Tail bounds for counts of zeros and eigenvalues, and an application to ratios

TL;DR

This paper establishes tail bounds for zero counts of the Riemann zeta function and eigenvalues of random matrices, and applies these results to ratios of zeta functions, linking zero distribution to ratio conjectures.

Contribution

It provides new probabilistic bounds on zero counts and eigenvalues, and connects zero distribution with ratio conjectures under the Riemann hypothesis.

Findings

Tail bounds decay as e^{-Cx log x} for zero counts

Ratios of zeta functions remain bounded on average under RH

Zero distribution controls ratio averages, supporting the GUE conjecture

Abstract

Let $t$ be random and uniformly distributed in the interval $[T,2T]$, and consider the quantity $N(t+1/\log T) - N(t)$, a count of zeros of the Riemann zeta function in a box of height $1/\log T$. Conditioned on the Riemann hypothesis, we show that the probability this count is greater than $x$ decays at least as quickly as $e^{-Cx\log x}$, uniformly in $T$. We also prove a similar results for the logarithmic derivative of the zeta function, and likewise analogous results for the eigenvalues of a random unitary matrix. We use results of this sort to show on the Riemann hypothesis that the averages $$ \frac{1}{T} \int_T^{2T} \Bigg| \frac{\zeta\Big(\frac{1}{2} + \frac{\alpha}{\log T} + it\Big)}{\zeta\Big(\frac{1}{2}+ \frac{\beta}{\log T} + it\Big)}\Bigg|^m\,dt $$ remain bounded as $T\rightarrow\infty$, for $\alpha, \beta$ complex numbers with $\beta\neq 0$. Moreover we show rigorously that the local distribution of zeros asymptotically controls ratio averages like the above; that is, the GUE Conjecture implies a (first-order) ratio conjecture.