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On the series expansion of a square-free zeta series

Artur Kawalec

Abstract

In this article, we develop a square-free zeta series associated with the Möbius function into a power series, and prove a Stieltjes like formula for these expansion coefficients. We also investigate another analytical continuation of these series and develop a formula for $ζ(\tfrac{1}{2})$ in terms of the Möbius function, and in the last part, we explore an alternating series version of these results.

On the series expansion of a square-free zeta series

Abstract

In this article, we develop a square-free zeta series associated with the Möbius function into a power series, and prove a Stieltjes like formula for these expansion coefficients. We also investigate another analytical continuation of these series and develop a formula for in terms of the Möbius function, and in the last part, we explore an alternating series version of these results.
Paper Structure (4 sections, 43 equations, 2 figures, 1 table)

This paper contains 4 sections, 43 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. On analytical continuation of the Dirichlet series
  3. On an alternating series representation
  4. Acknowledgement

Figures (2)

  • Figure 1: A plot of equation (28) for $x=10^8$ showing deviation near $s=\frac{1}{4}$
  • Figure 2: A plot of equation (32) for $\zeta(\frac{1}{2})$ as a function of limit variable $x$