On the mod-Gaussian convergence of a sum over primes

Abstract

We prove mod-Gaussian convergence for a Dirichlet polynomial which approximates ${\mathrm{Im }}\log \zeta (1/2+it)$. This Dirichlet polynomial is sufficiently long to deduce Selberg’s central limit theorem with an explicit error term. Moreover, assuming the Riemann hypothesis, we apply the theory of the Riemann zeta-function to extend this mod-Gaussian convergence to the complex plane. From this we obtain that ${\mathrm{Im }}\log \zeta (1/2+it)$ satisfies a large deviation principle on the critical line. Results about the moments of the Riemann zeta-function follow.

Appendices

Appendix 1: Selberg’s result

In this appendix we briefly discuss Selberg’s result about the rate of convergence in the central limit theorem of ${\mathrm{Im }}\log \zeta (1/2+it)$ (see [19, Theorem 2] and [21, Theorem 6.2]). From Theorem 1 we deduce:

Lemma 2

Let $x=e^{\log T/N}$ and $N$ such that $x\rightarrow \infty$ and $N/\log \log T\rightarrow \infty$ as $T\rightarrow \infty$. Suppose further that $N/\log \log T=O(\log \log T)$. Then

$$\begin{aligned}&\sup _{a<b}\left( \ \frac{1}{T}\lambda \left( \Big \{t\in [T,2T]:\frac{1}{\sqrt{(\log \log x +\gamma )/2}}\sum _{p\le x}\frac{\sin (t\log p)}{\sqrt{p}}\in [a,b]\Big \}\right) \right. \nonumber \\&\quad \left. -\int \limits _a^be^{-t^2/2}\frac{dt}{\sqrt{2\pi }}\ \right) =O(1/\sqrt{\log \log T}). \end{aligned}$$

(8.1)

Proof

We denote by $\varPhi _n(u)$ the left hand side of (1.3). Using [5], XVI.3, formula 3.13] we can bound the left hand side of (8.1) by

$$\begin{aligned} \frac{2}{\pi }\int \limits _{-c\sqrt{\log \log x}}^{c\sqrt{\log \log x}} e^{-u^2/2}|(\varPhi _n(u/\sqrt{(\log \log x +\gamma )/2})-1)/u|du+O\left( \frac{1}{c\sqrt{\log \log x}}\right) . \end{aligned}$$

(8.2)

An inspection of the proof of Proposition 2 combined with (4.2) shows that $\varPhi _n(u)=\varPhi (u)(1+O(1/\log x))+O(1/\log T), \,|u|\le c$. If we choose $c>0$ such that $\varPhi (u)$ has no zeros for $|u|\le c$, we obtain $\varPhi _n(u)=\varPhi (u)(1+O(1/\log x)), \,|u|\le c$. On the other hand, we have $\varPhi (u/\sqrt{(\log \log x +\gamma )/2})=1+O(u^2/\log \log x), \, |u|\le c\sqrt{\log \log x}$. Plugging in these estimates gives that (8.2) is $O(1/\sqrt{\log \log x})$. From $N/\log \log T=O(\log \log T)$ we conclude that $\log \log T/\log \log x\rightarrow 1$ and this completes the proof. $\square$

This lemma combined with the bound (see [21, Lemma 6.2])

$$\begin{aligned} |\{t\in [T,2T]:|r_{1,x}(t)|\ge c'\log \log \log T\}|=O(1/\sqrt{\log \log T}), \end{aligned}$$

where $c'>0$ is a constant, yields Selberg’s result

$$\begin{aligned}&\sup _{a<b}\Bigg (\ \frac{1}{T}\lambda \Big (\Big \{t\in [T,2T]: \frac{{\mathrm{Im }}\log \zeta (1/2+it)}{\sqrt{(\log \log T)/2}}\in [a,b]\Big \}\Big )\\&\quad -\int \limits _a^be^{-t^2/2}\frac{dt}{\sqrt{2\pi }}\ \Bigg )=O\bigg (\frac{\log \log \log T}{\sqrt{\log \log T}}\bigg ). \end{aligned}$$

Appendix 2: Mean value estimates

For completeness we present some standard mean value estimates which we applied in the proof of Corollary 2 (see [17, Lemma 3] and [20, Lemma 3]). For this purpose let $x$ and $y$ be positive real numbers, $a_p$ and $b_p$ be complex numbers with $|a_p|\le 1$ and $|b_p|\le \log p/\log x$, and $k$ be a nonnegative integer. By repeating the arguments in the proof of Proposition 1, we obtain

$$\begin{aligned}&\frac{1}{T}\int \limits _T^{2T}\bigg |\sum _{p\le x}{\frac{a_p}{p^{1+2it}}}\bigg |^{2k}dt \le k!\Big (\sum _{p\le x}\frac{1}{p^2}\Big )^k+2D k!(\pi (x))^k/T,\\&\frac{1}{T}\int \limits _T^{2T}\bigg |\sum _{y< p\le x}{\frac{a_p}{p^{1/2+it}}}\bigg |^{2k}dt \le k!\Big (\sum _{y< p\le x}\frac{1}{p}\Big )^k+2D k!(\pi (x)-\pi (y))^k/T,\\&\frac{1}{T}\int \limits _T^{2T}\bigg |\sum _{p\le x}{\frac{b_p}{p^{1/2+it}}}\bigg |^{2k}dt \le k!\frac{1}{(\log x)^k}\Big (\sum _{p\le x}\frac{\log p}{p}\Big )^k+2D k!(\pi (x))^k/T. \end{aligned}$$

