Hurwitz zeta function: Difference between revisions

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Fixing an integer ''Q'' ≥ 1, the [[Dirichlet L-function]]s for characters modulo Q are linear combinations, with constant coefficients, of the ζ(''s'',''q'') where ''q'' = ''r''/''Q'' and ''r'' = 1, 2, ..., ''Q''. This means that the Hurwitz zeta-functions for ''q'' a rational number have analytic properties that are closely related to that class of [[L-function]]s.

The [[discrete Fourier transform]] of the Hurwitz zeta function with respect to the order ''s'' is the [[Legendre chi function]]. That is, we can write

'''Hurwitz's formula''' is the theorem that

:<math>\zeta(1-s,x)=\frac{1}{2s}\left[e^{-i\pi s/2}\beta(x;s) + e^{i\pi s/2} \beta(1-x;s) \right]</math>

where

:<math>\beta(x;s)=2\Gamma(s+1)\sum_{n=1}^\infty \frac {\exp(2\pi inx) } {(2\pi n)^s}</math>

is a representation of the zeta that is valid for <math>0\le x\le 1</math> and <math>s>1</math>.

is a generalization of the [[Bernoulli polynomials]]:

isNote athat generalization<math>\beta</math> ofgeneralizes the [[Bernoulli polynomials]]:

:<math>B_n(x) = -\Re \left[ (-i)^n \beta(x;n) \right] </math>

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==References==

* Tom M. Apostol ''Introduction to Analytic Number Theory'', (1976) Springer-Verlag, New York. '' (See Chapter 12.7)

* Djurdje Cvijovic and Jacek Klinowski. Math. Comp. 68 (1999), 1623-1630, 1999. [http://www.ams.org/journal-getitem?pii=S0025-5718-99-01091-1 (abstract)]

* Linas Vepstas, [http://www.linas.org/math/chap-gkw/gkw.html The Bernoulli Operator, the Gauss-Kuzmin-Wirsing Operator, and the Riemann Zeta]