A Refined Conjecture for the Variance of Gaussian Primes Across Sectors

TL;DR

This paper refines the conjecture on the variance of Gaussian primes in sectors, incorporating lower order terms and bifurcation points, supported by both heuristic and unconditional results.

Contribution

It introduces a refined conjecture with a power saving error term, revealing a second bifurcation point overlooked by previous RMT models.

Findings

Identification of a second bifurcation point in the variance model

Unconditional proof for small sectors consistent with the conjecture

Application of the L-functions Ratios Conjecture to Gaussian primes

Abstract

We derive a refined conjecture for the variance of Gaussian primes across sectors, with a power saving error term, by applying the L-functions Ratios Conjecture. We observe a bifurcation point in the main term, consistent with the Random Matrix Theory (RMT) heuristic previously proposed by Rudnick and Waxman. Our model also identifies a second bifurcation point, undetected by the RMT model, that emerges upon taking into account lower order terms. For sufficiently small sectors, we moreover prove an unconditional result that is consistent with our conjecture down to lower order terms.

Figures

Equations

$$F_K(\theta ):=\sum _{k\in \mathbb{Z}}f\left(\frac{K}{\pi /2}\left(\theta -\frac{\pi }{2}\cdot k\right)\right),$$

$$F_K(\theta )=\sum _{k\in \mathbb{Z}}{\hat{F}}_K(k)e^{i4k\theta },{\hat{F}}_K(k)=\frac{1}{K}\hat{f}\left(\frac{k}{K}\right),$$

$$\tilde{\Phi }(s):={\int }_0^{\infty }\Phi (x)x^{s-1}dx.$$

$${\psi }_{K,X}(\theta ):=\sum _{\mathfrak{a}}\Phi \left(\frac{N(\mathfrak{a})}{X}\right)\Lambda (\mathfrak{a})F_K({\theta }_{\mathfrak{a}}-\theta ),$$

$$\begin{array}{rl}⟨{\psi }_{K,X}⟩& :=\frac{1}{\pi /2}{\int }_0^{\frac{\pi }{2}}\sum _{\mathfrak{a}}\Phi \left(\frac{N(\mathfrak{a})}{X}\right)\Lambda (\mathfrak{a})F_K({\theta }_{\mathfrak{a}}-\theta )d\theta \sim \frac{X}{K}\cdot C_{\Phi }\cdot c_f,\end{array}$$

$$c_f:=\frac{1}{4{\pi }^2}{\int }_{\mathbb{R}}f(x)dx,\text{ and }{C}_{\Phi }:=4{\pi }^2{\int }_0^{\infty }\Phi (u)du.$$

$${\psi }_{K,X}(\theta )\sim \frac{X}{K}\cdot c_f\cdot C_{\Phi }.$$

$$\text{Var}({\psi }_{K,X})$$

$$\text{Var}({\psi }_{K,X})\sim {\int }_{\mathbb{R}}f(y{)}^{2}dy{\int }_0^{\infty }\Phi (x{)}^{2}dx\cdot \text{min}(\log X,2\log K).$$

$$\frac{\text{Var}({\psi }_{K,X})}{C_f{X}^{1-\lambda }}=C_{\Phi }\log X+C_{\Phi }^{\prime }+{\pi }^2\tilde{\Phi }{\left(\frac{1}{2}\right)}^2+o\left(1\right),$$

$$\begin{array}{rl}{C}_f& :=\frac{1}{4{\pi }^2}{\int }_{\mathbb{R}}f(y{)}^{2}dy{C}_{\Phi }^{\prime }:=4{\pi }^2\cdot {\int }_0^{\infty }\log x\cdot \Phi (x{)}^{2}dx,\end{array}$$

$$\frac{\textnormal{Var}(\psi_{K,X})}{C_{f}X^{1-\lambda}}=\left\{\begin{array}[]{l l}C_{\Phi}\log X+\Delta_{\Phi}+O_{\Phi}\left(X^{-\epsilon}\right)&\textnormal{ if }\frac{1}{2}<\lambda<1\\ C_{\Phi}\left(2\lambda\log X\right)-K_{\Phi}+O_{\Phi}\left(X^{-\epsilon}\right)&\textnormal{ if }\lambda<\frac{1}{2},\end{array}\right.$$

$${\Delta }_{\Phi }:=C_{\Phi }^{\prime }-{\pi }^2\tilde{\Phi }{\left(\frac{1}{2}\right)}^2,$$

$$K_{\Phi }:=C_{\Phi ,\zeta }-C_{\Phi ,L}-A_{\Phi }^{\prime }+2{\pi }^2\tilde{\Phi }{\left(\frac{1}{2}\right)}^2+C_{\Phi }\left(\log \left(\frac{{\pi }^2}{4}\right)+2\right),$$

$$\displaystyle\begin{split}\Omega_{K,X}&:=\frac{1}{\pi/2}\int_{0}^{\frac{\pi}{2}}\bigg{|}\psi_{K,X}(\theta)\bigg{|}^{2}d\theta\\ &=\frac{2}{\pi}\sum_{\begin{subarray}{c}\mathfrak{a},\mathfrak{b}\subset\mathbb{Z}[i]\\ \end{subarray}}\Phi\left(\frac{N(\mathfrak{a})}{X}\right)\Phi\left(\frac{N(\mathfrak{b})}{X}\right)\Lambda(\mathfrak{a})\Lambda(\mathfrak{b})\int_{0}^{\frac{\pi}{2}}F_{K}(\theta_{\mathfrak{a}}-\theta)F_{K}(\theta_{\mathfrak{b}}-\theta)d\theta.\\ \end{split}$$

$$\begin{gathered} ⟨{\psi }_{K,X}⟩\sim \frac{X}{K}{\int }_{\mathbb{R}}f(x)dx{\int }_0^{\infty }\Phi (u)du \\ =O\left(\frac{X}{K}\right), \end{gathered}$$

$$\begin{array}{rl}\text{Var}({\psi }_{K,X})& ={\Omega }_{K,X}-⟨{\psi }_{K,X}{⟩}^2\\ & ={\Omega }_{K,X}+O\left(X^{1-ϵ}\right),\end{array}$$

$$|{\theta }_{\mathfrak{a}}-{\theta }_{\mathfrak{b}}|\ge \frac{1}{X}\gg \frac{1}{K}.$$

$${\int }_0^{\pi /2}{F}_K({\theta }_{\mathfrak{a}}-\theta )F_K({\theta }_{\mathfrak{b}}-\theta )d\theta$$

$$\displaystyle\begin{split}\Omega_{K,X}&=\frac{2}{\pi}\sum_{\begin{subarray}{c}\mathfrak{a}\subset\mathbb{Z}[i]\\ \theta_{\mathfrak{a}}\neq 0\end{subarray}}\Phi\left(\frac{N(\mathfrak{a})}{X}\right)^{2}\Lambda^{2}(\mathfrak{a})\int_{0}^{\frac{\pi}{2}}F_{K}(\theta)^{2}d\theta\\ &\phantom{=}+\frac{2}{\pi}\bigg{|}\sum_{\begin{subarray}{c}\mathfrak{a}\subset\mathbb{Z}[i]\\ \theta_{\mathfrak{a}}=0\end{subarray}}\Phi\left(\frac{N(\mathfrak{a})}{X}\right)\Lambda(\mathfrak{a})\bigg{|}^{2}\int_{0}^{\frac{\pi}{2}}F_{K}(\theta)^{2}d\theta.\end{split}$$

$$\begin{array}{rl}\frac{2}{\pi }{\int }_0^{\frac{\pi }{2}}|F_K(\theta ){|}^{2}d\theta & =\sum _{k\in \mathbb{Z}}|{\hat{F}}_K(k){|}^{2}d\theta =\frac{1}{K^2}\sum _{k\in \mathbb{Z}}\hat{f}{\left(\frac{k}{K}\right)}^2=4{\pi }^2\frac{C_f}{K},\end{array}$$

