  In this article, we develop a square-free zeta series associated with the
M\"obius 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 $\zeta(\tfrac{1}{2})$ in
terms of the M\"obius function, and in the last part, we explore an alternating
series version of these results.
