_1,r_2}(p_1,p_2)|\leq 2^{r_1+r_2}$ for all $* = (''), ('), (),$ $r_1,r_2 \geq 1,$ and $p_1,p_2$ primes then apply Lemma 39.

For example, when we are dealing with the term \[\widehat{\phi}_1(0)\sum_{p_1,p_2<R^\sigma}\sum_{m_1=0}^{\infty}\widehat{\phi}_2\left(\frac{\log(p_2)}{\log(R)} \right)\frac{\log(p_1) \log(p_2)}{\log^2(R)}B_{m_1,1}(p_1,p_2)\frac{P_{m_1}(p_1)}{\sqrt{p_2}}\] appearing in the computation of $S_{B_\infty}(\mathcal{F}_{N})$, the error term is \[\begin{aligned} & & \widehat{\phi}_1(0)\left(\sum_{p_1,p_2<R^\sigma}\sum_{m_1=0}^{2\log(R)}\widehat{\phi}_2\left(\frac{\log(p_2)}{\log(R)} \right)\frac{\log(p_1) \log(p_2)}{\log^2(R)}\text{Error}(B_{m_1,1}(p_1,p_2))\frac{P_{m_1}(p_1)}{\sqrt{p_2}} \right. \nonumber\\ & & \left. \ \ \ \ \ + \ \sum_{p_1,p_2<R^\sigma}\sum_{m_1=2\log(R)}^{\infty}(\text{Same Terms})\right). \end{aligned}\] For the first sum, we use bounds from Section 6 then apply Lemma 38: \[\begin{aligned} &\widehat{\phi}_1(0)\sum_{p_1,p_2}\sum_{m_1=0}^{2\log(R)}\widehat{\phi}_2\left(\frac{\log(p_2)}{\log(R)} \right)\frac{\log(p_1) \log(p_2)}{\log^2(R)}\text{Error}(B_{m_1,1}(p_1,p_2))\frac{P_{m_1}(p_1)}{\sqrt{p_2}}\notag \\ &\lesssim \frac{R^{\sigma/2}}{\log(R)}\left(\sum_{p_1}^{R^\sigma}\log(p_1)\sum_{m_1=0}^{2\log(R)} \frac{m_12^{m_1}p_1^{3m_1/4} \log(p_1N)}{N(p_1+1)^{m_1}}\right) \notag \\ &\lesssim \frac{R^{\sigma/2}\log(R)}{N}\left(\sum_{p_1<2027}\log^2(p_1)\sum_{m_1=0}^{2\log(R)} m_1\left(\frac{2p_1^{3/4}}{p_1+1}\right)^{m_1} + \sum_{p_1=2027}^{R^{\sigma}}\log^2(p_1)\sum_{m_1=0}^{2\log(R)} m_1\left(\frac{2p_1^{3/4}}{p_1+1}\right)^{m_1}\right) \notag \\ &\lesssim\frac{R^{\sigma/2}\log(R)}{N} \left(2027\left(\frac{2\cdot3^{3/4}}{4}\right)^{2\log(R)}\log^2(R) + \sum_{p=2027}^{R^\sigma} \frac{2p^{3/4}}{p_1+1}\right) \notag \\&\lesssim \frac{R^{\sigma/2}\log^2(R)\cdot R^{0.11}}{N} + \frac{R^{\sigma/2}}{N}\log(R) \sum_{p=2027}^{R^\sigma} p_1^{-1/4} \lesssim \frac{R^{\sigma/2}\log^2(N)R^{0.11}}{N} + \frac{R^{\sigma/2}\log(R)}{N}N^{3\sigma/4}. \end{aligned}\] This error is negligible as long as $\sigma < 0.8,$ which we assumed.

For the second sum, we bound $|B_{m_1,1}(p_1,p_2)|\lesssim 2^{m_1}$ then apply Lemma 39: \[\begin{aligned} &\widehat{\phi}_1(0)\sum_{p_1,p_2}\sum_{m_1=0}^{2\log(R)}\widehat{\phi}_2\left(\frac{\log(p_2)}{\log(R)} \right)\frac{\log(p_1) \log(p_2)}{\log^2(R)}\text{Error}(B_{m_1,1}(p_1,p_2))\frac{P_{m_1}(p_1)}{\sqrt{p_2}} \notag \\ &\lesssim\ \left(\frac{R^{\sigma/2}}{\log(R)}\right)\left(\sum_{p_1}\log(p)\sum_{m_1=2\log(R)}^\infty 2^{m_1}\frac{p_1^{m_1/2}}{(p_1+1)^{m_1}}\right) \notag \\ &\lesssim\ \left(\frac{R^{\sigma/2}}{\log(R)}\right)\left(\sum_{p_1}\log(p)\sum_{m_1=2\log(R)}^\infty \left(\frac{2p_1^{1/2}}{p_1+1}\right)^{m_1}\right) \notag \\ &\lesssim\ \left(\frac{R^{\sigma/2}}{\log(R)}\right)\sum_{p_1}\log(p)\left(\frac{2p_1^{1/2}}{p_1+1}\right)^{2\log(R)} \notag \\ &\lesssim\ \left(\frac{R^{\sigma/2}}{\log(R)}\right)\left(2027\left(\frac{2p_1^{1/2}}{p_1+1}\right)^{2\log(R)} + \sum_{p_1\geq 2026} \frac{\log(p_1)}{p_1^{2\log(R)/3}}\right) \ \lesssim \ \frac{R^{\sigma/2}}{R^{0.11}\log^2(R)}, \end{aligned}\] which is negligible as long as $\sigma < 0.22,$ which we assume.

Using Theorems 34 and 34 as a base case, we compute $S_{B''}$ and $S_{B'}$ in full generality.

Proof. Recall that $B''(p_1,p_2)$ is only well-defined for $p_1,p_2\mid N$ and $p_2\nmid N$. If $p_1^2\mid N$ then $B'(p_1,p_2)=0$. Let $N=q_1^{a_1}\cdots q_n^{a_n}$ where $q_i\asymp N^{\delta_i}$. Let $Q=\{q_i \mid a_i=1, \delta_i=0\}$. Note that $B' _{r_1,r_2}(p_1,p_2)$ will only have a main term if $p_1,p_2\in Q$. Thus, the main term of $S_{B'}$ is given by \[\begin{aligned} \text{Main}(S_{B''}(\mathcal{F}_{N}) \ =\ &\sum_{p_1,p_2\in Q}\sum_{m_1,m_2=1}^{\infty} MB''_{m_1,m_2}(p_1,p_2) \frac{\log(p_1)\log(p_2)}{p_1^{m_1/2}p_2^{m_2/2}\log^2(R)} \notag \\ &\cdot \widehat\phi_1 \left(m_1 \frac{\log(p_1)}{\log(R)}\right)\widehat\phi_2 \left(m_2 \frac{\log(p_2)}{\log(R)}\right). \end{aligned}\] Let us write this as \[\begin{aligned} \text{Main}(S_{B''}(\mathcal{F}_{N}) \ =\ \sum_{p_1\in Q}F''(p_1,p_2)+\sum_{\substack{p_1,p_2\in Q\\p_1\neq p_2}}F''(p_1,p_2). \end{aligned}\] For a fixed $p_1$, we note that \[F(p_1,p_2) \ = \ \frac{\log(p_1)\log(p_2)}{\log^2(R)} \widehat\phi_1(0)\widehat\phi_2(0) \frac{1}{(p_1^2-1)^2},\] and for fixed $p_1,p_2$ we obtain \[F(p_1,p_2) \ = \ \frac{\log(p_1)\log(p_2)}{\log^2(R)} \widehat\phi_1(0)\widehat\phi_2(0) \frac{1}{(p_1^2-1)(p_2-1)},\] so there is actually no difference between these two terms. Putting everything together yields our desired result. ◻