$$\displaystyle\begin{split}\Omega_{K,X}&=4\pi^{2}\frac{C_{f}}{K}\left(\sum_{\begin{subarray}{c}\mathfrak{a}\subset\mathbb{Z}[i]\\ \theta_{\mathfrak{a}}\neq 0\end{subarray}}\Phi\left(\frac{N(\mathfrak{a})}{X}\right)^{2}\Lambda^{2}(\mathfrak{a})+\bigg{|}\sum_{\begin{subarray}{c}\mathfrak{a}\subset\mathbb{Z}[i]\\ \theta_{\mathfrak{a}}=0\end{subarray}}\Phi\left(\frac{N(\mathfrak{a})}{X}\right)\Lambda(\mathfrak{a})\bigg{|}^{2}\right).\end{split}$$

$$\sum_{\begin{subarray}{c}\mathfrak{a}\subset\mathbb{Z}[i]\\ \theta_{\mathfrak{a}}\neq 0\end{subarray}}\Phi\left(\frac{N(\mathfrak{a})}{X}\right)^{2}\Lambda^{2}(\mathfrak{a})=\frac{1}{4\pi^{2}}\bigg{(}C_{\Phi}X\cdot\log X-XC^{\prime}_{\Phi}\bigg{)}+O_{\Phi}\left(Xe^{-c\cdot\sqrt{\log X}}\right),$$

$$\bigg{|}\sum_{\begin{subarray}{c}\mathfrak{a}\subset\mathbb{Z}[i]\\ \theta_{\mathfrak{a}}=0\end{subarray}}\Lambda(\mathfrak{a})\Phi\left(\frac{N(\mathfrak{a})}{X}\right)\bigg{|}^{2}=\frac{X}{4}\left(\tilde{\Phi}\left(\frac{1}{2}\right)\right)^{2}+O_{\Phi}\left(Xe^{-c\cdot\sqrt{\log X}}\right),$$

$$\begin{array}{rl}\sum _{\begin{array}{c}\mathfrak{a}\subset \mathbb{Z}[i]\\ {\theta }_{\mathfrak{a}}\ne 0\end{array}}& \Phi {\left(\frac{N(\mathfrak{a})}{X}\right)}^2{\Lambda }^2(\mathfrak{a})=\sum _{\mathfrak{p}|p\equiv 1(4)}\sum _{n=1}^{\infty }\Phi {\left(\frac{N({\mathfrak{p}}^n)}{X}\right)}^2{\Lambda }^2(\mathfrak{p})+\sum _{m=0}^{\infty }\Phi {\left(\frac{2^{2m+1}}{X}\right)}^2(\log 2{)}^2\\ & =\sum _{p\equiv 1(4)}2\cdot \Phi {\left(\frac{p}{X}\right)}^2(\log p{)}^2+\sum _{\mathfrak{p}|p\equiv 1(4)}\sum _{n=2}^{\infty }\Phi {\left(\frac{N({\mathfrak{p}}^n)}{X}\right)}^2{\Lambda }^2(\mathfrak{p})+O_{\Phi }\left(\log X\right),\end{array}$$

$$\sum _{\mathfrak{p}|p\equiv 1(4)}\Phi {\left(\frac{N({\mathfrak{p}}^n)}{X}\right)}^2{\Lambda }^2(\mathfrak{p})$$

$$\sum _{\mathfrak{p}|p\equiv 1(4)}\sum _{n=2}^{\infty }\Phi {\left(\frac{N({\mathfrak{p}}^n)}{X}\right)}^2{\Lambda }^2(\mathfrak{p})\ll \sum _{n=2}^{\log X}{X}^{\frac{1}{n}+ϵ}=O_{\Phi }\left(X^{\frac{2}{3}}\right),$$

$$\sum _{\begin{array}{c}\mathfrak{a}\subset \mathbb{Z}[i]\\ {\theta }_{\mathfrak{a}}\ne 0\end{array}}\Phi {\left(\frac{N(\mathfrak{a})}{X}\right)}^2{\Lambda }^2(\mathfrak{a})=\sum _{p\equiv 1(4)}2\cdot \Phi {\left(\frac{p}{X}\right)}^2(\log p{)}^2+O_{\Phi }\left(X^{\frac{2}{3}}\right).$$

$$f(t):=\log t\cdot \Phi {\left(\frac{t}{X}\right)}^2$$

$$a_{p}:=\left\{\begin{array}[]{l l}2\cdot\log p&\text{ if }p\equiv 1(4)\\ 0&\text{ otherwise,}\end{array}\right.$$

Full text

1. Introduction

Consider the ring of Gaussian integers $\mathbb{Z}[i]$, which is the ring of integers of the imaginary quadratic field $\mathbb{Q}(i)$. Let $\mathfrak{a}=\langle\alpha\rangle$ be an ideal in $\mathbb{Z}[i]$ generated by the Gaussian integer $\alpha\in\mathbb{Z}[i]$. The norm of the ideal $\mathfrak{a}$ is defined as $N(\mathfrak{a}):=\alpha\cdot\overline{\alpha}$, where $\alpha\mapsto\overline{\alpha}$ denotes complex conjugation. Let $\theta_{\alpha}$ denote the argument of $\alpha$. Since $\mathbb{Z}[i]$ is a principal ideal domain, and the generators of $\mathfrak{a}$ differ by multiplication by a unit $\{\pm 1,\pm i\}\in\mathbb{Z}[i]^{\times}$, we find that $\theta_{\mathfrak{a}}:=\theta_{\alpha}$ is well-defined modulo $\pi/2$. We may thus fix $\theta_{\mathfrak{a}}$ to lie in $[0,\pi/2)$, which corresponds to choosing a generator $\alpha$ that lies within the first quadrant of the complex plane.

We are interested in studying the angular distribution of $\{\theta_{\mathfrak{p}}\}\in[0,\pi/2)$, where $\mathfrak{p}\subsetneq\mathbb{Z}[i]$ are the collection of prime ideals with norm $N(\mathfrak{p})\leq X$. To optimize the accuracy of our methods, we employ several standard analytic techniques. In particular, we count the number of angles lying in a short segment of length $1/K$ in $[0,\pi/2]$ using a smooth window function, denoted by $F_{K}(\theta)$, and we count the number of ideals $\mathfrak{a}$ with norm $N(\mathfrak{a})\leq X$ using a smooth function, denoted by $\Phi$. We moreover count prime ideals using the weight provided by the Von Mangoldt function, defined as $\Lambda(\mathfrak{a})=\log N(\mathfrak{p})$ if $\mathfrak{a}=\mathfrak{p}^{r}$ is a power of a prime ideal $\mathfrak{p}$, and $\Lambda(\mathfrak{a})=0$ otherwise.

Let $f\in C_{c}^{\infty}(\mathbb{R})$ be an even, real-valued window function. For $K\gg 1$, define

[TABLE]

which is a $\pi/2$-periodic function whose support in $[0,\pi/2)$ is on a scale of $1/K$. The Fourier expansion of $F_{K}$ is given by

[TABLE]

where the normalization is defined to be $\widehat{f}\left(y\right):=\int_{\mathbb{R}}f(x)e^{-2\pi iyx}dx.$

Let $\Phi\in C_{c}^{\infty}(0,\infty)$ and denote the Mellin transform of $\Phi$ by

[TABLE]

Define

[TABLE]

where $\mathfrak{a}$ runs over all nonzero ideals in $\mathbb{Z}[i]$. We may then think of $\psi_{K,X}(\theta)$ as a smooth count for the number of prime power ideals less than $X$ lying in a window of scale $1/K$ about $\theta$. As in Lemma 3.1 of [16], the mean value of $\psi_{K,X}(\theta)$ is given by

[TABLE]

where

[TABLE]

For fixed $K>0$, then a smooth version of a result from Hecke [9] states that in the limit as $X\rightarrow\infty$,

[TABLE]

Alternatively, one may study the behavior of $\psi_{K,X}(\theta)$ for shrinking intervals, i.e. for large $K$. It follows from the work of Kubilius [13] that under the assumption of the Grand Riemann Hypothesis (GRH), (1.7) continues to hold for $K\ll X^{1/2-o(1)}$.

