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:
“Science is the language of the temporal world; love is that of the spiritual world. Man, indeed, describes more than he explains; while the angelic spirit sees and understands. Science saddens man; love enraptures the angel; science is still seeking, love has found. Man judges of nature in relation to itself; the angelic spirit judges of it in relation to heaven. In short to the spirits everything speaks.”
—Honoré De Balzac (1799–1850)
“Our sympathy is cold to the relation of distant misery.”
—Edward Gibbon (1737–1794)