Discretization of the maximum for the derivatives of a random model of $\log|\zeta|$ on the critical line

TL;DR

This paper investigates the maximum values of derivatives of a random model related to the logarithm of the Riemann zeta function on the critical line, revealing precise scaling behaviors and simplifying previous proofs.

Contribution

It extends previous results by analyzing the derivatives of the model, establishing their maximum's scale, and providing a simpler proof approach.

Findings

Maximum of derivatives varies on a specific logarithmic scale

Results improve and extend previous bounds from 2019

Progress towards the open problem of maximum tightness

Abstract

In this short note, we study the derivatives of all orders for the random field $$ X_T(h) = \sum_{p \leq T} \frac{\text{Re}(U_p \, p^{-i h})}{p^{1/2}}, \quad h\in [0,1], $$ where $(U_p, \, p ~\text{primes})$ is an i.i.d. sequence of uniform random variables on the unit circle in $\mathbb{C}$. We show that the maximum of $X_{T}$, and more generally the maximum of its $j$-th derivative, varies on a $(\log T)^{-\frac{1}{2}(j+2)}$ scale, which improves and extends the main result in Arguin & Ouimet (2019) and makes further progress towards the open problem of the tightness of the recentered maximum of $X_{T}$. Our proof is also much simpler and shorter.

Equations

$$X_T(h):=\sum _{p\le T}\frac{\text{Re}(U_p{p}^{-ih})}{p^{1/2}},h\in [0,1].$$

$$X_{r,k}(h)=\sum _{2^r<\log p\le 2^k}\frac{\mathrm{Re}(U_p{p}^{-\mathrm{i}h})}{p^{1/2}},h\in [0,1].$$

$$X_{r,k}^{(j)}(h):=\frac{{\mathrm{d}}^j}{\mathrm{d}{h}^j}{X}_{r,k}(h)=\{\begin{array}{ll}{\sum }_{2^r<\log p\le 2^k}(-1{)}^{j/2}\frac{(\log p{)}^j}{p^{1/2}}\mathrm{Re}(U_p{p}^{-\mathrm{i}h}),& \text{mbox}ifj\text{is even},\\ {\sum }_{2^r<\log p\le 2^k}(-1{)}^{(j-1)/2}\frac{(\log p{)}^j}{p^{1/2}}\mathrm{Im}(U_p{p}^{-\mathrm{i}h}),& \text{mbox}ifj\text{is odd},\end{array}$$

$$\mathbb{P}\Big{(}\big{|}\max_{h\in[0,1]}X_{r,k}^{(j)}(h)-\max_{h\in\mathcal{H}_{k}}X_{r,k}^{(j)}(h)\big{|}>K\Big{)}<\varepsilon.$$

$$h^{\star }:={\text{argmax}}_{h\in [0,1]}{X}_{r,k}^{(j)}(h),$$

$$x^{\star }:={\text{argmax}}_{x\in [0,1]:|x-h^{\star }|\le \alpha 2^{-\frac{1}{2}(j+2)k}}{X}_{r,k}^{(j+1)}(x).$$

$$\displaystyle\big{|}X_{r,k}^{(j+1)}(x^{\star})\big{|}^{2}=\Big{|}\int_{h^{\star}}^{x^{\star}}X_{r,k}^{(j+2)}(x){\rm d}x\Big{|}^{2}\leq|x^{\star}-h^{\star}|\int_{h^{\star}}^{x^{\star}}\big{|}X_{r,k}^{(j+2)}(x)\big{|}^{2}{\rm d}x,$$

$$\displaystyle\mathbb{E}\Big{[}\big{|}X_{r,k}^{(j+1)}(x^{\star})\big{|}^{2}\Big{]}$$

$$={\alpha }^2{2}^{-(j+2)k}\underset{x}{\max }\sum _{2^r<\log p\le 2^k}\frac{(\log p{)}^{2(j+2)}}{p}\left\{\frac{0\cdot p^{-2\mathrm{i}x}\pm 2\mathbb{E}[U_p\overline{U_p}]+0\cdot p^{2\mathrm{i}x}}{\pm 4}\right\}$$

$$={\alpha }^2{2}^{-(j+2)k}\sum _{2^r<\log p\le 2^k}\frac{(\log p{)}^{2(j+2)}}{2p}.$$

$$\mathbb{P}\Big{(}X_{r,k}^{(j+1)}(x^{\star})>A\alpha 2^{\frac{1}{2}(j+2)k}\Big{)}<\varepsilon.$$

$$0\le e^{X_{r,k}^{(j)}(h^{\star })}-e^{X_{r,k}^{(j)}(h)}$$

$$\le \alpha 2^{-\frac{1}{2}(j+2)k}\cdot X_{r,k}^{(j+1)}(x^{\star })e^{X_{r,k}^{(j)}(h^{\star })}$$

