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

TL;DR

This paper develops a Laurent series expansion for k-free Dirichlet series, provides a Stieltjes-like formula for its coefficients, and explores alternative analytical continuations and formulas for special values.

Contribution

It introduces a Laurent series expansion with a simple pole for k-free zeta Dirichlet series and derives a new formula for ta(1/k) using the k-free indicator function.

Findings

Laurent series expansion with a simple pole for k-free Dirichlet series

Stieltjes-like formula for expansion coefficients

New formula for ta(1/k) in terms of k-free indicator function

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 $\zeta(\tfrac{1}{k})$ for positive integer $k\geq 2$ in terms of the k-free indicator function.