Hurwitz zeta function

In mathematics, the Hurwitz zeta function is one of the many zeta functions. It defined as

${\displaystyle \zeta (s,q)=\sum _{k=0}^{\infty }(k+q)^{-s}.}$

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.

Hurwitz's formula is the theorem that

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

where

${\displaystyle \beta (x;s)=2\Gamma (s+1)\sum _{n=1}^{\infty }{\frac {\exp(2\pi inx)}{(2\pi n)^{s}}}}$

is a representation of the zeta that is valid for ${\displaystyle 0\leq x\leq 1}$ and ${\displaystyle s>1}$.

Note that ${\displaystyle \beta }$ generalizes the Bernoulli polynomials:

${\displaystyle B_{n}(x)=-\Re \left[(-i)^{n}\beta (x;n)\right]}$

where ${\displaystyle \Re z}$ denotes the real part of z.

The Hurwitz zeta is generalizes the polygamma function:

${\displaystyle \psi ^{(m))}=(-)^{m+1}m!\zeta (m+1,z)}$

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 − 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)
• 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. (abstract)
• Linas Vepstas, The Bernoulli Operator, the Gauss-Kuzmin-Wirsing Operator, and the Riemann Zeta