Proof. Recall that $B'(p_1,p_2)$ is only well-defined for $p_1\mid N$ and $p_2\nmid N$. If $p_1^2\mid N$ then $B'(p_1,p_2)=0$. Let $N=q_1^{a_1}\cdots q_n^{a_n}$ where $q_i\asymp N^{\delta_i}$. Let $Q=\{q_i \mid a_i=1, \delta_i=0\}$ and let $R=\{q_i\mid a_i=1\}$. Note that $B' _{r_1,r_2}(p_1,p_2)$ will only have a main term if $p_1\in Q$. Thus, the main term of $S_{B'}$ is given by \[\begin{aligned} \text{Main}(S_{B'}&(\mathcal{F}_{N})) \notag\\ \ = & \ \sum_{p_1\in Q}\sum_{p_2\not\in R}\sum_{m_1=1}^{\infty} M'_{m_1,1}(p_1,p_2) \frac{\log(p_1)\log(p_2)}{p_1^{m_1/2}\sqrt{p_2}\log^2(R)} \left(\sum_{(i,j) \in A}\widehat\phi_i\left(\frac{m_1\log(p_1)}{\log(R)}\right)\widehat\phi_j\left(\frac{\log(p_2)}{\log(R)}\right)\right)\notag\\ &+\sum_{p_1\in Q}\sum_{p_2\not\in R}\sum_{m_1=1}^{\infty} (M'_{m_1,2}(p_1,p_2)-2M'_{m_1,0}(p_1,p_2)) \frac{\log(p_1)\log(p_2)}{p_1^{m_1/2}p_2\log^2(R)}\notag \\ &\left(\sum_{(i,j) \in A}\widehat\phi_i\left(m_1\frac{\log(p_1)}{\log(R)}\right)\widehat\phi_j\left(2\frac{\log(p_2)}{\log(R)}\right)\right) \notag\\ &+\sum_{p_1\in Q}\sum_{p_2\not\in R}\sum_{m_1=1,m_2=0}^{\infty}M'_{m_1,m_2}(p_1,p_2)\frac{P_{m_2}(p_2)}{p_1^{m_1/2}}\frac{\log(p_1)\log (p_2)}{\log^2(R)}\left(\sum_{(i,j) \in A}\widehat\phi_i\left(\frac{m_1\log(p_1)}{\log(R)}\right)\widehat\phi_j\left(0\right)\right) \end{aligned}\] where $M'_{r_1,r_2}(p_1,p_2)$ denotes the main term of $B'_{r_1,r_2}(p_1,p_2)$. We have \[M'_{r_1,r_2}(p_1,p_2) \ = \ \begin{cases} p_1^{-r_1/2}C_{r_2/2} & r_1\equiv r_2\equiv0\\ 0 & \text{ otherwise} \end{cases}\] From Theorem 7.4, we have for $Q=\{q_1\}$ and $R=\{q_1,q_2\}$, we have \[\begin{aligned} \text{Main}(S_{B'}(\mathcal{F}_{N}) \ =& \ \frac{2\log(q_1)\widehat\phi_1(0)\widehat\phi_2(0)}{\log^2(R)(q_1^2-1)}\left(\frac{\log(q_1)}{q_1}-\frac{3\log(q_1)q_1}{(q_1+1)^3} + \gamma_{B',1}\right)\notag\\ &-\frac{\log(q_1)\left(\widehat\phi_1(0)\phi_2(0) + \phi_1(0)\widehat\phi_2(0)\right)}{4\log^2(R)(q_1^2-1)} -\frac{2\log(q_1)\widehat\phi_1(0)\widehat\phi_2(0)}{\log^3(R)(q_1^2-1)}\gamma_{PNT3}. \end{aligned}\] Note that $\text{Main}(S_{B'})(\mathcal{F}_{N})$ is of the form \[\sum_{p_1\in Q}\sum_{p_2\not\in R}F'(p_1,p_2)\] For a fixed $p_1\in Q$, let us compute a term of the first sum. We can write this term as \[\sum_{p}F'(p_1,p)-\sum_{p_2\in R}F'(p_1,p_2).\] and compute these parts individually. First, for some $p_2\in R$, let us find $F'(p_1,p_2)$. Note that $\text{Main}(B'_{m_1,1}(p_1,p_2)=0$ since $1$ is odd. Then, noting that $C_0=C_1=1$, we obtain that $\text{Main}(B'_{m_1,2}-2B'_{m_1,0})=-p_1^{m_1/2}$. Multiplying by the $p_1^{m_1/2}p_2$ in the denominator Summing over even $m_1$ yields $-\frac{1}{p_1^2-1}\frac{1}{p_2}$ for the second line. For the third line, we still get a factor of $-\frac{1}{p_1^2-1}$. We also sum \[\sum_{k=0}^\infty C_kP_{2k}(p_2)=\frac{3p_2}{(p_2+1)^3}.\] Thus, we have \[F'(p_1,p_2) \ = \ \frac{2\log(p_1)\log(p_2)\widehat\phi_1(0)\widehat\phi_2(0)}{\log^2(R)(p_1^2-1)} \left( -\frac{1}{p_2} + \frac{3p_2}{(p_2+1)^3} \right)\] Note that when $p_2\not\in Q$, this term will be negligible. Thus, we actually only need to find \[\sum_{p_1\in Q}\sum_{p}F'(p_1,p)-\sum_{p_2\in Q}F'(p_1,p_2).\] To find the first term, \[T_0(p_1) \ := \ \sum_{p}F'(p_1,p),\] we simply add $F'(p_1,p_2)$ back to the base case for $Q=\{q_1\}$ and $R=\{q_1,q_2\}$. This gives \[\begin{aligned} T_0(p_1) \ &= \ \frac{2\log(p_1)\widehat\phi_1(0)\widehat\phi_2(0)}{\log^2(R)(p_1^2-1)}\gamma_{B',1}\notag\\ &-\frac{\log(p_1)\left(\widehat\phi_1(0)\phi_2(0) + \phi_1(0)\widehat\phi_2(0)\right)}{4\log^2(R)(p_1^2-1)} -\frac{2\log(p_1)\widehat\phi_1(0)\widehat\phi_2(0)}{\log^3(R)(p_1^2-1)}\gamma_{PNT3}. \end{aligned}\] One can also verify that this result is correct by applying the Prime Number Theorem and explicitly computing this expression. Then, putting all of this together yields \[\begin{aligned} \text{Main}(S_{B'}(\mathcal{F}_{N}) &= \sum_{p_1 \in Q} \Bigg( \frac{2\log(p_1)\widehat\phi_1(0)\widehat\phi_2(0)}{\log^2(R)(p_1^2-1)}\left( \gamma_{B',1} + \sum_{p_2 \in Q} \left( \frac{\log(p_2)}{p_2} - \frac{3\log(p_2)p_2}{(p_2+1)^3} \right) \right) \notag\\ &\quad - \frac{\log(p_1)\left(\widehat\phi_1(0)\phi_2(0) + \phi_1(0)\widehat\phi_2(0)\right)}{4\log^2(R)(p_1^2-1)} - \frac{2\log(p_1)\widehat\phi_1(0)\widehat\phi_2(0)}{\log^3(R)(p_1^2-1)}\gamma_{PNT3} \Bigg). \end{aligned}\] The computation for the error term is the same and we are still guaranteed that it is $O\left ( \log^{-4}(R) \right )$. ◻

7.1 Breaking Universality

Recall from [eq:D1explicitformula] that $D_1(\mathcal{F},\phi) = \lim_{N \to \infty}D_1(\mathcal{F}_N,\phi)$, and \[\begin{aligned} D_1(\mathcal{F}_N,\phi) = \frac{U_{k,N}(\phi)}{\log(R)} + S_1(\mathcal{F}_N,\phi). \end{aligned}\]

Proof. Since $R = (64\pi^2)^{-1}N(k+1)(k+3)$ and $k$ is fixed, we have exactly \[\begin{aligned} \label{eq:logN-logR} \log(N)\ =\ \log(R)-\log\left ( \frac{(k+1)(k+3)}{64\pi^2} \right ) , \end{aligned}\] so Theorem 12 gives \[\begin{aligned} \label{eq:U-over-logR} \frac{U_{k,N}(\phi)}{\log(R)}\ =\ \widehat\phi(0)+\frac{\gamma_{k,1}\widehat\phi(0)}{\log(R)}-\frac{2\pi^2\gamma_{k,2}}{\log^3(R)}\int_{-\infty}^\infty \phi(x)x^2dx+O\left ( \log^{-5}(R) \right ) , \end{aligned}\] where \[\begin{aligned} \label{eq:gamma-k-def} \gamma_{k,1}\ &:=\ \psi\left ( \frac{k}{4} \right ) +\psi\left ( \frac{k}{4}+\frac 12 \right ) -2\log(\pi)-\log\left ( \frac{(k+1)(k+3)}{64\pi^2} \right ) ,\\ \gamma_{k,2}\ &:=\ \psi''\left ( \frac{k}{4} \right ) +\psi''\left ( \frac{k}{4}+\frac 12 \right ) . \end{aligned}\] By Theorem 15 we have $S_1(\mathcal{F}_{N},\phi) = S_{A'}(\mathcal{F}_{N})+S_A(\mathcal{F}_{N})+O\left ( \log^{-4}(R) \right )$, and substituting Theorem 33 for $S_A(\mathcal{F}_{N})$, \[\begin{aligned} \label{eq:S1-expanded} S_1(\mathcal{F}_{N},\phi)\ =\ S_{A'}(\mathcal{F}_{N})&+\frac{\phi(0)}{2}+\frac{\widehat\phi(0)}{\log(R)}\left ( 2\gamma_{PNT3}-\gamma_{A,2}+\gamma_{A,3}+\gamma_{A,6} \right ) \notag\\ &+\frac{\widehat\phi''(0)}{4\pi^2\log^3(R)}\left ( 4\gamma_{A,1}+\gamma_{A,4}+\gamma_{A,5}+\gamma_{A,7} \right ) +O\left ( \log^{-4}(R) \right ) . \end{aligned}\] Adding [eq:U-over-logR] and [eq:S1-expanded] and applying Theorem 32 to $S_{A'}(\mathcal{F}_{N})$, we obtain \[\begin{aligned} \label{eq:D1-expanded} D_1(\mathcal{F}_{N},\phi)\ =\ &\widehat\phi(0)+\frac{\phi(0)}{2} +\frac{\widehat\phi(0)}{\log(R)}\left ( \gamma_{k,1}+2\gamma_{PNT3}-\gamma_{A,2}+\gamma_{A,3}+\gamma_{A,6} \right ) \notag\\ &+\frac{1}{\log^3(R)}\left ( \frac{\widehat\phi''(0)}{4\pi^2}\left ( 4\gamma_{A,1}+\gamma_{A,4}+\gamma_{A,5}+\gamma_{A,7} \right ) -2\pi^2\gamma_{k,2}\int_{-\infty}^\infty \phi(x)x^2dx \right ) \notag\\ &-\frac{2\widehat\phi(0)\beta_{Q,1}}{\log(R)}-\frac{4\widehat\phi''(0)\beta_{Q,3}}{\log^3(R)}+O\left ( \log^{-4}(R) \right ) . \end{aligned}\]

The constants $\gamma_{k,1}, \gamma_{k,2}$ of [eq:gamma-k-def] are determined by the weight $k$, which is fixed throughout. The constants $\gamma_{PNT3}$ and $\gamma_{A,1},\dots,\gamma_{A,7}$ are also absolute. Hence the first two lines of [eq:D1-expanded] are unchanged if the sequence of levels is replaced by any other. The last line is not: by [eq:gammaQ-def] the constants $\beta_{Q,1}$ and $\beta_{Q,3}$ are sums over $Q$, and by Theorem 32 they are $0$ when $Q = \emptyset$ and positive when $Q \neq \emptyset$.

