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

Artur Kawalec

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arXiv · 2603.21535

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

Artur Kawalec

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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.

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Artur Kawalec. 2026-03-23. On the series expansion of the prime zeta function about $s=1$ and its coefficients. https://arxiv.org/abs/2603.21535

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