In this paper, we wish to study

[TABLE]

Such a quantity was investigated by Rudnick and Waxman [16], who, assuming GRH, obtained an upper bound for $\textnormal{Var}(\psi_{K,X})$.111See also [17]. They then used this upper bound to prove that almost all arcs of length $1/K$ contain at least one angle $\theta_{\mathfrak{p}}$ attached to a prime ideal with $N(\mathfrak{p})\leq K(\log K)^{2+o(1)}$.

Montogomery [14] showed that the pair correlation of zeros of $\zeta(s)$ behaves similarly to that of an ensemble of random matrices, linking the zero distribution of the zeta function to eigenvalues of random matrices. The Katz-Sarnak density conjecture [11, 12] extended this connection by relating the distribution of zeros across families of $L$-functions to eigenvalues of random matrices. Random matrix theory (RMT) has since served as an important aid in modeling the statistics of various quantities associated to $L$-functions, such as the spacing of zeros [10, 15, 18], and moments of $L$-functions [5, 6]. Motivated by a suitable RMT model for the zeros of a family of Hecke $L$-functions, as well as a function field analogue, Rudnick and Waxman conjectured that

[TABLE]

Inspired by calculations for the characteristic polynomials of matrices averaged over the compact classical groups, Conrey, Farmer, and Zirnbauer [3, 4] further exploited the relationship between $L$-functions and random matrices to conjecture a recipe for calculating the ratio of a product of shifted $L$-functions averaged over a family. The $L$-functions Ratios Conjecture has since been employed in a variety of applications, such as computing $n$-level densities across a family of $L$-functions, mollified moments of $L$-functions, and discrete averages over zeros of the Riemann Zeta function [7]. The Ratios Conjecture has also been extended to the function field setting [1]. While constructing a model using the Ratios Conjecture may pose additional technical challenges, the reward is often a more accurate model; RMT heuristics can model assymptotic behavior, but the Ratios Conjecture is expected to hold down to lower order terms. This has been demonstrated, for example, in the context of one-level density computations, by Fiorilli, Parks and Södergren [8].

This paper studies $\textnormal{Var}(\psi_{K,X})$ down to lower-order terms. Define a new parameter $\lambda$ such that $X^{\lambda}=K$. We prove the following theorem:

Theorem 1.1**.**

Fix $\lambda>1$. Then

[TABLE]

where

[TABLE]

and $C_{\Phi}$ is as in (1.6). Under GRH, the error term can be improved to $O_{\Phi}\left(X^{-\epsilon}\right)$ for some $\epsilon>0$ $($depending on $\lambda)$.

The proof of Theorem 1.1 is given in Section 2, and is obtained by classical methods. For $\lambda<1$ the computation is more difficult, and we use the Ratios Conjecture to suggest the following.

Conjecture 1.2**.**

Fix $0<\lambda<1$. We have

[TABLE]

where

[TABLE]

and

[TABLE]

for some constant $\epsilon>0$ $($depending on $\lambda)$. Here $C_{\Phi,\zeta}$, $C_{\Phi,L}$, and $A_{\Phi}^{\prime}$, are as in $(\ref{zeta constant})$, $(\ref{L constant})$, and $(\ref{A constant})$, respectively.

Conjecture 1.2 provides a refined conjecture for $\textnormal{Var}(\psi_{K,X})$ with a power saving error term (away from the bifurcation points). It moreover recovers the asymptotic prediction given by (1.9), which was initially obtained by completely different methods. Numerical data for $\textnormal{Var}(\psi_{K,X})$ is provided in Figure  1.

A saturation effect similar to the one above was previously observed by Bui, Keating, and Smith [2], when computing the variance of sums in short intervals of coefficients of a fixed $L$-function of high degree. There, too, the contribution from lower order terms must be taken into account in order to obtain good agreement with the numerical data.

A proof of Theorem 1.1 is provided in Section 2 below. When $\lambda>1$ the main contribution to the variance is given by the diagonal terms, which we directly compute by separately considering the weighted contribution of split primes (Lemma 2.1) and inert primes (Lemma 2.2). When $0<\lambda<1$ we may no longer trivially bound the off-diagonal contribution, and so we instead shift focus to the study of a relevant family of Hecke $L$-functions. In Section 3 we compute the ratios recipe for this family of $L$-functions, and in Section 4 we apply several necessary simplifications. Section 5 then relates the output of this recipe to $\textnormal{Var}(\psi_{K,X})$, resulting in Conjecture 5.1, which expresses $\textnormal{Var}(\psi_{K,X})$ in terms of four double contour integrals. Section 6 is dedicated to preliminary technical lemmas, and the double integrals are then computed in Sections 7$-$9. One finds that the main contributions to $\textnormal{Var}(\psi_{K,X})$ come from second-order poles, while first-order poles contribute a correction factor smaller than the main term by a factor of $\log X$.

The Ratios Conjectures moreover suggests an enlightening way to group terms. The first integral, which corresponds to taking the first piece of each approximate functional equation in the ratios recipe, corresponds to the contribution of the diagonal terms, computed in Theorem 1.1. In particular, we note that its contribution to $\textnormal{Var}(\psi_{K,X})$ is independent of the value of $\lambda$ (Lemma 5.2). In contrast, the contribution emerging from the second and third integrals depends on the value of $\lambda$ (Lemma 5.3). This accounts for the emergence of two bifurcation points in the lower order terms: one at $\lambda=1/2$ and another at $\lambda=1$. The fourth integral, corresponding to taking the second piece of each approximate functional equation in the ratios recipe, only makes a significantly contribution to $\textnormal{Var}(\psi_{K,X})$ when $\lambda<1/2$ (Lemma 5.4). This accounts for the bifurcation point in the main term, previously detected by the RMT model, as well as for the contribution of a complicated lower-order term, which appears to nicely fit the numerical data.

Acknowledgments: This work emerged from a summer project developed and guided by E. Waxman, as part of the 2017 SMALL Undergraduate Research Project at Williams College. We thank Zeev Rudnick for advice, and for suggesting this problem, as well as Bingrong Huang and J. P. Keating for helpful discussions. The summer research was supported by NSF Grant DMS1659037. Chen was moreover supported by Princeton University, and Miller was supported by NSF Grant DMS1561945. Waxman was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no 320755., as well as by the Czech Science Foundation GAČR, grant 17-04703Y.

2. Proof of Theorem 1.1

Recall that $X^{\lambda}=K$. To compute $\textnormal{Var}(\psi_{K,X})$ in the regime $\lambda>1$, it suffices to calculate the second moment, defined as

[TABLE]

Indeed, note that as in Lemma 3.1 of [16],

[TABLE]

so that for $\lambda>1$,

[TABLE]

where $\epsilon=2\lambda-1$.

Suppose $\mathfrak{a}\neq\mathfrak{b}$, and that at least one of $\theta_{\mathfrak{a}},\theta_{\mathfrak{b}}\neq 0$. Then by Lemma 2.1 in [16],

[TABLE]

Moreover, in order for the integral

[TABLE]

to be nonzero, we require that $\theta_{\mathfrak{a}}-\theta_{\mathfrak{b}}<\frac{\pi}{2K}$. Since $X=o(K)$, such off-diagonal terms contribute nothing, and the contribution thus only comes from terms for which $\theta_{\mathfrak{a}}=\theta_{\mathfrak{b}}$. We therefore may write

[TABLE]

By Parseval’s theorem we have that for sufficiently large $K$,

[TABLE]

and therefore

[TABLE]

Theorem 1.1 then follows from (2.3), (2.8), and the following two lemmas.

