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:
“A theory of the middle class: that it is not to be determined by its financial situation but rather by its relation to government. That is, one could shade down from an actual ruling or governing class to a class hopelessly out of relation to government, thinking of gov’t as beyond its control, of itself as wholly controlled by gov’t. Somewhere in between and in gradations is the group that has the sense that gov’t exists for it, and shapes its consciousness accordingly.”
—Lionel Trilling (1905–1975)
“... a worker was seldom so much annoyed by what he got as by what he got in relation to his fellow workers.”
—Mary Barnett Gilson (1877–?)