$$\le A{\alpha }^2\cdot e^{X_{r,k}^{(j)}(h^{\star })}.$$

$$e^{X_{r,k}^{(j)}(h)}\ge e^{X_{r,k}^{(j)}(h^{\star })}{e}^{-K}\text{on the event {0≤Xr,k(j+1)(x⋆)≤Aα221(j+2)k}},$$

$$\mathbb{P}\Big{(}X_{r,k}^{(j)}(h^{\star})\geq\max_{h\in\mathcal{H}_{k}}X_{r,k}^{(j)}(h)\geq X_{r,k}^{(j)}(h^{\star})-K\Big{)}\geq 1-\varepsilon,$$

$$G_{r,k}(h)=\sum _{2^r<\log p\le 2^k}\frac{\mathrm{Re}(Z_p{p}^{-\mathrm{i}h})}{p^{1/2}},h\in [0,1],$$

$$\bigg{|}\sum_{P<p\leq Q}\frac{(\log p)^{j}}{p}-\left(\frac{(\log Q)^{j}}{j}-\frac{(\log P)^{j}}{j}\right)\bigg{|}\leq D_{j},$$

Full text

Résumé

Dans cette courte note, nous étudions un champ aléatoire qui approxime la partie réelle du logarithme de la fonction zêta de Riemann sur la ligne critique, introduit par Harper, (2013); Arguin et al., (2017), et ses dérivées de tous les ordres. Nous montrons que le maximum du champ aléatoire, et plus généralement le maximum de sa $j$-ième dérivée, varie sur une échelle $(\log T)^{\scriptscriptstyle-\frac{1}{2}(j+2)}$, ce qui améliore et étend le résultat principal dans Arguin & Ouimet, (2019) et nous fait progresser concernant le problème ouvert de la tension du maximum recentré. Notre preuve est aussi beaucoup plus simple et courte.

1 Model and background

Let $(U_{p},\,p~{}\text{primes})$ be an i.i.d. sequence of uniform random variables on the unit circle in $\mathbb{C}$. The random field of interest is

[TABLE]

(A sum over the variable $p$ always denotes a sum over primes.) Harper, (2013) showed that $(X_{\scriptscriptstyle T}(h),\,h\in[0,1])$ is a good model for the large values of $(\log|\zeta(\tfrac{1}{2}+\mathrm{i}T+\mathrm{i}h)|,\,h\in[0,1])$ when $T$ is large, if we assume the Riemann hypothesis. The second order of the maximum was shown in Arguin et al., (2017), but the tightness of the recentered maximum of $X_{\scriptscriptstyle T}$ is still open. This is the motivation behind this paper. Our result shows that the maximum of $X_{\scriptscriptstyle T}$ can be discretized to a number of points, $\log T$, that coincides with the number of leaves in the approximate branching structure underlying $\log|\zeta|$. This is a non-trivial improvement over the main result in Arguin & Ouimet, (2019). Our method yields similar discretizations for all the derivatives of $X_{\scriptscriptstyle T}$.

For various asymptotic results of interest on the extreme values of the model in (1.1), see Harper, (2013); Arguin et al., (2017); Arguin & Tai, (2019); Arguin et al., (2019a); Arguin & Ouimet, (2019); Ouimet, (2018, 2019); Saksman & Webb, (2016, 2018). For asymptotic results on the maximum of the Riemann zeta function on the critical line, we refer the reader to Najnudel, (2018); Arguin et al., (2019b, c); Harper, (2019); Bondarenko & Seip, (2017); de la Bretèche & Tenenbaum, (2019) and references therein. Related conjectures can be found in Farmer et al., (2007); Fyodorov et al., (2012); Fyodorov & Keating, (2014).

2 Result

Below, we work with the increments of the field $X_{\scriptscriptstyle T}$. For $-1\leq r\leq k$, let

[TABLE]

The $j$-th derivative of $X_{\scriptscriptstyle r,k}$ is

[TABLE]

which can be seen as a toy model for the real part of the $j$-th logarithmic derivative of the Riemann zeta function on the critical line.

The theorem below says that if we want to find the asymptotics of the maximum of $X_{\scriptscriptstyle r,k}^{\scriptscriptstyle(j)}$ up to a constant, we can restrict the maximum to a discrete set containing $\mathcal{O}(2^{\scriptscriptstyle\frac{1}{2}(j+2)k})$ equidistant points.

Theorem 2.1** (Discretization).**

Fix $j\in\mathbb{N}_{0}$ and let $\mathcal{H}_{k}\vcentcolon=\alpha 2^{-\frac{1}{2}(j+2)k}\hskip 0.85358pt\mathbb{N}_{0}\cap[0,1]$ for $\alpha>0$. Then, for all $\varepsilon,K>0$, there exists $\alpha=\alpha(j,\varepsilon,K)>0$ small enough that

[TABLE]

Similarly, if $\alpha>0$ is small enough with respect to $j$ and $\varepsilon$, there exists $K=K(j,\varepsilon,\alpha)>0$ large enough that (2.2) holds.