Lemma 2.1**.**

We have

[TABLE]

while under GRH, the error term has a power saving, say, to $O_{\Phi}\left(X^{2/3}\right)$.

Lemma 2.2**.**

Unconditionally we have that

[TABLE]

while, again, under GRH, the error term has a power saving.

Proof of Lemma 2.1:

Proof.

Consider the quantity

[TABLE]

where we note that since $\Phi$ is compactly supported, the sum on the far right has at most $O_{\Phi}\left(\log X\right)$ terms. Moreover,

[TABLE]

since the sum has at most $O_{\Phi}(X^{1/n})$ terms. It follows that

[TABLE]

and therefore

[TABLE]

Upon setting

[TABLE]

and

[TABLE]

it follows from Abel’s Summation Formula and the Prime Number Theorem that

[TABLE]

where the error term assumes RH. Applying the change of variables $u:=t/X$, we then obtain that for sufficiently large $X$,

[TABLE]

Under RH, the error term is then given as

[TABLE]

while unconditionally it is as in (2.9). Combining the results of (2.14), (2.17), (2.18), and (2.19), we then obtain Lemma 2.1. ∎

Proof of Lemma 2.2:

Proof.

Next, we consider the quantity

[TABLE]

Since

[TABLE]

we have that

[TABLE]

and therefore

[TABLE]

Moreover, since

[TABLE]

we obtain

[TABLE]

By the Mellin inversion theorem, we find that

[TABLE]

Let $\chi_{0}\in\left(\mathbb{Z}/4\mathbb{Z}\right)^{\times}$ denote the principal character, and $\chi_{1}\in\left(\mathbb{Z}/4\mathbb{Z}\right)^{\times}$ denote the non-principal character, with corresponding $L$-functions given by $L(s,\chi_{0})$ and $L(s,\chi_{1})$, respectively. Upon noting that

[TABLE]

we obtain

[TABLE]

It follows that

[TABLE]

Moreover, we compute

[TABLE]

where $\gamma_{0}$ is the Euler-Mascheroni constant, while $L^{\prime}/L(s,\chi_{1})$ is holomorphic about $s=1$. Shifting integrals, we pick up a pole at $s=1$ and find that

[TABLE]

for some $c>0$. Squaring this then yields

[TABLE]

As above, we note that under the assumption of GRH the error term can be improved to have a power-saving. ∎

3. Implementing the Ratios Conjecture

Throughout this section, and the remainder of the paper, we will assume GRH.

3.1. The Recipe

The $L$-Functions Ratios Conjecture described in [3], provides a procedure for computing an average of $L$-function ratios over a designated family. Let $\mathcal{L}(s,f)$ be an $L$-function, and $\mathcal{F}=\{f\}$ a family of characters with conductors $c(f)$, as defined in section 3 of [4]. $\mathcal{L}(s,f)$ has an approximate functional equation given by

[TABLE]

Moreover, one may write

[TABLE]

where the series converges absolutely for Re$(s)>1$. To conjecture an asymptotic formula for the average

[TABLE]

the Ratios Conjecture suggests the following recipe.

Step One: Start with

[TABLE]

Replace each $L$-function in the numerator with the two terms from its approximate functional equation, ignore the remainder terms and allow each of the four resulting sums to extend to infinity. Replace each $L$-function in the denominator by its series (3.2). Multiply out the resulting expression to obtain 4 terms. Write these terms as

[TABLE]

Step Two: Replace each product of $\epsilon(f,s)$ factors by its expected value when averaged over the family.

Step Three: Replace each summand by its expected value when averaged over the family.

Step Four: Call the total $M_{f}:=M_{f}(\alpha,\beta,\gamma,\delta)$, and let $F=|\mathcal{F}|$. Then for

[TABLE]

and

[TABLE]

the conjecture is that

[TABLE]

for all $\epsilon>0$, where $g$ is a suitable weight function.

3.2. Hecke $L$-functions

We are interested in applying the ratios recipe to the following family of $L$-functions. Consider the Hecke character

[TABLE]

which provides a well-defined function on the ideals of $\mathbb{Z}[i]$. To each such character we may associate an $L$-function

[TABLE]

Note that $L_{k}(s)=L_{-k}(s)$, and that

[TABLE]

Moreover, when $k\neq 0$, then $L_{k}(s)$ has an analytic continuation to the entire complex plane, and satisfies the functional equation

[TABLE]

3.3. Step One: Approximate Function Equation

We seek to apply the above procedure to compute the average

[TABLE]

for specified values of $\alpha,\beta,\gamma,\delta$. For this particular family of $L$-functions, we have

[TABLE]

and

[TABLE]

which is a multiplicative function defined explicitly on prime powers by

[TABLE]

where, for prime $p\equiv 1(4)$, we define $\theta_{p}:=\theta_{\mathfrak{p}}$, where $\mathfrak{p}\subset\mathbb{Z}[i]$ is a prime ideal lying above $p$. Note, moreover, that the above formula is independent of our specific choice of $\mathfrak{p}$.

As per the recipe, we ignore the remainder term and allow both terms in the approximate functional equation to be summed to infinity. This allows us to write

[TABLE]

upon noting that $\overline{A_{k}(n)}=A_{k}(n)$ for all $A_{k}(n)$.

To compute the inverse coefficients, write

[TABLE]

We then obtain

[TABLE]

where

[TABLE]

Multiplying out the resulting expression gives

[TABLE]

[TABLE]

where the above follows upon noting that

[TABLE]

The algorithm now dictates that we compute the $\Gamma$-average

[TABLE]

as well as an average for the quantity coming from the first piece of each functional equation, namely

[TABLE]

Here we write $\langle\cdot\rangle_{K}$ to denote the average over all $0<|k|\leq K$. The average of the remaining three pieces will then follow similarly upon applying the appropriate change of variables.

3.4. Step Two: Averaging the Gamma Factors

The gamma factor averages over the family of Hecke $L$-functions are provided by the following lemma.

Lemma 3.1**.**

Fix $0<\alpha,\beta<\frac{1}{2}$. We find that

[TABLE]

and similarly

[TABLE]

Proof.

A proof of (3.26) is given in [19], and the proof of (3.27) is essentially identical. Specifically, one may use Stirling’s approximation and Taylor expansion to demonstrate that

[TABLE]

and then average over $0<|k|\leq K$ to obtain (3.27). ∎

3.5. Step Three: Coefficient Average

In this section, we seek to compute the coefficient average

[TABLE]

To do so, we must consider several cases depending on the value of $p$ mod 4. Define

[TABLE]

and write

[TABLE]

3.5.1.

p $\equiv$ 1(4): By (3.20), we may restrict to the case in which $h,l\in\{0,1,2\}$. If $h,l\in\{0,2\}$, then $\delta_{1(4)}(m,n,h,l)$ reduces to $\left<A_{k}(p^{m})A_{k}(p^{n})\right>_{K}$, where

[TABLE]

Expanding the product $A_{k}(p^{m})A_{k}(p^{n})$ yields a double sum of points on the unit circle, and averaging over $k\leq K$ then eliminates, in the limit, any such terms which are not identically equal to 1. Collecting the significant terms, we find that

[TABLE]

If either $h=1$ and $l\in\{0,2\}$, or $l=1$ and $h\in\{0,2\}$, then the product $\mu_{k}(p^{h})\mu_{k}(p^{l})=-A_{k}(p)=-(e^{4ki\theta_{p}}+e^{-4ki\theta_{p}})$, so that (3.29) reduces to

[TABLE]

Expanding out this product yields again a sum of points on the unit circle, which upon averaging over $k\leq K$ eliminates, in the limit, any such terms not identically equal to 1. We then obtain

[TABLE]

Finally, suppose $h=l=1$. In this case, the product $\mu_{k}(p^{h})\mu_{k}(p^{l})=A_{k}(p)^{2}=e^{2\cdot 4ki\theta_{p}}+2+e^{-2\cdot 4ki\theta_{p}}$, so that (3.29) reduces to

[TABLE]

Collecting significant contributions as before, we conclude that

[TABLE]

3.5.2.

p $\equiv$ 3(4): Again we may restrict to the case in which $h,l\in\{0,2\}$. If $h=l\in\{0,2\}$, then $\mu_{k}(p^{h})\mu_{k}(p^{l})=1$, and therefore

[TABLE]

Likewise, if $(h,l)=(0,2)$ or $(h,l)=(2,0)$ then $\mu_{k}(p^{h})\mu_{k}(p^{l})=-1$ and

[TABLE]

3.5.3.

p = 2: When $p=2$, we may restrict to the case in which $h,l\in\{0,1\}$. If, moreover, $h=l$, then

[TABLE]

while if $h\neq l$,

[TABLE]

3.5.4.