Let $R$ and $\widetilde R$ denote the conductors of the two families in the theorem statement. By [eq:logN-logR] the hypothesis $N \asymp \widetilde N$ gives $\log(R)-\log(\widetilde R) = O(1)$, so $\log(R)/\log(\widetilde R) \to 1$. Subtracting the two instances of [eq:D1-expanded], the terms in the first two lines occur in both with the same constants, and for such a constant $c$, \[\begin{aligned} \label{eq:conductor-mismatch} \left\lvert \frac{c}{\log(R)}-\frac{c}{\log(\widetilde R)} \right\rvert\ =\ \left\lvert c \right\rvert\frac{\left\lvert \log(R)-\log(\widetilde R) \right\rvert}{\log(R)\log(\widetilde R)}\ \lesssim\ \frac{1}{\log^2(R)}, \end{aligned}\] while the terms of order $\log^{-3}(R)$ differ by $O\left ( \log^{-4}(R) \right )$. Only the last line survives, giving \[\begin{aligned} D_1(\mathcal{F}_{N},\phi)-D_1(\widetilde\mathcal{F}_{\widetilde N},\phi)\ =\ -\frac{2\widehat\phi(0)\left ( \beta_{Q,1}-\beta_{\widetilde Q,1} \right ) }{\log(R)}+O\left ( \log^{-2}(R) \right ) , \end{aligned}\] and multiplying by $\log(R)$ yields [eq:D1-comparison]. If exactly one of $Q,\widetilde Q$ is empty then $\beta_{Q,1} \neq \beta_{\widetilde Q,1}$, since one of the two is $0$ and the other is positive, and $\widehat\phi(0) \neq 0$ by hypothesis. ◻

We move on to the second level. Recall from [eq:D2explicitformula] that $D_2(\mathcal{F},\phi_1,\phi_2) = \lim_{N \to \infty}D_2(\mathcal{F}_N,\phi_1,\phi_2)$, and \[\begin{aligned} \label{eq:finite-level-D2} D_2(\mathcal{F}_{N},\phi_1,\phi_2)\ =\ &\frac{U_{k,N}(\phi_1)U_{k,N}(\phi_2)}{\log^2(R)} +\frac{U_{k,N}(\phi_{2})}{\log(R)}S_1(\mathcal{F}_{N},\phi_1)+\frac{U_{k,N}(\phi_{1})}{\log(R)}S_1(\mathcal{F}_{N},\phi_2)\notag\\ &+S_2(\mathcal{F}_{N},\phi_1,\phi_2)-2\left ( \frac{U_{k,N}(\phi_1\phi_2)}{\log(R)}+S_1(\mathcal{F}_{N},\phi_1\phi_2) \right ) \notag\\ &+\frac{\phi_1(0)\phi_2(0)}{2}\left ( 1-\mathcal{W}(\mathcal{F}_{N}) \right ) , \qquad \mathcal{W}(\mathcal{F}_{N})\ :=\ \frac{1}{W_R(\mathcal{F}_{N})}\sum_{f \in \mathcal{F}_{N}}w_R(f)\varepsilon_f . \end{aligned}\]

Proof. For squarefree $N$, every $f \in H^*_k(N)$ has $\varepsilon_f = i^k\mu(N)\sqrt{N}\lambda_f(N)$ [ILS], so by [def:Deltastar], \[\begin{aligned} \mathcal{W}(\mathcal{F}_{N})\ =\ i^k\mu(N)\sqrt{N}\cdot\frac{\Delta^*_{k,N}(1,N)}{W_R(\mathcal{F}_{N})}. \end{aligned}\] Since $N$ is squarefree, we have $N_1 = N$, so [eq:normalized-general-trace-bound] applies with $D = N$ and $a = b = 1$ and gives $\Delta^*_{k,N}(1,N)/W_R(\mathcal{F}_{N}) \ll_{k,\varepsilon} N^{-3/4+\varepsilon}$. Multiplying by $\sqrt N$ gives the claim. ◻

Proof. As $\widehat\phi_1,\widehat\phi_2$ are supported in $[-\sigma,\sigma]$ with $\sigma<0.11$, the function $\phi_1\phi_2$ is even and Schwartz with $\widehat{\phi_1\phi_2} = \widehat\phi_1 * \widehat\phi_2$ supported in $[-2\sigma,2\sigma]$ and $2\sigma<0.22$. Theorems 15 and 33 therefore apply to each of $\phi_1$, $\phi_2$ and $\phi_1\phi_2$, and we may treat the three occurrences of $S_1$ in [eq:finite-level-D2] alike. By Theorem 15, $S_1(\mathcal{F}_{N},\phi) = S_{A'}(\mathcal{F}_{N})+S_A(\mathcal{F}_{N})+O\left ( \log^{-4}(R) \right )$, where $S_A(\mathcal{F}_{N})$ is given by [eq:SA_asymp]. Theorem 32 gives, for each of $\phi = \phi_1,\phi_2,$ and $\phi_1\phi_2$, \[\begin{aligned} \label{eq:SA'-two-level} S_{A'}(\mathcal{F}_{N})\ =\ -\frac{2\widehat\phi(0)\beta_{Q,1}}{\log(R)}+O\left ( \log^{-3}(R) \right ) . \end{aligned}\] By Theorem 17, $S_2(\mathcal{F}_{N},\phi_1,\phi_2) = S_{B''}(\mathcal{F}_{N})+S_{B'}(\mathcal{F}_{N})+S_{B_f}(\mathcal{F}_{N})+S_{B_\infty}(\mathcal{F}_{N})+O\left ( \log^{-4}(R) \right )$. Recall that $S_{B''}(\mathcal{F}_{N})$ and $S_{B'}(\mathcal{F}_{N})$ are $O\left ( \log^{-2}(R) \right )$ by Theorems 40 and 41 respectively, regardless of the factorization of $N$. Moreover, $S_{B_f}(\mathcal{F}_{N})$ and $S_{B_\infty}(\mathcal{F}_{N})$ depend on $N$ only through $R$ and not its factorization by Theorems 36 and 37. Finally $\mathcal{W}(\mathcal{F}_{N})$ is negligible by Lemma 44.

Hence every term of [eq:finite-level-D2] depends on $N$ only through $R$, apart from an error of $O\left ( \log^{-2}(R) \right )$ and the three contributions of [eq:SA'-two-level]. Since $U_{k,N}(\phi_j)/\log(R) = \widehat\phi_j(0)+O\left ( \log^{-1}(R) \right )$, those three contributions are \[\begin{aligned} -\frac{2\widehat\phi_1(0)\widehat\phi_2(0)\beta_{Q,1}}{\log(R)},\qquad -\frac{2\widehat\phi_1(0)\widehat\phi_2(0)\beta_{Q,1}}{\log(R)},\qquad \frac{4\widehat{\phi_1\phi_2}(0)\beta_{Q,1}}{\log(R)}, \end{aligned}\] each up to $O\left ( \log^{-2}(R) \right )$, the last carrying the factor $-2$ of the inclusion–exclusion correction. As $\widehat\phi_i(0) = \int_{-\infty}^\infty\phi_i(x)dx$ and $\widehat{\phi_1\phi_2}(0) = \int_{-\infty}^\infty\phi_1(x)\phi_2(x)dx$, their sum is $-4\beta_{Q,1}\Xi(\phi_1,\phi_2)/\log(R)$, and we obtain \[\begin{aligned} \label{eq:D2-expanded} D_2(\mathcal{F}_{N},\phi_1,\phi_2)\ =\ G(R)-\frac{4\beta_{Q,1}\Xi(\phi_1,\phi_2)}{\log(R)}+O\left ( \log^{-2}(R) \right ) , \end{aligned}\] where $G$ is the same function of $k$, $\phi_1$, $\phi_2$ and $R$ for both families. Moreover $G(R) = g_0+g_1\log^{-1}(R)+O\left ( \log^{-2}(R) \right )$ for constants $g_0,g_1$ depending only on $k$ and the test functions, by [eq:U-over-logR], [eq:SA_asymp] and Theorems 36 and 37.

