Mean values of the Riemann zeta function at shifted zeros under the Riemann Hypothesis

TL;DR

This paper derives precise asymptotic formulas for sums involving the Riemann zeta function at shifted zeros under the Riemann Hypothesis, improving error estimates and correcting previous inaccuracies.

Contribution

It provides sharper asymptotic formulas for zeta sums at shifted zeros assuming RH, refining previous unconditional results and correcting errors.

Findings

Sharper error terms under RH

Correction of previous unconditional error estimates

Asymptotic formulas valid in specified complex regions

Abstract

Assuming the Riemann hypothesis, we obtain asymptotic formulas for $\sum_{0<\gamma<T}\zeta(\rho+\delta)\zeta(1-\rho+\overline{\delta})$ in the region $-\frac{a}{\log T} \leq \Re \delta \leq \frac{1}{2}+\frac{a}{\log T}$, $|\Im \delta|\ll 1$. Unconditionally, this asymptotic formula was recently obtained by Garunk\v{s}tis and Novikas in essentially the same region, with a slight incompleteness. Assuming RH, we obtain a sharper error term, and we also correct an inaccuracy in the unconditional error term there.