In mathematics, the Hurwitz zeta function is one of the many zeta functions. It defined as
When q = 1, this coincides with Riemann's zeta function.
Fixing an integer Q ≥ 1, the Dirichlet L-functions 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-functions.
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
where
is a representation of the zeta that is valid for and .
Note that generalizes the Bernoulli polynomials:
where denotes the real part of z.
Although Hurwitz's zeta function is thought of by mathematicians as being relevant to the "purest" of mathematical disciplines − 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.
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. (abstract)
- Linas Vepstas, The Bernoulli Operator, the Gauss-Kuzmin-Wirsing Operator, and the Riemann Zeta