Let $R$ and $\widetilde R$ denote the conductors of the two families. Subtracting the two instances of [eq:D2-expanded], the constant $g_0$ cancels identically, and $g_1\log^{-1}(R)$ and $g_1\log^{-1}(\widetilde R)$ differ by $O\left ( \log^{-2}(R) \right )$ by [eq:conductor-mismatch]. Only the remaining term survives, giving \[\begin{aligned} D_2(\mathcal{F}_{N},\phi_1,\phi_2)-D_2(\widetilde\mathcal{F}_{\widetilde N},\phi_1,\phi_2)\ =\ -\frac{4\Xi(\phi_1,\phi_2)\left ( \beta_{Q,1}-\beta_{\widetilde Q,1} \right ) }{\log(R)}+O\left ( \log^{-2}(R) \right ) , \end{aligned}\] and multiplying by $\log(R)$ yields [eq:D2-comparison]. If exactly one of $Q,\widetilde Q$ is empty then $\beta_{Q,1} \neq \beta_{\widetilde Q,1}$, since one of the two is $0$ and the other is positive, and $\Xi(\phi_1,\phi_2) \neq 0$ by hypothesis. ◻

8 Proof of Lemma 16

Lemma 16. For prime $p$, we have \[\begin{aligned} M_{3,2}(p)\ =& \ \frac{32p^2+24p+8}{p(p+1)^3} - \frac{27p^3-17p^2+5p+1}{\sqrt{p}(p+1)^4}\lambda_f(p)-\frac{64p^4-4p^3+44p^2+20p+4}{p(p+1)^5} \lambda_f(p)^2 \notag\\&+ \sum_{r=3}^{\infty}\frac{(p-1)(r^2(p-1)^2-12rp-8p)p^{r/2}\lambda_f(p)^r}{(p+1)^{r+3}}. \end{aligned}\]

Proof We recall $M_{3,2}(p)\ =\ \sum_{m=3}^{\infty}m^2\left(\frac{\alpha_f(p)}{p^{1/2}}\right)^m+\sum_{m=3}^{\infty}m^2\left(\frac{\beta_f(p)}{p^{1/2}}\right)^m$. We note that since \[\begin{aligned} \frac{1}{1-x}\ &=\ \sum_{n=0}^{\infty}x^n, \notag \\ \frac{x}{(1-x)^2}\ &=\ \sum_{n=1}^{\infty}nx^n, \text{ and }\notag \\ 2\frac{x^2}{(1-x)^3}\ &=\ \sum_{n=2}^{\infty }n(n-1)x^{n}\ =\ \sum_{n=2}^{\infty }n^2x^{n}-\sum_{n=2}^{\infty }nx^{n}, \end{aligned}\] we have \[\begin{aligned} \sum_{n=2}^{\infty}n^2x^n\ &=\ 2\frac{x^2}{(1-x)^3}+\frac{x}{(1-x)^2}-x =\frac{x^2+x}{(1-x)^3}-x. \end{aligned}\] Thus, \[\begin{aligned} \sum_{m=3}^{\infty}m^2\left(\frac{\alpha_f(p)}{p^{1/2}}\right)^m+\sum_{m=3}^{\infty}m^2\left(\frac{\beta_f(p)}{p^{1/2}}\right)^m \ = \ &\frac{\left(\frac{\alpha_f(p)}{p^{1/2}}\right)^2+\frac{\alpha_f(p)}{p^{1/2}}}{\left(1- \frac{\alpha_f(p)}{p^{1/2}}\right)^3}+\frac{\left(\frac{\beta_f(p)}{p^{1/2}}\right)^2+\frac{\beta_f(p)}{p^{1/2}}}{\left(1- \frac{\beta_f(p)}{p^{1/2}}\right)^3} \notag \\ &-4\left(\left(\frac{\alpha_f(p)}{p^{1/2}}\right)^2+\left(\frac{\beta_f(p)}{p^{1/2}}\right)^2\right)-\left(\frac{\alpha_f(p)}{p^{1/2}}+\frac{\beta_f(p)}{p^{1/2}}\right) . \end{aligned}\] Multiplying by $p^{3/2}$ gives \[\frac{\left(\frac{\alpha_f(p)}{p^{1/2}}\right)^2+\frac{\alpha_f(p)}{p^{1/2}}}{\left(1- \frac{\alpha_f(p)}{p^{1/2}}\right)^3}\ =\ \frac{\alpha_f(p)^2p^{1/2}+\alpha_f(p)p}{(p^{1/2}-\alpha_f(p))^3}\] and we obtain a similar result for the corresponding $\beta_f(p)$ term. We aim to combine \[\frac{\alpha_f(p)^2p^{1/2}+\alpha_f(p)p}{(p^{1/2}-\alpha_f(p))^3}+\frac{\beta_f(p)^2p^{1/2}+\beta_f(p)p}{(p^{1/2}-\beta_f(p))^3}.\] The denominator becomes \[\begin{aligned} (p^{1/2}-\alpha_f(p))(p^{1/2}-\beta_f(p))\ &= \ p+\alpha_f(p)\beta_f(p)-(\alpha_f(p)+\beta_f(p))p^{1/2} \notag \\ \ &= \ p+1-\lambda_f(p)p^{1/2}. \end{aligned}\] We note we are leaving the aforementioned expression inside the cube. Thus we have \[\begin{aligned} \label{eq:S1 second} \frac{\lambda_f(p)^2(p^2-p)+\lambda_f(p)(p^2\sqrt{p}-\sqrt{p})-8(p^2-p)}{(p+1-\lambda_f(p)\sqrt{p})^3}-\frac{4(\lambda_f(p)^2-2)}{p}-\frac{\lambda_f(p)}{p^{1/2}}. \end{aligned}\] Looking at the first term, we express the denominator $(p+1-\lambda_f(p)\sqrt{p})^{3}$ as $(p+1)^{3}\left(1-\frac{\lambda_f(p)\sqrt{p}}{p+1} \right)^{3}$ and use the power series $\frac{1}{(1-x)^3}\ = \ \frac{1}{2}\sum_{n=2}^{\infty}n(n-1)x^{n-2}$ to obtain \[\frac{1}{\left(1-\frac{\lambda_f(p)p^{1/2}}{p+1}\right)^3}\ =\ \frac{1}{2}\sum_{r=0}^{\infty}(r+2)(r+1)\left(\frac{\lambda_f(p)p^{1/2}}{p+1} \right)^{r}.\] Thus we obtain \[\left(\lambda_f(p)^2(p^2-p)+\lambda_f(p)(p^2\sqrt{p}-\sqrt{p})-8(p^2-p)\right)\sum_{r=0}^{\infty}\frac{(r+2)(r+1)}{2}\frac{\lambda_f(p)^rp^{r/2}}{(p+1)^{r+3}},\] which is equal to \[\begin{aligned} &-\frac{8(p^2-p)}{(p+1)^3}-\frac{24(p^2-p)p^{1/2}}{(p+1)^4}\lambda_f(p)-\frac{48(p^2-p)p}{(p+1)^5}\lambda_f(p)^2-\sum_{r=3}^{\infty}\frac{(r+2)(r+1)}{2}\frac{8(p^2-p)\lambda_f(p)^rp^{r/2}}{(p+1)^{r+3}} \notag \\ &+\frac{p^{5/2}-p^{1/2}}{(p+1)^3}\lambda_f(p)+3\frac{(p^{5/2}-p^{1/2})p^{1/2}}{(p+1)^4}\lambda_f(p)^2+\sum_{r=2}^{\infty}\frac{(r+2)(r+1)}{2}\frac{(p^{5/2}-p^{1/2})\lambda_f(p)^{r+1}p^{r/2}}{(p+1)^{r+3}} \notag \\ &+\frac{p^2-p}{(p+1)^3}\lambda_f(p)^2+\sum_{r=1}^{\infty}\frac{(r+2)(r+1)}{2}\frac{(p^2-p)\lambda_f(p)^{r+2}p^{r/2}}{(p+1)^{r+3}}. \end{aligned}\] Combining like terms yields \[\begin{aligned} &-\frac{8(p^2-p)}{(p+1)^3} + \frac{\sqrt{p}(p-1)(p^2-22p+1)}{(p+1)^4}\lambda_f(p)+\frac{4p(p-1)(p^2-10p+1)}{(p+1)^5}\lambda_f(p)^2 \\ &+\sum_{r=3}^{\infty}\frac{(p-1)(r^2(p-1)^2-12rp-8p)p^{r/2}\lambda_f(p)^r}{(p+1)^{r+3}} . \end{aligned}\] Plugging this back into equation [eq:S1 second] we obtain \[\begin{aligned} \label{eq:S1 third} \notag M_{3,2}(p) \ =& \ \frac{-8(p^2-p)}{(p+1)^3} + \frac{\sqrt{p}(p-1)(p^2-22p+1)\lambda_f(p)}{(p+1)^4}+\frac{4p(p-1)(p^2-10p+1)\lambda_f(p)^2}{(p+1)^5} \notag \\&-\frac{4(\lambda_f(p)^2-2)}{p}-\frac{\lambda_f(p)}{\sqrt{p}}+ \sum_{r=3}^{\infty}\frac{(p-1)(r^2(p-1)^2-12rp-8p)p^{r/2}\lambda_f(p)^r}{(p+1)^{r+3}} \notag\\ =&\ \frac{32p^2+24p+8}{p(p+1)^3}-\frac{27p^3-17p^2+5p+1}{(p+1)^4}\lambda_f(p) -\frac{64p^4-4p^3+44p^2+20p+4}{p(p+1)^5}\lambda_f(p)^2 \notag\\ &+\sum_{r=3}^{\infty}\frac{(p-1)(r^2(p-1)^2-12rp-8p)p^{r/2}\lambda_f(p)^r}{(p+1)^{r+3}}. \end{aligned}\]0◻

