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 $$\frac{1}{T}{\int }_T^{2T}|\frac{\zeta (\frac{1}{2}+\frac{\alpha }{\log T}+it)}{\zeta (\frac{1}{2}+\frac{\beta }{\log T}+it)}{|}^{m}dt$$ remain bounded as $T\to \infty$, for $\alpha ,\beta$ complex numbers with $\beta \ne 0$. 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.

arXiv comment37 pages. Incorporates referee suggestions

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