Hurwitz Zeta Function - Relation To Dirichlet L-functions

Relation To Dirichlet L-functions

At rational arguments the Hurwitz zeta function may be expressed as a linear combination of Dirichlet L-functions and vice versa: The Hurwitz zeta function coincides with Riemann's zeta function ζ(s) when q = 1, when q = 1/2 it is equal to (2s−1)ζ(s), and if q = n/k with k > 2, (n,k) > 1 and 0 < n < k, then

the sum running over all Dirichlet characters mod k. In the opposite direction we have the linear combination

There is also the multiplication theorem

of which a useful generalization is the distribution relation

(This last form is valid whenever q a natural number and 1 − qa is not.)