9 Proof of Lemma 19

Lemma 19 Suppose $\phi_1$ and $\phi_2$ are even Schwartz function with $\widehat\phi_1$ and $\widehat\phi_2$ having support in $[-\sigma,\sigma].$ We then have the following estimate: \[\begin{aligned} \label{lem:phi0.estimate.statement2} &\frac{1}{W_R(\mathcal{F}_{N})}\sum_{\substack{p_1,p_2}}\sum_{m_1,m_2\geq 3} C(m_1,m_2)\notag \\ & \ = \ \frac{1}{W_R(\mathcal{F}_{N})}\sum_{p_1,p_2<R^\sigma}\sum_{m_1,m_2\geq 3}\sum_{\substack{f\in \mathcal{F}_N\\p_1\nmid N\\p_2\nmid N }}w_R(f)\frac{\sum_{j=1}^2\left(\alpha_f(p_j)^{m_j}+\beta_f(p_j)^{m_j}\right)}{p_1^{m_1/2}p_2^{m_2/2}} \frac{\log(p_1)\log (p_2)}{\log^2(R)}\widehat\phi_1(0) \widehat\phi_2(0)\notag\\ & + O\left ( \log^{-4}(R) \right ) . \end{aligned}\]

Proof. Let us write \[\begin{aligned} &(A) \ :=\ \frac{1}{W_R(\mathcal{F}_{N})}\sum_{p_1,p_2}\sum_{m_1,m_2\geq 3} C(m_1,m_2),\\ & (B)\ :=\ \frac{1}{W_R(\mathcal{F}_{N})}\sum_{p_1,p_2<R^\sigma}\sum_{m_1,m_2 \geq 3}\sum_{\substack{f\in \mathcal{F}_N\\p_1\nmid N\\p_2\nmid N }}w_R(f)\frac{\sum_{j=1}^2\left(\alpha_f(p_j)^{m_j}+\beta_f(p_j)^{m_j}\right)}{p_1^{m_1/2}p_2^{m_2/2}} \frac{\log(p_1)\log (p_2)}{\log^2(R)}\widehat\phi_1(0) \widehat\phi_2(0), \\ &(C)\ := \ \frac{1}{W_R(\mathcal{F}_{N})}\sum_{p_1,p_2<R^\sigma}\sum_{m_1,m_2 \geq 3 }\notag\\ &\qquad\sum_{\substack{f\in \mathcal{F}_N\\p_1\nmid N\\p_2\nmid N }}w_R(f)\frac{\sum_{j=1}^2\left(\alpha_f(p_j)^{m_j}+\beta_f(p_j)^{m_j}\right)}{p_1^{m_1/2}p_2^{m_2/2}} \frac{\log(p_1)\log (p_2)}{\log^2(R)} \widehat\phi_1 \left(m_1 \frac{\log(p_1)}{\log(R)}\right) \widehat\phi_2(0). \end{aligned}\] Notice that the statement of the lemma is equivalent to saying that $(A)-(B) = O\left ( \log^{-4}(R) \right )$. We first show that $(A)- (C) =O\left ( \log^{-4}(R) \right )$, then argue that $(C) - (B) = O\left ( \log^{-4}(R) \right ) .$ \[\begin{aligned} \label{lemma.A-C} &\frac{1}{W_R(\mathcal{F}_{N})}\sum_{p_1,p_2<R^\sigma}\sum_{m_1,m_2 \geq 3 }\notag\\ &\qquad\sum_{\substack{f\in \mathcal{F}_N\\p_1\nmid N\\p_2\nmid N }}w_R(f)\frac{\sum_{j=1}^2\left(\alpha_f(p_j)^{m_j}+\beta_f(p_j)^{m_j}\right)}{p_1^{m_1/2}p_2^{m_2/2}} \frac{\log(p_1)\log (p_2)\widehat\phi_1 \left(m_1 \frac{\log(p_1)}{\log(R)}\right)}{\log^2(R)} \left[\widehat\phi_2 \left(m_2 \frac{\log(p_2)}{\log(R)}\right) - \widehat\phi_2(0)\right]\notag \\ &\ \lesssim \ \frac{1}{\log^4(R)}\sum_{p_1<R^\sigma}\sum_{m_1\geq 3} \frac{\log(p_1)}{p_1^{m_1/2}}\widehat\phi_1\left(m_1\frac{\log(p_1)} {\log(R)}\right)\notag\\ & \lesssim \frac{1}{\log^4(R)}\sum_{p_1}\frac{\log(p_1)}{p_1(\sqrt{p_1}-1)}\notag\\ &\ \lesssim \ \frac{1}{\log^4(R)}\sum_{p_1}\frac{\log(p_1)}{p^{3/2}}\notag\\ &\ \lesssim \ \frac{1}{\log^4(R)}. \end{aligned}\] The first approximation follows from using Taylor expansion, bounding $|\alpha_f(p)^m + \beta_f(p)^m|$ by 2, and finally observing that $\sum_{m_2\geq3 } {m_2^2}/{p_2^{m_2/2}} = O(p_2^{-3/2})$ and $\sum_{p_2}\log(p_2)^3/p_2^{3/2} < \infty.$ Therefore $(A)-(C) = O\left ( \log^{-4}(R) \right )$.

we show that $(C)-(B)= O\left ( \log^{-4}(R) \right )$. By a similar approximation, \[\begin{aligned} &\frac{1}{W_R(\mathcal{F}_{N})}\sum_{p_1,p_2<R^\sigma}\sum_{m_1,m_2 \geq 3 }\notag\\ &\qquad\sum_{\substack{f\in \mathcal{F}_N\\p_1\nmid N\\p_2\nmid N }}w_R(f)\frac{\sum_{j=1}^2\left(\alpha_f(p_j)^{m_j}+\beta_f(p_j)^{m_j}\right)}{p_1^{m_1/2}p_2^{m_2/2}} \frac{\log(p_1)\log (p_2)}{\log^2(R)} \left(\widehat\phi_1 \left(m_1 \frac{\log(p_1)}{\log(R)}\right)-\widehat\phi_1(0)\right)\widehat\phi_2(0) \notag \\ \ &\lesssim \ \frac{1}{\log^4(R)} \left(\sum_{p_1<R^\sigma}\sum_{m_1\geq 3}\frac{m_1^2\log(p_1)^3}{p_1^{m_1/2}}\right)\left(\sum_{p_2<R^\sigma}\sum_{m_2\geq 3}\frac{\log(p_2)^3}{p_2^{m_2/2}}\right)\\ \ &\lesssim\ \frac{1}{\log^4(R)}\left(\sum_{p_1<R^\sigma}{p_1}^{-3/2}\right) \notag \\ \ &\lesssim \ \frac{1}{\log^4(R)}. \end{aligned}\] With this, the proof of Lemma 19 is complete. ◻

10 Lemmas used for main terms in theorems 34, 35, 36, and 37.

In this section, we state and prove various lemmas used to compute the main Terms in Theorems 34, 35, 36, and 37.

Proof. By Abel summation \[\begin{aligned} \label{l1theta} S(R) & \ = \ \sum_{p}\frac{2\log(p)}{p\log(R)}\widehat\phi\Bigl(\frac{2\log(p)}{\log(R)}\Bigr) =\frac{2}{\log(R)}\sum_{p}\frac{\log(p)}{p}\widehat\phi\Bigl(\frac{2\log(p)}{\log(R)}\Bigr)\notag\\ &\ = \ \frac{2}{\log(R)}\lim_{x\to\infty}\Bigl[\theta(x)\frac{1}{x}\widehat\phi\Bigl(\frac{2\log(x)}{\log(R)}\Bigr) -\int_{1}^{x}\theta(t)\frac{d}{dt}\Bigl(\frac{1}{t}\widehat\phi\Bigl(\frac{2\log(t)}{\log(R)}\Bigr)\Bigr)dt\Bigr]\notag \\ & \ = \ -\frac{2}{\log(R)}\int_{1}^{\infty}\theta(t)\frac{d}{dt}\Bigl(\frac{1}{t}\widehat\phi\Bigl(\frac{2\log(t)}{\log(R)}\Bigr)\Bigr)dt. \end{aligned}\]

