Hurwitz zeta function

In mathematics, the Hurwitz zeta function is defined as

$\zeta (s,q)=\sum _{k=0}^{\infty }(k+q)^{-s}.$

When q = 1, this coincides with Riemann's zeta function.

Fixing an integer Q ≥ 1, the Dirichlet L-functions for characters modulo Q are linear combinations, with constant coefficients, of the ζ(s,q) where q = r/Q and r = 1, 2, ..., Q. This means that the Hurwitz zeta-functions for q a rational number have analytic properties that are closely related to that class of L-functions.

Although Hurwitz's zeta function is thought of by mathematicians as being relevant to the "purest" of mathematical disciplines, number theory, it also occurs in applied statistics; see Zipf's law and Zipf-Mandelbrot law.

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