  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.