Define $u=2\log(t)/\log(R)$. Then \[\frac{d}{dt}\Bigl(\frac{1}{t}\widehat\phi(u)\Bigr) \ = \ -\frac{1}{t^{2}}\widehat\phi(u)+\frac{2}{(\log(R))t^{2}}\widehat\phi'(u).\] Write $\theta$ as $t + E(t)$. The first term in [l1theta] is \[\begin{aligned} \frac{2}{\log(R)}\int_{1}^{\infty}\frac{1}{t}\Bigl(\widehat\phi\Bigl(\frac{2\log(t)}{\log(R)}\Bigr) -\frac{2}{\log(R)}\widehat\phi'\Bigl(\frac{2\log(t)}{\log(R)}\Bigr)\Bigr)dt \notag \\ & \ = \ \frac{2}{\log(R)}\int_{1}^{\infty}\frac{1}{t}\widehat{\phi}\Bigl(\frac{2\log(t)}{\log(R)}\Bigr)dt-\frac{4}{(\log(R))^{2}}\int_{1}^{\infty}\frac{1}{t}\widehat\phi'\Bigl(\frac{2\log(t)}{\log(R)}\Bigr)dt \notag \\ & \ = \ \frac{2}{\log(R)}\cdot\frac{\log(R)}{2}\int_{0}^{\infty}\widehat\phi(u)du-\frac{4}{(\log(R))^{2}}\cdot\frac{\log(R)}{2}\int_{0}^{\infty}\widehat\phi'(u)du \notag \\ & \ = \ \frac{2}{\log(R)}\cdot\frac{\log(R)}{2}\cdot\frac{\phi(0)}{2}-\frac{4}{(\log(R))^{2}}\cdot\frac{\log(R)}{2}\cdot\bigl(-\widehat\phi(0)\bigr) \notag \\ & \ = \ \frac{\phi(0)}{2}+\frac{2\widehat\phi(0)}{\log(R)}. \label{l1first} \end{aligned}\] The second term in [l1theta] is \[\frac{2}{\log(R)}\int_{1}^{\infty}\frac{E(t)}{t^{2}}\Bigl(\widehat\phi\Bigl(\frac{2\log(t)}{\log(R)}\Bigr) -\frac{2}{\log(R)}\widehat\phi'\Bigl(\frac{2\log(t)}{\log(R)}\Bigr)\Bigr)dt.\] Because $\widehat\phi$ is even, the Taylor expansions are \[\widehat\phi(u) \ = \ \widehat\phi(0)+\frac{\widehat\phi''(0)}{2}u^{2}+O(u^{4}), \quad \widehat\phi'(u) \ = \ \widehat\phi''(0)u+O(u^3).\] Thus, the second term in [l1theta] becomes \[\begin{aligned} & \ = \ \frac{2}{\log(R)}\int_{1}^{\infty}\frac{E(t)}{t^{2}} \Bigl[\widehat\phi(0) +\frac{\widehat\phi''(0)}{2}\Bigl(\frac{2\log(t)}{\log(R)}\Bigr)^{2} -\frac{2\widehat\phi''(0)}{\log(R)}\Bigl(\frac{2\log(t)}{\log(R)}\Bigr) +O\Bigl(\Bigl(\frac{\log(t)}{\log(R)}\Bigr)^{3}\Bigr) \Bigr]dt \notag \\ & \ = \ \frac{2\widehat\phi(0)}{\log(R)}\int_{1}^{\infty}\frac{E(t)}{t^{2}}dt +\frac{4\widehat\phi''(0)}{(\log(R))^{3}}\int_{1}^{\infty}\frac{E(t)(\log(t))^{2}}{t^{2}}dt -\frac{8\widehat\phi''(0)}{(\log(R))^{3}}\int_{1}^{\infty}\frac{E(t)\log(t)}{t^{2}}dt\notag\\ & +O\left ( \log^{-4}(R) \right ) . \label{l1second} \end{aligned}\] Combining the terms in [l1first] and [l1second] yields the lemma. ◻

Proof. By the Convolution Theorem, we have \[\widehat\phi_1\left(\frac{\log(p)}{\log(R)}\right) \widehat\phi_2\left(\frac{\log(p)}{\log(R)}\right) \ = \ \widehat\phi\left(\frac{\log(p)}{\log(R)}\right), \quad \phi \ := \ \phi_1*\phi_2.\] Define \[f(t) \ := \ \frac{\log(t)}{t\log^2(R)}\widehat\phi\left(\frac{\log(t)}{\log(R)}\right), \quad \theta(t) \ := \ t + E(t).\] \[S_1(R) \ := \ \sum_{p} \frac{\log^2(p)}{p\log^2(R)} \widehat\phi\left(\frac{\log(p)}{\log(R)}\right) \ = \ \sum_p \log(p)f(p).\] By Abel summation, we write \[S_1(R) \ = \ \lim_{x \to \infty}\theta(x)f(x) -\int_1^{\infty}\theta(t)f'(t)dt = -\int_1^{\infty}\theta(t)f'(t)dt. \label{l2sum}\] Differentiate $f$ with respect to $u := \frac{\log(t)}{\log(R)}$. Then we change variables and calculate [l2sum] \[\begin{aligned} f'(t) \ =& \ \frac{1}{\log^2(R)}\left(\frac{1-\log(t)}{t^{2}}\widehat\phi(u) + \frac{\log(t)}{t^2 \log(R)} \widehat\phi'(u)\right),\notag \\ S_1(R) \ =& \ \int_0^\infty \theta(t)\left(\frac{\log(t) - 1}{t^2\log^2(R)} \widehat\phi(u) - \frac{1}{\log(R)} \frac{\log(t)}{t^2 \log^2(R)}\widehat\phi'(u)\right)dt. \label{l2sumc} \end{aligned}\] Write $\theta$ as $t + E(t)$, by change of variable, the first term in [l2sumc] is \[\int_0^\infty u\widehat\phi(u)du - \frac{1}{\log(R)} \int_0^\infty \left(\widehat\phi(u) + u\widehat\phi'(u)\right)du \ = \ \int_0^\infty u\widehat\phi(u)du.\] Since $\widehat\phi$ is even. The Taylor expansions are \[\widehat\phi(u) \ = \ \widehat\phi(0)+O(u^2), \quad \widehat\phi'(u) \ = \ \widehat\phi''(0)u+O(u^3). \label{l2taylor}\] Substituting [l2taylor] into $f'(t)$ yields \[f'(t) \ = \ \frac{\widehat\phi(0)}{t^{2}\log^2(R)}(1-\log(t)) +O\Bigl(\frac{\log^2t}{t^{2}\log^4(R)}\Bigr). \label{l2firstt}\] The second term is then equal to \[-\frac{\widehat\phi(0)}{\log^2(R)}\int_1^{\infty}\frac{E(t)}{t^2}(1-\log(t))dt + O\left ( \log^{-4}(R) \right ) . \label{l2secondt}\] Combining [l2firstt] and [l2secondt] yields the lemma. ◻

Proof. As before, we appeal to the convolution theorem, writing $\phi := \phi_1*\phi_2$ so that \[\sum_{p}\frac{\log^2 (p)}{p^2\log^2(R)}\widehat\phi_1\left(\frac{2\log(p)}{\log(R)}\right)\,\,\widehat\phi_2\left(\frac{2\log(p)}{\log(R)}\right)\,=\,\sum_p \frac{\log^2(p)}{p^2\log^2(R)}\widehat\phi\left(\frac{2\log(p)}{\log(R)}\right).\] Recall that we define $\theta(t):=\sum_{p\leq t} \log(p)$ and let \[f(x) \ := \ \frac{\log(x)}{x^2}\widehat\phi\left(\frac{2\log(x)}{\log(R)}\right).\] by the Abel summation formula and the fact that $\mathrm{supp}(\widehat\phi)$ is bounded, \[S_2(R) \ = \ \frac{1}{\log^2(R)}\lim_{x\to \infty}\left[ \theta(x) f(x)-\int_1^x \theta(t)f'(t)dt\right]=-\frac{1}{\log^2(R)}\int_1^\infty \theta(t) f'(t)dt.\] To ease notation, substitute $u=2\log(t)/\log(R)$ and differentiate to get \[\begin{aligned} f'(t)\ =& \ \frac{1-2\log(t)}{t^3}\widehat\phi(u)+\frac{2\log(t)}{t^3\log(R)}\widehat\phi'(u) \notag \\ \int_1^\infty t f'(t)dt \ =& \ -\int_1^\infty f(t)dt. \end{aligned}\] Splitting the error in $\theta(t)=t+E(t),$ we first consider the main term. Substituting in $u$ again gives \[\int_1^\infty f(t)dt \ = \ \frac{\log ^2 (R)}{4}\int_0^\infty u e^\frac{-u\log(R)}{2}\widehat\phi(u)du.\] using the evenness of $\widehat\phi$, we Taylor expand and find \[\widehat\phi(u) \ = \ \widehat\phi(0)+\frac{\widehat\phi''(0)}{2}u^2+O(u^4).\] and note that $\int_0^\infty ue^{-\alpha u} \ = \ 1/\alpha^2$ and $\int_0^\infty u^3e^{-\alpha u} \ = \ 6/\alpha^4.$ Thus, the main term is, after bringing in the factor of $-1/\log^2(R)$, \[\frac{\widehat\phi(0)}{\log^2(R)}+\frac{12 \widehat\phi''(0)}{\log^4(R)}+O\left ( \log^{-6}(R) \right ) \ = \ \frac{\widehat\phi(0)}{\log^2(R)}+O\left ( \log^{-4}(R) \right ) .\] We state the error term \[-\frac{1}{\log^2(R)}\int_1^\infty E(t) f'(t)dt \ = \ -\frac{1}{\log^2(R)}\int_1^\infty E(t) \left(\frac{1-2\log(t)}{t^3}\widehat\phi(u)+\frac{2\log(t)}{t^3\log(R)}\widehat\phi'(u)\right)dt.\] Since we also have from evenness that \[\widehat{\phi}'(u) \ = \ \widehat{\phi}''(0)u+O(u^3),\] we expand each of $\widehat\phi$ and $\widehat\phi'$ to see that the above is equal to \[\begin{aligned} -\frac{1}{\log^2(R)}\int_1^\infty E(t)\left(\frac{1-2\log(t)}{t^3}\widehat\phi(0)+\left(\frac{2\log^2(t)}{t^3\log^2(R)}+\frac{1-2\log(t)}{t^3}\right)\widehat\phi''(0)+O\left(\frac{\log^3 (t)}{\log^3(R)}\right)\right)\nonumber\\ =-\frac{\widehat\phi(0)+\widehat\phi''(0)}{\log^2(R)}\int_1^\infty \frac{E(t)(1-2\log(t))}{t^3}dt+O\left ( \log^{-4}(R) \right ) . \end{aligned}\] ◻

