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On the series expansion of k-free Dirichlet series and its analytical continuation

Artur Kawalec

Abstract

In this article, we develop a k-free zeta Dirichlet series into a Laurent series with a simple pole, and prove a Stieltjes like formula for the expansion coefficients of the regular part. We also investigate another analytical continuation of these series and develop a formula for $ζ(\tfrac{1}{k})$ for positive integer $k\geq 2$ in terms of the k-free indicator function.

On the series expansion of k-free Dirichlet series and its analytical continuation

Abstract

In this article, we develop a k-free zeta Dirichlet series into a Laurent series with a simple pole, and prove a Stieltjes like formula for the expansion coefficients of the regular part. We also investigate another analytical continuation of these series and develop a formula for for positive integer in terms of the k-free indicator function.
Paper Structure (4 sections, 1 theorem, 60 equations, 2 figures, 1 table)

This paper contains 4 sections, 1 theorem, 60 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Proof of Theorem 1
  3. On analytical continuation of the Dirichlet series
  4. A special case around the first zero

Key Result

Theorem 1

For $n\geq 0$, the coefficients are:

Figures (2)

  • Figure 1: A plot of equation (57) for $\zeta(\frac{1}{2})$ value $(k=2)$ as a function of limit variable $x$
  • Figure 3: A plot of equation (59) for $\zeta(\frac{1}{4})$ value $(k=4)$ as a function of limit variable $x$

Theorems & Definitions (1)

  • Theorem 1