Hurwitz zeta function

In mathematics, the Hurwitz zeta function is one of the many zeta functions. It is formally defined for complex variables s with Re(s) > 1 and a ≠ 0, −1, −2, ... by $$\zeta (s,a)=\sum _{n=0}^{\infty }{\frac {1}{(n+a)^{s}}}.$$

This series is absolutely convergent for the given values of s and a and can be extended to a meromorphic function defined for all s ≠ 1. The Riemann zeta function is ζ(s, 1). The Hurwitz zeta function is named after Adolf Hurwitz, who introduced it in 1882.

Hurwitz zeta function corresponding to a = 1/3, shown using domain coloring.

Hurwitz zeta function corresponding to a = 24/25.

Hurwitz zeta function as a function of a with s = 3 + 4i.