On the series expansion of the secondary zeta function about $s=1$ and its coefficients

TL;DR

This paper derives a new formula for the Laurent series coefficients of the secondary zeta function at $s=1$, verifies it numerically, and applies convergence improvement techniques.

Contribution

It introduces a novel formula for the secondary zeta function's expansion coefficients, analogous to Stieltjes constants, and enhances computational convergence.

Findings

Derived a new formula for the Laurent expansion coefficients.

Numerically verified the formula for multiple test cases.

Applied Brent's Theorem to improve convergence.

Abstract

The secondary zeta function is defined as a generalized zeta series over the imaginary parts of non-trivial zeros assuming (RH). This function admits Laurent series expansion at the double pole at $s=1$. In this article, we derive a new formula for the expansion coefficients of the regular part, which is similar to the Stieltjes constants formula for the Riemann zeta function. We also numerically verify and compute the new formula to high precision for several test cases. Lastly, we also apply the Brent's (BPT) Theorem for improving convergence of the main formula.