On sums of squares of $|\zeta(\frac12+i\gamma)|$ over short intervals

TL;DR

This paper investigates the sum of squared magnitudes of the Riemann zeta-function at its zeros over short intervals, providing an unconditional upper bound involving logarithmic factors for a broad range of H.

Contribution

It establishes an unconditional upper bound for the sum of squared zeta values over zeros in short intervals, extending previous results with explicit logarithmic factors.

Findings

Sum of squared zeta values is bounded by H log^2 T log log T

Bound holds unconditionally for T^{2/3} log^4 T extless H extless T

Provides insights into the distribution of zeta zeros over short intervals

Abstract

A discussion involving the evaluation of the sum $$\sum_{T<\g\le T+H}|\zeta(1/2+i\gamma)|^2$$ and some related integrals is presented, where $\gamma\,(>0)$ denotes imaginary parts of complex zeros of the Riemann zeta-function $\zeta(s)$. It is shown unconditionally that the above sum is $\,\ll H\log^2T\log\log T\,$ for $\,T^{2/3}\log^4T \ll H \le T$.