Asymptotic formulas for harmonic series in terms of a non-trivial zero on the critical line

TL;DR

This paper derives asymptotic formulas for harmonic series linked to non-trivial zeros of the Riemann zeta function, providing insights into their behavior and numerical evaluations.

Contribution

It introduces new asymptotic formulas connecting harmonic series to non-trivial zeros of the zeta function and explores recursive relations for these zeros.

Findings

Asymptotic formulas relate harmonic series to zeta zeros

Numerical computations for various zeros are provided

A recursive formula for non-trivial zeros is investigated

Abstract

In this article, we develop two types of asymptotic formulas for harmonic series in terms of single non-trivial zeros of the Riemann zeta function on the critical line. The series is obtained by evaluating the complex magnitude of an alternating and non-alternating series representation of the Riemann zeta function. Consequently, if the asymptotic limit of the harmonic series is known, then we obtain the Euler-Mascheroni constant with $\log(k)$. We further numerically compute these series for different non-trivial zeros. We also investigate a recursive formula for non-trivial zeros.