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.
Categories: math.NT