Summary: Summarizing the above results, we then conclude that

[TABLE]

3.6. Step Four: Conjecture

Upon applying the averages, the Ratios Conjecture recipe claims that for $\alpha,\beta,\gamma,\delta$ satisfying the conditions specified in (3.6), we have

[TABLE]

where

[TABLE]

and

[TABLE]

4. Simplifying the Ratios Conjecture Prediction

In this section we seek a simplified form of $M_{K}(\alpha,\beta,\gamma,\delta)$. First, we again consider several separate cases, depending on the value of $p$ mod 4.

4.1. Pulling out Main Terms

Suppose $p\equiv 3(4)$. By (3.42), we expand each local factor as

[TABLE]

Assuming small positive fixed values of $\text{Re}(\alpha),\text{Re}(\beta),\text{Re}(\gamma),\text{Re}(\delta)$, we factor out all terms which, for fixed $p$, converge substantially slower than $1/p^{2}$ and note that

[TABLE]

In fact we write

[TABLE]

where

[TABLE]

and $A_{p}(\alpha,\beta,\gamma,\delta):=G_{p}(\alpha,\beta,\gamma,\delta)/Y_{p}(\alpha,\beta,\gamma,\delta)$ is another local function, which converges like $1/p^{2}$ for sufficient small $\text{Re}(\alpha),\text{Re}(\beta),\text{Re}(\gamma),$ and $\text{Re}(\delta)$.

Next, suppose $p\equiv 1(4)$. Factoring out terms with slow convergence as above, we expand $G_{p}(\alpha,\beta,\gamma,\delta)$ as

[TABLE]

Since

[TABLE]

[TABLE]

and

[TABLE]

we conclude that, for $p\equiv 1(4)$, we may write

[TABLE]

where

[TABLE]

and $A_{p}(\alpha,\beta,\gamma,\delta)$ is a function that converges sufficiently rapidly.

Finally, note that

[TABLE]

We therefore may write

[TABLE]

where

[TABLE]

and $A_{2}(\alpha,\beta,\gamma,\delta):=G_{2}(\alpha,\beta,\gamma,\delta)/Y_{2}(\alpha,\beta,\gamma,\delta)$.

4.2. Expanding the Euler Product

Recall that for Re$(x)>0$,

[TABLE]

and

[TABLE]

where $L(s):=L(s,\chi_{1})$. Incorporating the above simplifications, and again collecting only terms which converge substantially slower that $p^{-3/2}$, we arrive at the following conjecture.

Conjecture 4.1**.**

With constraints on $\alpha,\beta,\gamma,\delta$ as described in (3.6) and (3.7), we have

[TABLE]

where

[TABLE]

[TABLE]

and $A(\alpha,\beta,\gamma,\delta):=\prod_{p}A_{p}(\alpha,\beta,\gamma,\delta)$ is an Euler product that converges for sufficiently small fixed values of $\textnormal{Re}(\alpha),\textnormal{Re}(\beta),\textnormal{Re}(\gamma),\textnormal{Re}(\delta)$.

In further calculations, it will be helpful to define

[TABLE]

as well as

[TABLE]

It will also be necessary to make use of the following lemma.

Lemma 4.2**.**

We have that

[TABLE]

Proof.

Since $Y(\alpha,\beta,\alpha,\beta)=\mathcal{Y}(\alpha,\beta,\alpha,\beta)=1$, it suffices to show that $\hfill\break G(\alpha,\beta,\alpha,\beta)=1$. Note that $G_{2}(\alpha,\beta,\alpha,\beta)=1$, and upon writing

[TABLE]

we similarly obtain that $G_{p}(\alpha,\beta,\alpha,\beta)=1$ whenever $p\equiv 3(4)$. Moreover, we rewrite

[TABLE]

and

[TABLE]

as well as

[TABLE]

and

[TABLE]

so that for $p\equiv 1(4)$,

[TABLE]

Upon setting $\alpha=\gamma$ and $\beta=\delta$, we then have $G_{p}(\alpha,\beta,\alpha,\beta)=1$. The lemma then follows from (4.16). ∎

Lemma 4.3**.**

Define $A_{\beta}(\alpha):=A(-\alpha,-\beta,\alpha,\beta).$ Then

[TABLE]

Proof.

Write

[TABLE]

where

[TABLE]

are the local factors of $A_{\beta}(\alpha)$, and note that $p_{\beta}(-\beta)=1$ at each prime $p$. By the product rule,

[TABLE]

Note that

[TABLE]

so that

[TABLE]

from which the result follows. ∎

5. The Ratios Conjecture Prediction for $\text{Var}(\psi_{K,X})$:

Let $F_{K}(\theta)$ be as in (1.1). By the Fourier expansion of $F_{K}$, we may write

[TABLE]

Since the mean value is given by the zero mode $k=0$, the variance may be computed as

[TABLE]

By applying the Mellin Inversion Formula

[TABLE]

we obtain

[TABLE]

Inserting this into (5.2), we find that

[TABLE]

Upon recalling that

[TABLE]

$\text{Var}(\psi_{K,X})$ can be restricted to terms for which the Fourier coefficients are equal, i.e.,

[TABLE]

by Fubini’s theorem. Moreover, under GRH, $\frac{L_{k}^{\prime}}{L_{k}}(s)$ is holomorphic in the half-plane Re$(s)>\frac{1}{2}$, and thus we may shift the vertical integrals to Re$(s)=\frac{1}{2}+\epsilon$, and Re$(s^{\prime})=\frac{1}{2}+\epsilon^{\prime}$, for any $\epsilon,\epsilon^{\prime}>0$. Upon making the change of variables $\alpha:=s-\frac{1}{2}$ and $\beta:=s^{\prime}-\frac{1}{2}$ we find that

[TABLE]

Note by (3.7) that the substitution of the ratios conjecture is only valid when Im$(\alpha)$, Im$(\beta)\ll_{c}K^{1-c}$, for small $c>0$. If either Im$(\alpha)>K^{1-c}$ or Im$(\beta)>K^{1-c}$, we use the rapid decay of $\tilde{\Phi}$, as well as upper bounds on the growth of $\frac{L^{\prime}_{k}}{L_{k}}$ within the critical strip, to show that the contribution to the double integral coming from these tails is bounded by $O_{c}\left(K^{-1+c}\right)$. For Im$(\alpha)$, Im$(\beta)<K^{1-c}$, we take the derivative of (3.43) to obtain

[TABLE]

where222Here, and elsewhere, we allow for a slight abuse of notation: $\alpha$ and $\beta$ denote coordinates of $M_{K}$, as well as coordinates of the point at which the derivative is then evaluated.

[TABLE]

Plugging (5.11) into (5.10) for Im$(\alpha)$, Im$(\beta)<K^{1-c}$, and using a similar argument as above to bound the tails, we then arrive at the following conjecture:

Conjecture 5.1**.**

We have that

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

Conjecture 1.2 now follows from Conjecture 5.1 as a consequence of the following three lemmas:

Lemma 5.2**.**

We have

[TABLE]

Lemma 5.3**.**

We have

[TABLE]

where $\epsilon>0$ is a constant $($depending on $\lambda)$.

