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

Let $t$ be random and uniformly distributed in the interval $[T,2T]$, and consider the quantity $N(t+1\mathrm{/}\log T)-N(t)$, a count of zeros of the Riemann zeta function in a box of height $1\mathrm{/}\log T$. 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 $$\frac{1}{T}{\int }_T^{2T}\mid \frac{\zeta (\frac{1}{2}+\frac{\alpha }{\log T}+it)}{\zeta (\frac{1}{2}+\frac{\beta }{\log T}+it)}{\mid }^{m}dt$$ remain bounded as $T\to \mathrm{\infty }$, for $\alpha, \beta$complexnumberswith$\beta\neq 0$. 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.