Hurwitz zeta function: Difference between revisions

Revision as of 04:01, 2 January 2005

Revision as of 02:52, 29 January 2005

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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]].

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

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Note that <math>\beta</math> generalizes the [[Bernoulli polynomials]]:

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

where <math>\Re z</math> denotes the real part of ''z''.

The Hurwitz zeta is generalizes the [[polygamma function]]:

:<math>\psi^{(m))}= (-)^{m+1} m! \zeta (m+1,z)</math>

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

==Applications==

Although Hurwitz's zeta function is thought of by mathematicians as being relevant to the "purest" of mathematical disciplines &minus; [[number theory]], it also occurs in the study of [[fractals]] and [[dynamical systems]] and in applied [[statistics]]; see [[Zipf's law]] and [[Zipf-Mandelbrot law]].

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

* Milton Abramowitz and Irene A. Stegun, ''Handbook of Mathematical Functions'', (1964) Dover Publications, New York. ISBN 486-61272-4 . See paragraph 6.4.10.

* 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]