Lemma 5.4**.**

We have

[TABLE]

where

[TABLE]

Here $\epsilon>0$ is a constant $($depending on $\lambda)$, and $C_{\Phi,\zeta}$, $C_{\Phi,L}$, and $A_{\Phi}^{\prime}$, are as in $(\ref{zeta constant})$, $(\ref{L constant})$, and $(\ref{A constant})$, respectively.

Conjecture 1.2 follows upon inserting the results from Lemma 5.2, Lemma 5.3, and Lemma 5.4, into Conjecture 5.1. Note that when $\lambda>1$, Conjecture 5.1 moreover agrees with Theorem 1.1.

6. Auxiliary Lemmas

Before proceeding to the proofs of Lemmas 5.2, 5.3, and 5.4, we will prove a few auxiliary lemmas that will be used frequently in the rest of the paper.

Lemma 6.1**.**

Let $h(\alpha)$ be holomorphic in $\Omega:=\left\{-\frac{1}{4}<\textnormal{Re}(\alpha)<\epsilon\right\}$ for some $\epsilon>0$, except for possibly at a finite set of poles. Moreover, suppose that $h(\alpha)$ does not grow too rapidly in $\Omega$, i.e., there exists a fixed $d>0$ such that $h(\alpha)\ll|\alpha|^{d}$ away from the poles in $\Omega$. Set

[TABLE]

where $\alpha,\beta,$ and $\tilde{\Phi}$ are as above. Then

[TABLE]

where $\textnormal{Res}(f,a_{k})$ denotes the residue of $f$ at each pole $a_{k}\in\Omega$.

Proof.

Consider the contour integral drawn counter-clockwise along the closed box

[TABLE]

where

[TABLE]

By Cauchy’s residue theorem,

[TABLE]

Set $\alpha=\sigma+iT$. By the properties of the Mellin transform, we find that for any fixed $A>0$,

[TABLE]

Since moreover $h(\alpha)$ does not grow too rapidly, we bound

[TABLE]

so that

[TABLE]

and similarly

[TABLE]

Finally, we bound

[TABLE]

from which the theorem then follows. ∎

Lemma 6.2**.**

Let $\alpha,\beta,\tilde{\Phi}$ be as above. Suppose $h(\alpha,\beta)$ is holomorphic333A function $f:\Omega\subset\mathbb{C}^{2}\mapsto\mathbb{C}$ is said to be homolorphic if it is holomorphic in each variable separately. in the region

[TABLE]

for some $\epsilon>0$, and moreover that $h(\alpha,\beta)$ does not grow too rapidly in $\Omega\times\Omega$, i.e., does not grow too rapidly in each variable, separately. Then

[TABLE]

Proof.

Set

[TABLE]

where $h_{\beta}(\alpha):=h(\alpha,\beta)$. Since $f_{\beta}$ is holomorphic, by an application of Lemma 6.1 we write

[TABLE]

where $g$ does not grow too rapidly as a function of $\beta$. By another application of Lemma 6.1, it then follows that

[TABLE]

∎

Lemma 6.3**.**

Let $\alpha,\beta$,$\tilde{\Phi}$, and $f_{\beta}$ be as above. Suppose $f_{\beta}(\alpha)$ has a finite pole at $a_{k}(\beta)$ with residue $\textnormal{Res}(f_{\beta},a_{k}(\beta))$. Moreover, suppose that for each $a_{k}(\beta)$, $\textnormal{Res}(f_{\beta},a_{k}(\beta))$ is holomorphic in $\Omega:=\left\{-\frac{1}{4}<\textnormal{Re}(\beta)<\epsilon\right\}$ for some $\epsilon>0$, and that $\textnormal{Res}(f_{\beta},a_{k}(\beta))$ does not grow too rapidly in $\Omega$. Then

[TABLE]

Proof.

By Lemma 6.1, we write

[TABLE]

where, as in the proof of Lemma 6.2, we explicitly note the dependence of the error term on $\beta$. Applying Lemma 6.2 to the error term in (6.21), we obtain

[TABLE]

and finally by another application of Lemma 6.1,

[TABLE]

∎

Lemma 6.4**.**

Let $C_{\Phi}$ and $C^{\prime}_{\Phi}$ be as in $(\ref{mean value constants})$ and $(\ref{C constant 1})$, respectively. Then

[TABLE]

and

[TABLE]

Proof.

Set $\phi(y)=\Phi(e^{y})e^{y/2}$ so that

[TABLE]

and similarly $\tilde{\Phi}\left(\frac{1}{2}-it\right)=\widehat{\phi}\left(\frac{t}{2\pi}\right).$ By shifting the integral to Re$(\beta)=0$ we obtain

[TABLE]

Since $\overline{\widehat{\phi}\left(-\frac{t}{2\pi}\right)}=\widehat{\phi}\left(\frac{t}{2\pi}\right)$, we moreover have that

[TABLE]

i.e.,

[TABLE]

Next, note that

[TABLE]

Upon setting $g(y)=y\cdot\Phi(e^{y})e^{y/2}$, we write

[TABLE]

so that by shifting to the half-line Re$(\beta)=1/2$, it follows that

[TABLE]

∎

7. Proof of Lemma 5.2

In this section we seek to compute

[TABLE]

Note that

[TABLE]

where we recall that $\tilde{\mathcal{A}}(\alpha,\beta,\alpha,\beta)=1$. Since

[TABLE]

is holomorphic in $\Omega\times\Omega$, by Lemma 6.2 we find that the integral corresponding to this term is bounded by $O\left(X^{-2/5}\right)$. Moreover, by an application of Lemma 6.3, the integrals corresponding to

[TABLE]

are each bounded by $O\left(X^{-1/5}\right)$. The main contributions to (7.1) thus come from

[TABLE]

and we now proceed to separately compute each of the three corresponding integrals.

7.1. Computing $\frac{\zeta^{\prime\prime}}{\zeta}(1+\alpha+\beta):$

The first double integral we would like to compute is

[TABLE]

where

[TABLE]

Since $f_{(\ref{1st})}$ has one double pole at $\alpha=-\beta$, it follows from Lemma 6.1 that

[TABLE]

To compute $\text{Res}(f_{(\ref{1st})},-\beta)$, we split $f_{(\ref{1st})}(\alpha)$ into two parts.

i) First, we expand $\frac{\zeta^{\prime\prime}}{\zeta}(1+\alpha+\beta)$ about the point $\alpha=-\beta$, yielding

[TABLE]

where $\gamma_{i}$ are Stieltjes constants, not to be confused with the variable $\gamma$ used previously.

ii) Next, we expand $g(\alpha)=\tilde{\Phi}\left(\frac{1}{2}+\alpha\right)X^{\alpha}$ about the point $\alpha=-\beta$. Since

[TABLE]

it follows that

[TABLE]

Multiplying the two Taylor expansions above, we find that

[TABLE]

and therefore

[TABLE]

By an application of Lemma 6.1, it follows that

[TABLE]

i.e.,

[TABLE]

7.2. Computing $-\frac{\zeta^{\prime}}{\zeta}(1+\alpha+\beta)^{2}$

Next, we are interested in the integral

[TABLE]

where

[TABLE]

Since $f_{(\ref{2nd})}(\alpha)$ has a single pole at $\alpha=-\beta$, it follows from Lemma 6.1 that

[TABLE]

To determine the residue of this integral at the point $\alpha=-\beta$, we expand $\frac{\zeta^{\prime}}{\zeta}(1+\alpha+\beta)^{2}$ and $g(\alpha):=\tilde{\Phi}\left(\frac{1}{2}+\alpha\right)X^{\alpha}$ about the point $\alpha=-\beta$, yielding

[TABLE]

and

[TABLE]

so that

[TABLE]

It follows that

[TABLE]

from which we obtain

[TABLE]

7.3. Computing $\left(\frac{\zeta^{\prime}}{\zeta}(1+2\alpha)\right)\left(\frac{\zeta^{\prime}}{\zeta}(1+2\beta)\right)$

Next we are interested in the integral

[TABLE]

where

[TABLE]

Since

[TABLE]

$f$ has a simple pole at $\alpha=0$ with residue

[TABLE]

It thus follows from Lemma 6.1 that

[TABLE]

and similarly

[TABLE]

from which we conclude that

[TABLE]

Lemma 5.2 then follows upon combing the results of (7.14), (7.21), and (7.28).

