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

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.