Hurwitz zeta function

generalised zeta function

An Dirichlet series related to the Riemann zeta function which may be used to exhibit properties of various Dirichlet L-functions.

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 $s=1$.

References

• Tom M. Apostol, "Introduction to Analytic Number Theory", Undergraduate Texts in Mathematics, Springer (1976) ISBN 0-387-90163-9 Zbl 0335.10001