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

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.