A weighted one-level density of the non-trivial zeros of the Riemann zeta-function

TL;DR

This paper analyzes the distribution of non-trivial zeros of the Riemann zeta-function weighted by specific powers, revealing how often these zeros are near points where the zeta function attains large values, under the Riemann hypothesis.

Contribution

It introduces a weighted one-level density for the zeros of the zeta-function, extending previous results to include weights involving powers of |6(rac12+it)|, for k=1 and 2.

Findings

For k=1, about T(log T)^{0} zeros are near large zeta values.

For k=2, about T(log T)^{-3} zeros are near large zeta values.

Under RH, zeros cluster near points where |6| attains size (6(rac12+it)|^{k+o(1)}.

Abstract

We compute the one-level density of the non-trivial zeros of the Riemann zeta-function weighted by $|\zeta(\frac12+it)|^{2k}$ for $k=1$ and, for test functions with Fourier support in $(-\frac12,\frac12)$, for $k=2$. As a consequence, for $k=1,2$, we deduce under the Riemann hypothesis that $T(\log T)^{1-k^2+o(1)}$ non-trivial zeros of $\zeta$, of imaginary parts up to $T$, are such that $\zeta$ attains a value of size $(\log T)^{k+o(1)}$ at a point which is within $O(1/\log T)$ from the zero.