8. Proof of Lemma 5.3

Next, we consider the quantity

[TABLE]

coming from the integral $I_{2}$, as well as the symmetric quantity

[TABLE]

coming from the integral $I_{3}$. As before, we approach this term by term, and note that by an application of Lemma 6.3, the integrals over

[TABLE]

may be bounded by $O\left(X^{-\frac{1}{5}}\right)$. Significant contributions then come from integration against the following integrands:

$i)-\frac{\zeta^{\prime}}{\zeta}(1+2\beta)$ and $-\frac{\zeta^{\prime}}{\zeta}(1+2\alpha)$,

$ii)-\frac{\zeta^{\prime}}{\zeta}(1-\alpha+\beta)$ and $-\frac{\zeta^{\prime}}{\zeta}(1+\alpha-\beta)$,

$iii)\hskip 5.69054pt2\cdot\frac{\zeta^{\prime}}{\zeta}(1+\alpha+\beta)$.

8.1. Computing $-\frac{\zeta^{\prime}}{\zeta}(1+2\beta)$ and $-\frac{\zeta^{\prime}}{\zeta}(1+2\alpha)$:

Combining the discussion above with (5.1), we seek to compute the following integral:

[TABLE]

where

[TABLE]

Note that since

[TABLE]

$f_{(\ref{4th})}$ has a simple pole at $\beta=0$ with residue

[TABLE]

so that by Lemma 6.1,

[TABLE]

Inserting this back into the outer integral, we find that

[TABLE]

where

[TABLE]

If $\lambda>\frac{1}{2}$, we shift to the vertical line Re$(\alpha)=1/5$, so that

[TABLE]

Since the integrand decays rapidly as a function of $t$, the integral is bounded absolutely by a constant that is independent of $\lambda$. It follows that for any fixed $\lambda>\frac{1}{2}$,

[TABLE]

If $\lambda<\frac{1}{2}$ we shift to the vertical line Re$(\alpha)=-1/5$, pick up a residue at $\alpha=0$, and bound the remaining contour by $O\left(X^{\left(-\frac{1}{5}\right)\left(1-2\lambda\right)}\right)$. Since

[TABLE]

the residue is given by

[TABLE]

where we make use of Lemma 4.2. Since

[TABLE]

it follows that

[TABLE]

Upon including the contribution from the integral over $-\frac{\zeta^{\prime}}{\zeta}(1+2\alpha)$ coming from the third piece of the Ratios Conjecture, we conclude that the combined contribution from these two symmetric pieces together is equal to

[TABLE]

8.2. Computing $-\frac{\zeta^{\prime}}{\zeta}(1-\alpha+\beta)$ and $-\frac{\zeta^{\prime}}{\zeta}(1+\alpha-\beta)$

In this section we assume that $0<\textnormal{Re}(\alpha)<\textnormal{Re}(\beta)=\epsilon^{\prime}$. The integral that we are interested in computing is

[TABLE]

where

[TABLE]

Recalling that

[TABLE]

we find that $f_{(\ref{5th})}$ has a simple pole at $\alpha=\beta$. Under the assumption that $0<\textnormal{Re}(\alpha)<\textnormal{Re}(\beta)=\epsilon^{\prime}$, this pole is picked up upon shifting the contour to the line Re$(\alpha)=-1/5$, and the residue is

[TABLE]

It follows that

[TABLE]

where

[TABLE]

If $\lambda>1$, we shift to the vertical line Re$(\alpha)=1/5$, and bound

[TABLE]

while if $\lambda<1$, we shift to the vertical line Re$(\alpha)=-\frac{1}{5}$, pick up a pole at $\alpha=0$, and bound the remaining contour by $O\left(X^{\left(-1/5\right)\left(2-2\lambda\right)}\right)$. Since

[TABLE]

we conclude that

[TABLE]

Lastly, we consider the integral

[TABLE]

where

[TABLE]

which is the symmetry quantity corresponding to $I_{(\ref{5th},\textnormal{sym})}$ coming from (8.2) above. Under the assumption that $0<\textnormal{Re}(\alpha)<\textnormal{Re}(\beta)$, the inner integral is holomorphic in the region $-\frac{1}{5}<\textnormal{Re}(\alpha)<\epsilon$, from which it follows that

[TABLE]

Note that had we instead assumed $0<\textnormal{Re}(\beta)<\textnormal{Re}(\alpha)<1/5$, we would obtain a significant contribution from $I_{(\ref{5th},\textnormal{sym})}$ and a negligible contribution from $I_{(\ref{5th})}$. In this way, the symmetry between $\alpha$ and $\beta$ is preserved.

8.3. Computing $\frac{\zeta^{\prime}}{\zeta}(1+\alpha+\beta)$

Next, we compute

[TABLE]

where

[TABLE]

Since

[TABLE]

the residue at $\beta=-\alpha$ is

[TABLE]

It follows that

[TABLE]

and thus upon shifting the line of integration to Re$(\alpha)=1/5$, we conclude that

[TABLE]

Lemma 5.3 then follows upon combining the computations in (8.17), (8.26), (8.29), and (8.35).

9. Proof of Lemma 5.4

Since

[TABLE]

we write

[TABLE]

where

[TABLE]

Suppose $\lambda>1/2$. We then shift to the vertical line Re$(\alpha)=1/5$, so that

[TABLE]

By the decay properties of $\Phi$, the integral is bounded by a constant (depending on $\beta$) that is independent of $\lambda$. It follows that

[TABLE]

where $g$ does not grow too rapidly as a function of $\beta$. Inserting this back into the outer integral, and shifting the line of integration to Re$(\beta)=1/5$, we obtain

[TABLE]

Next, suppose $\lambda<1/2$. We shift the line of integration to $\textnormal{Re}(\alpha)=-1/5$, and pick up a simple at $\alpha=0$, and a double pole at $\alpha=-\beta$. By an application of Lemma 6.1, we then find

[TABLE]

It remains to compute these two residue contributions.

9.1. Simple Pole at $\alpha=0$:

Note that $f_{4}$ has a simple pole at $\alpha=0$ with residue

[TABLE]

which contributes when $\lambda<1/2$. Inserting this into the outer integral, we find that

[TABLE]

where

[TABLE]

The integral in (9.9) has a simple pole at $\beta=0$ with residue

[TABLE]

so that the total contribution from this pole is

[TABLE]

9.2. Double Pole at $\alpha=-\beta$:

To compute the residue of $f_{4}$ at the point $\alpha=-\beta$, we split $f_{4}(\alpha)$ into three components.

i) First, define

[TABLE]

Since $h(\alpha)$ is holomorphic at $\alpha=-\beta$, we may expand it as a power series of the form

[TABLE]

ii) Next, we expand

[TABLE]

about the point $\alpha=-\beta$, where

[TABLE]

The expansion is given as

[TABLE]

iii) Finally, we note that

[TABLE]

The total residue is then found to be the full coefficient of $(\alpha+\beta)^{-1}$, i.e.,

[TABLE]

We now compute these two contributions separately.

9.2.1. First Piece

The total contribution from the first piece is

[TABLE]

where we note that $\mathcal{A}(\beta,-\beta,-\beta,\beta)=1$. Inserting this into the outer integral of (9.2), we find that the main contribution of this piece is

[TABLE]

i.e., the total contribution is given by

[TABLE]

9.2.2. Second Piece

One directly computes

[TABLE]

upon noting that $A_{\beta}(-\beta)=A(\beta,-\beta,-\beta,\beta)=1.$ Inserting this expression back into the outer integral of (9.2), we find that the total contribution from this piece is

[TABLE]

where

[TABLE]

[TABLE]

and

[TABLE]

where we have made use of Lemma 4.3. Lemma 5.4 then follows upon combining the results of (9.12), (9.22), and (9.24).

