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
Famous quotes containing the words relation to and/or relation:
“Unaware of the absurdity of it, we introduce our own petty household rules into the economy of the universe for which the life of generations, peoples, of entire planets, has no importance in relation to the general development.”
—Alexander Herzen (1812–1870)
“The proper study of mankind is man in his relation to his deity.”
—D.H. (David Herbert)