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.)
Read more about this topic: Hurwitz Zeta Function
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