Computation of the secondary zeta function

TL;DR

This paper introduces an algorithm to compute the secondary zeta function's analytic continuation for all complex s, regardless of the Riemann hypothesis, enabling precise calculations of its meromorphic extension.

Contribution

It provides a novel algorithm for the analytic continuation of the secondary zeta function without assuming the Riemann hypothesis.

Findings

Algorithm successfully computes the secondary zeta function for all s.

Enables high-precision calculations of the secondary zeta function.

Works regardless of the truth of the Riemann hypothesis.

Abstract

The secondary zeta function $Z(s)=\sum_{n=1}^\infty\alpha_n^{-s}$, where $\rho_n=\frac12+i\alpha_n$ are the zeros of zeta with $\Im(\rho)>0$, extends to a meromorphic function on the hole complex plane. If we assume the Riemann hypothesis the numbers $\alpha_n=\gamma_n$, but we do not assume the RH. We give an algorithm to compute the analytic prolongation of the Dirichlet series $Z(s)=\sum_{n=1}^\infty \alpha_n^{-s}$, for all values of $s$ and to a given precision.