# Hurwitz zeta function

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The Hurwitz zeta function $\zeta(\alpha,s)$ is defined for real $\alpha$, $0 < \alpha \le 1$ as $$\zeta(\alpha,s) = \sum_{n=0}^\infty (n+\alpha)^{-s} \ .$$ The series is convergent, and defines an analytic function, for $\Re s > 1$. The function possesses an analytic continuation to the whole $s$-plane except for a simple pole of residue 1 at $s=1$.