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

TL;DR

This paper derives a series expansion of the prime zeta function around its singularity at s=1, providing formulas for the coefficients and generalizing Mertens's Theorems, with numerical verification.

Contribution

It introduces a new series expansion for the prime zeta function near s=1 and generalizes Mertens's Theorems with explicit coefficient formulas.

Findings

Derived a series expansion of the prime zeta function at s=1

Proved formulas for the expansion coefficients

Numerically verified the formulas with high precision

Abstract

In this article, we derive a series expansion of the prime zeta function about the $s=1$ logarithmic singularity and prove general formula for its expansion coefficients, which is similar to the Stieltjes expansion coefficients for the Riemann zeta function. These results can also be viewed as a generalization of Mertens's Theorems to higher order. We also numerically verify and compute the presented formulas to high precision for several test cases.