The Lindel\"of Hypothesis for Zeta Zero Ordinates

TL;DR

This paper derives asymptotic formulas for exponential sums over the zeros of the Riemann zeta-function, connecting to the Lindelöf Hypothesis regarding the distribution of these zeros.

Contribution

It provides new conditional and unconditional asymptotic results for sums over zeta zeros, advancing understanding of the Lindelöf Hypothesis.

Findings

Asymptotic formulas for sums over zeta zeros are established.

Results relate to the Lindelöf Hypothesis for zeta zeros.

Both conditional and unconditional results are presented.

Abstract

We provide conditional and unconditional asymptotic formulae for the exponential sums $\sum_\gamma\,\gamma^{-i\tau}$, where the summation is over the ordinates of the nontrivial zeros $\rho=\beta+i\gamma$ of the Riemann zeta-function. In particular, the obtained results are related to the Lindel\"of Hypothesis for these ordinates (in the sense of Gonek et al. [10]).