Remark 2.2**.**

When $r=-1$ and $2^{k}=\log T$, $X_{\scriptscriptstyle r,k}$ is the full model $X_{\scriptscriptstyle T}$, and $|\mathcal{H}_{k}|=\mathcal{O}(\log T)$. Theorem 2.1 improves the main result in Arguin & Ouimet, (2019), where only the case $j=0$ was treated and the discrete set had $\mathcal{O}(\sqrt{\log\log T}\log T)$ points instead. The proof rested on estimates of the joint Laplace transform of $X_{\scriptscriptstyle r,k}^{\scriptscriptstyle(1)}$ and continuity estimates derived from a chaining argument of Arguin et al., (2017). Our proof here is much simpler, much shorter, and also applies for the derivatives of all orders. The main idea is to control the maximum of $X_{\scriptscriptstyle r,k}^{\scriptscriptstyle(j+1)}$ around the point $h^{\star}$ where $X_{\scriptscriptstyle r,k}^{\scriptscriptstyle(j)}$ maximizes by applying a mean value theorem to $X_{\scriptscriptstyle r,k}^{\scriptscriptstyle(j+1)}$ followed by Jensen’s inequality and a bound on the variance of $X_{\scriptscriptstyle r,k}^{\scriptscriptstyle(j+2)}$ using prime number theorem estimates.

3 Proof of Theorem 2.1

Fix $j\in\mathbb{N}_{0}$ and let

[TABLE]

(Note that $h^{\star},x^{\star}\in(0,1)$ with probability $1$.) By the mean value theorem, the fact that $X_{\scriptscriptstyle r,k}^{\scriptscriptstyle(j+1)}(h^{\star})=0$, and by Jensen’s inequality, we have

[TABLE]

which implies, using $(\mathrm{Re}\,z)^{2}=(z+\overline{z})^{2}/4$, $(\mathrm{Im}\,z)^{2}=(z-\overline{z})^{2}/(-4)$, $\mathbb{E}[U_{p}^{2}]=\mathbb{E}[\overline{U_{p}}^{2}]=0$ and the independence of the $U_{p}$’s,

[TABLE]

(Here, $\pm$ means $+$ when $j$ is even and $-$ when $j$ is odd.) By the prime number theorem estimates in Lemma 4.1, the last sum of primes is bounded above by $(2^{\scriptscriptstyle 2(j+2)k}-2^{\scriptscriptstyle 2(j+2)r})/(4(j+2))+D_{j}$, where $D_{j}>0$ is a constant that only depends on $j$. Hence, by Chebyshev’s inequality, we can take $A=A(j,\varepsilon)>0$ large enough that

[TABLE]

Now, on the complementary event $\{0\leq X_{\scriptscriptstyle r,k}^{\scriptscriptstyle(j+1)}(x^{\star})\leq A\alpha 2^{\scriptscriptstyle\frac{1}{2}(j+2)k}\}$, apply the mean value theorem for $\exp(X_{\scriptscriptstyle r,k}^{\scriptscriptstyle(j)}(h^{\star}))$ around the point $h\in\mathcal{H}_{k}$ that is closest on the left-hand side of $h^{\star}$ (this choice implies $0\leq h^{\star}-h\leq\alpha 2^{\scriptscriptstyle-\frac{1}{2}(j+2)k}$), then

[TABLE]

For any given $K>0$, choose $\alpha=\alpha(j,\varepsilon,K)>0$ small enough that $A\alpha^{2}\leq 1-e^{\scriptscriptstyle-K}$. We find that

[TABLE]

and thus

[TABLE]

by (3.4). This ends the proof.

Remark 3.1**.**

It is straightforward to verify that the proof of Theorem 2.1 carries over to the Gaussian model

[TABLE]

from Saksman & Webb, (2016, 2018), where $Z_{p}=W_{p}^{\scriptscriptstyle(1)}+\mathrm{i}W_{p}^{\scriptscriptstyle(2)}$ and $W_{p}^{\scriptscriptstyle(1)},\hskip 1.42262ptW_{p}^{\scriptscriptstyle(2)}\sim\mathcal{N}(0,1/2)$ are i.i.d. This is because the variance bound in (3) is also valid if we replace $X$ by $G$.

4 Tool

The estimates below are simple consequences of the prime number theorem and integration by parts.

Lemma 4.1** (Lemma A.1 in Arguin & Ouimet, (2019)).**

Let $j\in\mathbb{N}^{*}$ and $1\leq P<Q$, then

[TABLE]

for some constant $D_{j}>0$ that only depends on $j$.