Appendix A Obtaining Numerical Evidence for Conjecture 1.2

The data provided in Figure 1 was obtained using the Mathematica code provided below. Fix $X=10^{9}$, $\Phi=1_{(0,1]}$, and $f=1_{[-\frac{1}{2},\frac{1}{2}]}$. The code outputs $\mathrm{Var}(\psi_{K,X})/(\langle\psi_{K,X}\rangle\log X)$ as a function of $\lambda:=\log K/\log X$, for values of $\lambda$ ranging between $0.1\leq\lambda\leq 0.7$ with step size $0.025$. For simplicity, we ignore the small contributions coming from prime powers, as well as from the unique prime $(1+i)\subset\mathbb{Z}[i]$ lying above 2.

• In[1]:=

X = 10^9; ( This size took a long time for Mathematica to run.)A = 1; ( We count primes in Z[i] with norm from A to B )B = X;Roundmod[\mmaPat{m_}, \mmaPat{res_}, \mmaPat{N_}] = Ceiling[(\mmaPat{m} - \mmaPat{res})/\mmaPat{N}]\mmaPat{N} + \mmaPat{res}; ( An auxiliary function which finds the smallest integer n >= m such that n=res (mod N). )gauss = Take[   Ratios[Flatten[     Table[PowersRepresentations[p, 2, 2], {p,       Select[Range[Roundmod[A, 1, 4], B, 4], PrimeQ]}]]], {1, -1,*    2}];****    (* For primes p which are 1 modulo 4 between the specified ranges A and B, we compute the unique representation p = a^2 + b^2 for a,b nonnegative integers with a < b. Then we return the list of numbers b/a *)***gauss2 = Table[N[ArcTan[theta]], {theta, gauss}]; ( Using the list "gauss" we compute the angles associated to Gaussian primes (a + bi) lying over a rational prime congruent to 1 modulo 4, for 0 <= a < b. *)***gauss3 = Table[N[ArcTan[theta]], {theta, Table[1/gauss[[i]], {i, Length[gauss]}]}]; ( Using the list "gauss" we compute the angles associated to Gaussian primes (a + bi) lying over a rational prime congruent to 1 modulo 4, for 0 <= b < a. These are complex conjugates of the primes giving angles in the "gauss2" list. *)***primes1 = Select[Range[Roundmod[A, 1, 4], B, 4], PrimeQ]; ( We find the primes which are 1 modulo 4, between the ranges A and B. *)*primes3 = Select[Range[Roundmod[Sqrt[A], 3, 4], Sqrt[B], 4], PrimeQ]; ( We find the primes which are 3 modulo 4, between the ranges A and B. )trivial = Table[0., Length[primes3]]; ( The rational primes which are 3 modulo 4 remain prime in the Gaussian integers, and have an angle of zero. This list contains one zero for each prime congruent to 3 modulo 4, between A and B. )allAngles = Join[trivial, gauss2, gauss3]; ( This is a list, with multiplicity, of the angles of Gaussian primes with norm between A and B. By convention, the angle is in the interval [0,Pi/2). )allPrimes = N[Join[2 Log[primes3], Log[primes1], Log[primes1]]]; ( The elements of this list correspond to Gaussian primes P with norm between A and B. The Gaussian prime P appears as the number log(N(P)), which is the von Mangoldt function evaluated at P. Suppose P lies over a rational prime p. If p is 3 modulo 4 then N(P) = p^2, and P is the unique Gaussian prime lying over p. If p is 1 modulo 4, then we have N(P)=p and there is exactly one other Gaussian prime P’ lying over the same prime p. )anglesWeights = WeightedData[allAngles, allPrimes];Do[Print[{j,    Divide[Variance[      Last[HistogramList[        anglesWeights, {0, Divide[Pi, 2],         Divide[Pi, 2 Round[X^j]]}]]], X^{1 - j}Log[X]]}], {j, .1,  .7, .025}] ( This outputs pairs {lambda, Var(psi_{K,X})/(<psi_{K,X}> log(X))} for .1 <= lambda <= .7, with step size .025 for lambda.)

The following is used to compute a numerical approximation for $C_{\Phi,\zeta}$, when $\Phi=1_{(0,1]}$:

• In[2]:=

<< NumericalCalculus‘ ( imports a package that allows us to take numerical limits and derivatives )**PhiTilde[\mmaPat{s_}] := (1/\mmaPat{s}) ( Mellin transform of Phi. )PhiTildeProduct[\mmaPat{t_}] := PhiTilde[1/2 + I\mmaPat{t}]PhiTilde[1/2 - I\mmaPat{t}]ZetaPrime[\mmaPat{s_}] := ND[Zeta[t], t, \mmaPat{s}] ( Using Mathematica’s in-built Zeta function. We take a derivative )2PiI(I   NIntegrate[    PhiTildeProduct[t](ZetaPrime[1 + .2 + 2It]/Zeta[1 + .2 + 2It] +****       ZetaPrime[1 - .2 - 2It]/Zeta[1 - .2 - 2It]), {t, -25, 25}])**

The following is used to compute a numerical approximation for $C_{\Phi,L}$, when $\Phi=1_{(0,1]}$:

• In[3]:=

<< NumericalCalculus‘ ( imports a package that allows us to take numerical limits and derivatives )PhiTilde[\mmaPat{s_}] := (1/\mmaPat{s}) ( Mellin transform of Phi. )PhiTildeProduct[\mmaPat{t_}] := PhiTilde[1/2 + I\mmaPat{t}]PhiTilde[1/2 - I\mmaPat{t}]L[\mmaPat{s_}] := N[DirichletL[4, 2, \mmaPat{s}]] ( This is the Dirichlet L-function for the non-trivial character modulo 4. )LPrime[\mmaPat{s_}] := ND[L[t], t, \mmaPat{s}] ( Takes a derivative of the L-function. )-2PiI(I   NIntegrate[    PhiTildeProduct[t](LPrime[1 + .2 + 2It]/L[1 + .2 + 2It] +****       LPrime[1 - .2 - 2It]/L[1 - .2 - 2It]), {t, -25, 25}])**

The following is used to compute a numerical approximation for $A^{\prime}_{\Phi}$, when $\Phi=1_{(0,1]}$:

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h[β_, p_] = \displaystyle\frac{\text{\ (p^2\ -\ 2\ p^(4\ \color[rgb]{0,0.171875,0.765625}{β})\ +\ p^(2\ +\ 8\ \color[rgb]{0,0.171875,0.765625}{β}))\ Log[p]\ }}{\text{\ p^2\ -\ p^(4\ \color[rgb]{0,0.171875,0.765625}{β})\ -\ p^(4\ +\ 4\ \color[rgb]{0,0.171875,0.765625}{β})\ +\ p^(2\ +\ 8\ \color[rgb]{0,0.171875,0.765625}{β})\ }}

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*PhiTilde[\mmaPat{s_}] := (1/\mmaPat{s}) ( Mellin transform of Phi. *)***PhiTildeProduct[\mmaPat{t_}] := PhiTilde[1/2 + I\mmaPat{t}]PhiTilde[1/2 - I\mmaPat{t}]***qn[\mmaPat{p_}] = NIntegrate[h[I\mmaPat{t}, p]*PhiTildeProduct[\mmaPat{t}], {\mmaPat{t}, 0, Infinity}]*primes3 = Select[Range[3, 1000, 4], PrimeQ]; ( Selects the primes congruent to 3 modulo 4 which are less than 1000. )output = 0;For[i = 1, i <= Length[primes3], i++, output += 2qn[primes3[[i]]]]   Print["Range is ", j, ". Integral is ", output]