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

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.