Proof. Define \[S(x):=\sum_{p\leq x}\frac{\log(p)}{p^\alpha}, \quad \theta(t) := t + E(t).\] By Abel summation \[\begin{aligned} S(x) &\ = \ x^{-\alpha}\theta(x)-2^{-\alpha}\theta(2)+\alpha\int_{2}^{x}\theta(t) t^{-\alpha-1} dt.\\ &\ = \ x^{-\alpha}\bigl(x+E(x)\bigr)-2^{-\alpha}\log (2) +\alpha\int_{2}^{x} \bigl(t+E(t)\bigr) t^{-\alpha-1} dt\\ &\ = \ x^{1-\alpha} + \alpha\int_{2}^{x} t^{-\alpha} dt -2 ^{-\alpha}\log (2) + x^{-\alpha}E(x) + \alpha\int_{2}^{x} E(t) t^{-\alpha-1} dt. \end{aligned}\] After calculating the main terms, we have \[S(x)\ = \ \frac{x^{1-\alpha}}{1-\alpha} +x^{-\alpha}E(x) +\alpha\int_{2}^{x} E(t) t^{-\alpha-1} dt -2^{-\alpha}\log (2)-\frac{\alpha 2^{1-\alpha}}{1-\alpha}. \label{l4ppower}\] By the Prime Number Theorem, $E(t)=\theta(t)-t=O\bigl(t/ \log(t)\bigr)$, substituting in [l4ppower] yields \[x^{-\alpha}E(x)\ = \ \Bigl(\frac{x^{1-\alpha}}{\log(x)}\Bigr),\qquad \int_{2}^{x} E(t) t^{-\alpha-1}dt\ = \ O\Bigl(\int_{2}^{x}\frac{t^{-\alpha}}{\log(t)}dt\Bigr) \ = \ O\left(\frac{x^{1-\alpha}}{\log(x)}\right).\] The constants are absorbed into the big O term, gives \[S(x) \ = \ \frac{x^{1-\alpha}}{1-\alpha}+O\left(\frac{x^{1-\alpha}}{\log(x)}\right).\] ◻

Proof. As in [Sl], the generating function for the Catalan numbers can be written in closed-form for $|z|<1/4$: \[\label{catSum} \sum_{i=0}^\infty C_i z^i \ = \ \frac{1-\sqrt{1-4z}}{2z}.\] Letting $z=p/(p+1)^2$, we note that \[|z|\leq \frac{2}{(2+1)^2} \ = \ \frac{2}{9}<\frac{1}{4}\] so we always have convergence to the above formula. Factoring and subtracting off the first two terms, \[\sum_{i=2}^\infty C_i\frac{p^{i-1}(p-1)}{(p+1)^{2i}} \ = \ \frac{p-1}{p}\left(\sum_{i=0}^\infty C_iz^i-C_0-C_1z\right).\] Since \[1-4z \ = \ \frac{(p+1)^2}{(p+1)^2}-\frac{4p}{(p+1)^2} \ = \ \left(\frac{p-1}{p+1}\right)^2,\] we have, using $C_0=C_1=1$, \[\sum_{i=0}^\infty C_i z^i-C_0-C_1z \ = \ \frac{1-\sqrt{1-4z}}{2z}-1-z=\frac{p+1}{p}-1-z \ = \ \frac{1}{p}-z.\] We multiply through by the extra factors to get \[\sum_{i=2}^\infty C_i \frac{p^{i-1}(p-1)}{(p+1)^{2i}} \ = \ \frac{p-1}{p}\left(\frac{1}{p}-z\right) \ = \ \frac{p-1}{p^2}-\frac{p-1}{p}z \ = \ \frac{p-1}{p^2}-\frac{p-1}{(p+1)^2}.\] Furthering our effort to simplify the bracket in the sum, note that \[\frac{-3p-1}{p(p+1)^2}+\frac{p-1}{p^2}-\frac{p-1}{(p+1)^2}\ = \frac{-3p^2-p+(p-1)(p+1)^2-(p-1)p^2}{p^2(p+1)^2}\ = \ -\frac{1}{p^2},\] hence, the sum becomes \[-\sum_p\widehat\phi_2\left(\frac{\log(p)}{\log(R)}\right) \frac{\log^2(p)}{p^2\log ^2 (R)}\] which is amenable to our techniques involving the Prime Number Theorem. Let $u=\log(t)/\log(R)$ and apply the Abel summation formula with the sequence $a_p=\log(p)$ to get the main term \[-\int_1^\infty \frac{\log(t)}{t^2\log^2(R)}\widehat\phi_2\left(\frac{\log(t) }{\log(R)}\right)dt \ = \ -\int_0^\infty u e^{-u\log(R)} \widehat\phi_2(u)du\] arising from $\theta(t)=t+E(t).$ Writing $\widehat\phi(u)=\widehat\phi(0)+O(u^2)$ and plugging back into the integral, the main term becomes \[-\frac{\widehat\phi(0)}{\log^2(R)}+O\left ( \log^{-4}(R) \right )\] since $\int_0^\infty ue^{-u\alpha}du=1/\alpha^2.$ We proceed to compute the error term. We have the error is \[\frac{1}{\log^2(R)}\int_1^\infty E(t)\dfrac{d}{dt}\left[ \frac{\log^2 t}{t^2}\phi\left(\frac{\log(t)}{\log(R)}\right)\right].\] Now, \[\dfrac{d}{dt}\left[ \frac{\log^2 t}{t^2}\phi\left(\frac{\log(t)}{\log(R)}\right)\right] \ = \ \frac{1-2\log(t)}{t^3}\widehat\phi(u)+\frac{\log(t)}{t^3\log(R)}\widehat\phi'(u)\] so the error term is \[\frac{\widehat\phi(0)+\widehat\phi''(0)}{\log^2(R)}\int_1^\infty \frac{E(t)(1-2\log(t))}{t^3}dt+O\left ( \log^{-4}(R) \right ) .\] ◻

Proof. Let $f(x)=\log(x)\,\widehat\phi_2\left(\frac{2\log(x)}{\log(R)}\right) \frac{x^2+3x+1}{x^2(x+1)^3}$ so that by our standard tricks, we have \[\label{line4error} \sum_p \frac{\log^2(p)}{\log^2(R)}\widehat\phi_2\left(\frac{2\log(p)}{\log(R)}\right) \frac{p^2+3p+1}{p^2(p+1)^3}\ = \ -\frac{1}{\log^2(R)}\int_1^\infty tf'(t)dt-\frac{1}{\log^2(R)}\int_1^\infty E(t)f'(t)dt.\] Integrating the first term by parts and letting $s=2\log(t)$, it suffices to compute \[\int_1^\infty f(t)dt \ = \ \frac{1}{4}\int_0^\infty s \widehat\phi\left(\frac{s}{\log(R)}\right) \frac{e^s+3+e^{-s}}{(e^s+1)^3}ds.\] Expanding $\widehat\phi$ using evenness, this gives that the main term is \[\frac{I\widehat\phi(0)}{4\log^2(R)}+O\left ( \log^{-4}(R) \right ) ,\] where we let \[I \ := \ \int_0^\infty s\frac{e^s+3+e^{-s}}{(e^s+1)^3}ds\] to later be integrated numerically. We compute the error term in [line4error]. Differentiating, we have \[f'(x) \ = \ \frac{A(x)\widehat\phi'\left(\frac{2\log(x)}{\log(R)}\right)+\log(R)\,\,B(x)\widehat\phi\left(\frac{2\log(x)}{\log(R)}\right)}{x^3(x+1)^4\log(R)}\] where \[\begin{aligned} &A(x)\ \coloneq \ (2x^3+8x^2+8x+2)\log(x) \notag \\ &B(x)\ \coloneq \ (-3x^3-12x^2-8x-2)\log(x)+x^3+4x^2+4x+1. \end{aligned}\] Using evenness to expand $\widehat\phi$ and $\widehat\phi'$ to compute the integral \[-\frac{1}{\log^2(R)}\int_1^\infty E(t)f'(t)dt,\] we have the first term is absorbed into the $O(\log^{-4}R)$ error. Considering the second term, only the constant term in the expansion of $\widehat\phi$ matters for us, yielding an error term of \[-\frac{\widehat\phi(0)}{\log^2(R)}\int_1^\infty \frac{E(t)B(t)}{t^3(t+1)^4}dt.\] ◻

Proof. Rearranging \[\begin{aligned} &\sum_{p_1,p_2}\frac{\log(p_1)\log(p_2)}{p_2\log^2(R)}\widehat\phi\left(\frac{2\log(p_2)}{\log(R)}\right)\frac{2}{p_1(p_1+1)} \notag \\ \ = \ &\sum_{p_1} \frac{\log(p_1)}{p_1(p_1+1)\log(R)} \sum_{p_2}\frac{2\log(p_2)}{p_2\log(R)}\widehat\phi\left(\frac{2\log(p_2)}{\log(R)}\right). \end{aligned}\] We apply Lemma 46 to the sum over $p_2$, getting that the above equals \[\frac{S(R)}{\log(R)}\sum_{p_1} \frac{\log(p_1)}{p_1(p_1+1)}\] where we can compute $S(R)$ up to $O(\log^{-4}R)$ numerically, leaving only the consideration of the sum over $p_1.$ We apply the standard method of Abel summation, setting $g(x)=\frac{1}{x(x+1)}$ so that \[\label{line1split} \sum_{p_1}\frac{\log(p_1)}{p_1(p_1+1)}\ =\ -\int_1^\infty tg'(t)dt -\int_1^\infty E(t)g'(t)dt.\] Integrating by parts, the first term gives a contribution of \[_1^\infty+\int_1^\infty g(t)dt \ = \ -\frac{1}{2}+\log 2.\] Since \[g'(x) \ = \ -\frac{2x+1}{(x^2+x)^2},\] the error term in [line1split] \[\int_1^\infty \frac{E(t)(2t+1)}{(t^2+t)^2}dt \ \eqqcolon \ F_1 \label{llF1}\] which is a constant that we can compute numerically. Hence, after substituting in our expression for $S(R)$ from Lemma 46, we have in total \[\begin{aligned} &\sum_{p_1,p_2}\frac{\log(p_1)\log(p_2)}{p_2\log^2(R)}\widehat\phi\left(\frac{2\log(p_2)}{\log(R)}\right)\frac{2}{p_1(p_1+1)} \ = \ \frac{S(R)}{\log(R)}\left[-\frac{1}{2}+\log(2)+F_1\right]\\ &=\ \bigg[\frac{\widehat\phi(0)}{2\log(R)}+\frac{2\widehat\phi(0)}{\log(R)}\left(1+\int_0^\infty\frac{E(t)}{t^2}dt\right)+O\left ( \log^{-4}(R) \right ) \bigg]\nonumber\\ &\qquad \cdot \left[\left(-\frac{1}{2}+\log(2)\right)+\int_1^\infty \frac{E(t)(2t+1)}{(t^2+t)^2}dt \right]. \label{l7sum} \end{aligned}\] Let \[\begin{aligned} F_2 \ \coloneqq \ \int_1^\infty \frac{E(t)(2t+1)}{(t^2+t)^2}dt. \label{llF2} \end{aligned}\] Substituting [llF1] and [llF2] into [l7sum] yields the lemma. ◻