If $x\le T^{1/k}$, the first and the third term are bounded by $(Ak)^k$ and the second by $(k(\log \log x-\log \log y+A))^k, \,A>0$ some constant.

For example, we obtain for a function $|g(u)|\le 1$

$$\begin{aligned}&\frac{1}{T}\int \limits _T^{2T}\bigg | \frac{1}{\log T^{1/V}}\sum _{n\le T^{1/V}}\frac{\varLambda (n)}{n^{1/2+it}}g\left( \frac{\log n}{\log T^{1/V}}\right) \bigg |^{2\lfloor V\rfloor }dt\nonumber \\&\quad =\frac{1}{T}\int \limits _T^{2T}\bigg |\sum _{p\le T^{1/V}}{\frac{b_p}{p^{1/2+it}}}+\sum _{p^2\le T^{1/V}}{\frac{a_p}{p^{1+2it}}}+O(1)\bigg |^{2\lfloor V\rfloor }dt\nonumber \\&\quad \le 3^{2V}\big ((AV)^V+(AV)^V+O(1)^V\big ). \end{aligned}$$

(9.1)

Appendix 3: Large deviation theory

In this appendix we give the definition of the large deviation principle and state two important results which we used in the proofs of Corollary 2 and 3 (see [4]).

A function $I:\mathbb{R }\rightarrow [0,\infty ]$ is called a rate function (resp. good rate function), if for all $\alpha \in [0,\infty )$, the sets $\{x:I(x)\le \alpha \}$ are closed (resp. compact). A family $\{Z_\epsilon \}$ of real-valued random variables satisfies the large deviation principle with the speed $\epsilon$ and the rate function $I$, if

• (a) For any closed set $F\subseteq \mathbb{R }$

  $$\begin{aligned} \limsup _{\epsilon \rightarrow 0}\epsilon \log \mathbb{P }(Z_\epsilon \in F)\le -\inf _{x\in F}I(x). \end{aligned}$$

• (b) For any open set $G\subseteq \mathbb{R }$

  $$\begin{aligned} \liminf _{\epsilon \rightarrow 0}\epsilon \log \mathbb{P }(Z_\epsilon \in G)\ge -\inf _{x\in G}I(x). \end{aligned}$$

Theorem 3

(Gärtner-Ellis, see Theorem 2.3.6 or 4.5.20 in [4]) Suppose that for each $\lambda \in \mathbb{R }$

$$\begin{aligned} \varLambda (\lambda ):=\lim _{\epsilon \rightarrow 0}\epsilon \log \mathbb{E }\big [ e^{\lambda Z_{\epsilon }/\epsilon }\big ] \end{aligned}$$

exists and that $\varLambda$ is differentiable. Then the family $\{Z_\epsilon \}$ satisfies the large deviation principle with the good rate function $I(x)=\sup _{\lambda \in \mathbb{R }}(\lambda x-\varLambda (\lambda ))$.

Theorem 4

(Varadhan, see Theorem 4.3.1 in [4]) Suppose that $\{Z_\epsilon \}$ satisfies the large deviation principle with a good rate function $I$ and let $h\in \mathbb{R }$. Assume further that for some $\gamma >1$

$$\begin{aligned} \limsup _{\epsilon \rightarrow 0}\epsilon \log \mathbb{E }\big [ e^{\gamma h Z_{\epsilon }/\epsilon }\big ]<\infty . \end{aligned}$$

(10.1)

Then

$$\begin{aligned} \lim _{\epsilon \rightarrow 0}\epsilon \log \mathbb{E }\big [ e^{h Z_{\epsilon }/\epsilon }\big ]=\sup _{x\in \mathbb{R }}(xh-I(x)). \end{aligned}$$

Definition 1

(see Definition 4.2.10 in [4]) Let $\{Z_\epsilon \}$ and $\{{\tilde{Z}}_\epsilon \}$ be two families of real-valued random variables, defined on the same probability space. Then $\{Z_\epsilon \}$ and $\{{\tilde{Z}}_\epsilon \}$ are called exponentially equivalent if for each $\delta >0,$

$$\begin{aligned} \limsup _{\epsilon \rightarrow 0}\epsilon \log \mathbb{P }(|Z_\epsilon -{\tilde{Z}}_\epsilon |>\delta )=-\infty . \end{aligned}$$

(10.2)