On the series expansion of the secondary zeta function

Abstract

In article, we explore the secondary zeta function $Z(s)$, which is defined as a generalized zeta type of series over imaginary parts of non-trivial zeros of the Riemann zeta function $\zeta(s)$. This function has been analytically continued as a meromorphic function in $\mathbb{C}$ with one double pole and an infinity of simple poles. The secondary zeta function is of interest because it can naturally represent an analytical formula for non-trivial zeros of the Riemann zeta function that we will explore, and we show that the non-trivial zeros can be generated directly from primes by introducing a new form of an explicit formula written in terms of the prime zeta function. Additionally, we will also give several new series expansions for $Z(s)$ and numerically compute these coefficients to high precision, and also develop several new methods to analytically extend $Z(s)$ to larger domains and develop algorithms to compute them.