Proof. The proof is the same partial-summation calculation as in Lemma 52. Factoring the $p_2$-sum gives \[\frac{S(R)}{2\log(R)} \sum_{p_1}\log(p_1)\frac{p_1^2+3p_1+1}{p_1(p_1+1)^3}.\] With \[h(x)=\frac{x^2+3x+1}{x(x+1)^3},\] the Prime Number Theorem and partial summation give \[\sum_p\log(p)h(p)=\log 2+\frac34+F_3, \qquad F_3=\int_1^\infty E(t)\frac{2t^3+8t^2+4t+1}{t^2(t+1)^4}\,dt.\] Substituting the expansion for $S(R)$ from Lemma 46 yields the stated formula. The intermediate steps are identical to those in Lemma 52. ◻

Proof. The proof follows the same strategy as Lemmas 52 and 53. Summing the Catalan series first gives \[\sum_{\ell=2}^\infty C_\ell \frac{p^\ell(p-1)}{(p+1)^{2\ell+1}} = \frac{2p^2-p+1}{p(p+1)^3}.\] The remaining prime sum is then treated by the same Prime-Number-Theorem partial-summation argument. With \[f(x)=-\frac{2x^2-x+1}{x(x+1)^3},\] one obtains \[\sum_p\log(p)\frac{2p^2-p+1}{p(p+1)^3} = \log 2+\frac14+F_4,\] where $F_4$ is the error integral appearing in the statement. Substitution of Lemma 46 completes the calculation; we omit the repeated intermediate partial-summation steps. ◻

Proof. We first simplify the sum over $\ell$. Let \[G(x) \ := \ \sum_{\ell=0}^\infty C_\ell x^\ell \ = \ \frac{1-\sqrt{1-4x}}{2x}\] and \[S\ :=\ \sum_{\ell=2}^\infty (C_{\ell+1}-C_\ell)z^\ell\] where $z=p/(p+1)^2.$ Since $C_0=C_1=1$ and $C_2=2$, it follows that \[S\ =\ \frac{G(z)-1-z-2z^2}{z}-G+1+z=\frac{(1-z)G-1-z^2}{z}.\] Plug in $z=p/(p+1)^2$, noting that $\sqrt{1-4z}=(p-1)/(p+1)$ to get that \[G(z)\ =\ \frac{p+1}{p}\] and thus \[S=\frac{3p^2+3p+1}{p^2(p+1)^2}.\] Multiplying through by what we factored out, we obtain \[\sum_{\ell=2}^\infty(C_{\ell+1}-C_\ell) \frac{p^\ell(p-1)}{p(p+1)^{2\ell +1}} \ = \ \frac{(p-1)(3p^2+3p+1)}{p^3(p+1)^3}.\] The rest of the argument is analogous to [line4]. We define \[g(x)\ :=\ \widehat\phi\left(\frac{2\log(x)}{\log(R)}\right)G(x) \log(x),\] where \[G(x)\ :=\ \frac{(x-1)(3x^2+3x+1)}{x^3(x+1)^3}.\] Hence, we can rewrite the sum we want as \[\frac{1}{\log^2(R)}\sum_p \log(p)\, g(p) \ = \ -\frac{1}{\log^2(R)}\int_1^\infty tg'(t)dt -\frac{1}{\log^2(R)}\int_1^\infty E(t)g'(t)dt.\] Letting $u=2\log(p)/\log(R)$, \[-\frac{1}{\log^2(R)}\int_1^\infty tg'(t)dt\ =\ \frac{1}{\log^2(R)}\int_1^\infty g(t)dt \ = \ \frac{1}{\log^2(R)}\int_1^\infty \log(t)\,\widehat\phi_2\left(\frac{2\log(t)}{\log(R)}\right) G(t) dt.\] This is equal to \[\frac{\widehat\phi_2(0)}{\log^2(R)}\int_1^\infty \frac{\log(t) (t-1)(3t^2+3t+1)}{t^3(t+1)^3}+O\left ( \log^{-4}(R) \right ) \ =\ \frac{\widehat\phi_2(0)}{4\log^2(R)}+O\left ( \log^{-4}(R) \right ) . \label{ll10f}\] We consider the error term. We have \[g'(x)\ =\ \frac{A(x)\log(x) \widehat\phi_2'\left(\frac{2\log(x)}{\log(R)}\right)+(B(x)\log(x)+C(x))\log(R) \widehat\phi_2\left(\frac{2\log(x)}{\log(R)}\right)}{\log(R)x^4(x+1)^4}\] where $A(x)$, $B(x)$, and $C(x)$ are polynomials. Expanding $\widehat\phi$ and $\widehat\phi'$, the main term doesn’t contribute, so the error is \[-\frac{\widehat\phi_2(0)}{\log^2(R)}\int_1^\infty E(t)\frac{B(t)\log(t)+C(t)}{t^4(t+1)}dt\] which is explicitly \[-\frac{\widehat\phi_2(0)}{\log^2(R)}\int_1^\infty E(t)\frac{(-9t^4+10t^2+10t+3)\log(t)+3t^4+3t^3-2t^2-3t-1}{t^4(t+1)^4}dt. \label{ll10s}\] Combining [ll10f] and [ll10s] yields the lemma. ◻

We do not need the GRH for any of the calculations. However, assuming the GRH allows us to order the zeros and relate the $n$-level density with the spacing between the zeros.↩︎
In general, we do not need the test function $\Phi$ to be a product of coordinate-wise test functions [cheek2026densitylowlyingzeros].↩︎
[Miller2009_LOTerms_1LevelDensity] uses the notation $A_{k,N}$ instead of $U_{k,N}$.↩︎
Since we are looking at cusp forms, $a_f(0)=0$; for general forms, this is not necessarily true.↩︎
[Miller2009_LOTerms_1LevelDensity] denotes these sums $A_{r, \mathcal{F}}, A'_{r, \mathcal{F}}$, and the $B$-sums are denoted similarly.↩︎
The harmonic weights are nearly constant across the family: by [I1, HL, Miller2009_LOTerms_1LevelDensity] we have $w_R(f)= N^{\varepsilon}$ uniformly for $f$ in the family. If we allow ineffective constants, the factor $N^{\varepsilon}$ can be replaced by $\log(N)$ for sufficiently large $N$. There are other ways of normalizations. [Miller2009_LOTerms_1LevelDensity] and our paper averages with $w_R(f)\ =\ \frac{\zeta_N(2)}{Z(1,f)} = \frac{\zeta(2)}{L(1,\mathrm{sym}^2 f)}$. Rouymi [Rouymi2011TraceNonAnnulation] uses the Petersson-harmonic weight $w_R'(f)=\frac{\Gamma(k-1)}{(4\pi)^{k-1}}\langle f,f\rangle_N^{-1}$. Rankin-Selberg shows $\langle f,f\rangle_N \asymp_{k,N} L(1,\mathrm{sym}^2 f)$, so $w_R(f)\propto_{k,N} w_R'(f)$. In addition, using this weight, Rouymi derived formulas for level $N = p^{\alpha}$ for fixed prime $p$ and $\alpha \to \infty$. We also note that different weights exhibit different behaviors as seen in [KnightlyReno2019] and [dillon2025centered].↩︎

Breaking Universality in the Lower Order Terms in the 1-level and 2-level Density of Holomorphic Cusp Newforms

Abstract

Contents

1 Introduction

1.1 Main results

1.2 The explicit formula

2 Preliminaries

3 Computing \({U_{k,N}(\phi)}\)

4 Computing \({S_1((\mathcal{F}_{N},\phi)}\) up to \(O\left ( \log^{-4}(R) \right )\) error

5 Computing \({S_2(\mathcal{F}_{N},\phi_1,\phi_2)}\) up to \(O\left ( \log^{-4}(R) \right )\) error

5.1 First line of [eq:S_2 formula]: \(p_1 \mid N\), \(p_2 \mid N\)

5.2 Sum of second and third lines of [eq:S_2 formula]: \(p_1 \mid N\), \(p_2 \nmid N\) and \(p_2 \mid N\), \(p_1 \nmid N\)