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

Artur Kawalec

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.

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

Abstract

In this article, we derive a series expansion of the prime zeta function about the 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.
Paper Structure (2 sections, 3 theorems, 49 equations, 1 table)

This paper contains 2 sections, 3 theorems, 49 equations, 1 table.

Table of Contents

  1. Introduction
  2. Proof of Theorem 1

Key Result

Theorem 1

For $n=0$, the coefficient is: where $\gamma$ is Euler's constant, and for $n\geq 1$ the $n$th coefficient is:

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3