Hurwitz zeta function

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

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

When q = 1, this coincides with Riemann's zeta function.

1 Series representation
2 Relation to Dirichlet L-functions
3 Hurwitz's formula
4 Relation to Bernoulli polynomials
5 Relation to the polygamma function
6 Relation to the Lerch transcendent
7 Functional equation
8 Taylor series
9 Fourier transform
10 Relation to Jacobi theta function
11 Applications
12 References

Series representation

The Hurwitz zeta function has a globally convergent series representation defined on the entire complex s-plane, excluding s=1. Due to Helmut Hasse in 1930, it is

$\zeta(s,q)=\frac{1}{s-1} \sum_{n=0}^\infty \frac{1}{n+1} \sum_{k=0}^n (-1)^k {n \choose k} (q+k)^{1-s}.$

This series converges uniformly on compact subsets of the s-plane to an entire function.

Relation to Dirichlet L-functions

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

Specifically, let $χ$ be a character mod N. Then we can write the Dirichlet L-function as

$L(s,\chi) = \sum_{n=1}^\infty \frac {\chi(n)}{n^s} = \frac {1}{N^s} \sum_{k=1}^N \chi(k)\; \zeta \left(s,\frac{k}{N}\right) .$

In particular, the trivial character yields the Riemann zeta:

$\zeta(s) = \frac {1}{N^s} \sum_{k=1}^N \zeta \left(s,\frac{k}{N}\right) .$

Hurwitz's formula

Hurwitz's formula is the theorem that

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

where

$\beta(x;s)= 2\Gamma(s+1)\sum_{n=1}^\infty \frac {\exp(2\pi inx) } {(2\pi n)^s}= \frac{2\Gamma(s+1)}{(2\pi)^s} \mbox{Li}_s (e^{2\pi ix})$

is a representation of the zeta that is valid for $0\le x\le 1$ and $s > 1$ . Here, $Li s (z)$ is the polylogarithm.

Relation to Bernoulli polynomials

The function $β$ defined above generalizes the Bernoulli polynomials:

$B_n(x) = -\Re \left[ (-i)^n \beta(x;n) \right]$

where $\Re z$ denotes the real part of z. Alternately,

$\zeta(-n,x)=-{B_{n+1}(x) \over n+1}.$

Relation to the polygamma function

The Hurwitz zeta generalizes the polygamma function:

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

Relation to the Lerch transcendent

The Lerch transcendent generalizes the Hurwitz zeta:

$\Phi(z, s, q) = \sum_{k=0}^\infty \frac { z^k} {(k+q)^s}$

and thus

$\zeta (s,q)=\Phi(1, s, q).\,$

Functional equation

The functional equation relates values of the zeta on the left- and right-hand sides of the complex plane. For integers $1\leq m \leq n$ ,

$\zeta \left(1-s,\frac{m}{n} \right) = \frac{2\Gamma(s)}{ (2\pi n)^s } \sum_{k=1}^n \cos \left( \frac {\pi s} {2} -\frac {2\pi k m} {n} \right)\; \zeta \left( s,\frac {k}{n} \right)$

holds for all values of s.

Taylor series

The derivative of the zeta in the second argument is a shift:

$\frac {\partial} {\partial q} \zeta (s,q) = -s\zeta(s+1,q).$

Thus, the Taylor series can be written as

$\zeta(s,x+y) = \sum_{k=0}^\infty \frac {y^k} {k!} \frac {\partial^k} {\partial x^k} \zeta (s,x) = \sum_{k=0}^\infty {s+k-1 \choose s-1} (-y)^k \zeta (s+k,x).$

Closely related is the Stark-Keiper formula:

$\zeta(s,N) = \sum_{k=0}^\infty \left[ N+\frac {s-1}{k+1}\right] {s+k-1 \choose s-1} (-1)^k \zeta (s+k,N)$

which holds for integer N and arbitrary s.

Fourier transform

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

Relation to Jacobi theta function

If $\vartheta (z,\tau)$ is the Jacobi theta function, then

$\int_0^\infty \left[\vartheta (z,it) -1 \right] t^{s/2} \frac{dt}{t}= \pi^{-(1-s)/2} \Gamma \left( \frac {1-s}{2} \right) \left[ \zeta(1-s,z) + \zeta(1-s,1-z) \right]$

holds for $\Re s > 0$ and z complex, but not an integer. For z=n an integer, this simplifies to

$\int_0^\infty \left[\vartheta (n,it) -1 \right] t^{s/2} \frac{dt}{t}= 2\ \pi^{-(1-s)/2} \ \Gamma \left( \frac {1-s}{2} \right) \zeta(1-s) =2\ \pi^{-s/2} \ \Gamma \left( \frac {s}{2} \right) \zeta(s).$

where ζ here is the Riemann zeta function. Note that this later form is the functional equation for the Riemann zeta, as originally given by Riemann. The distinction based on z being an integer or not accounts for the fact that the Jacobi theta function converges to the Dirac delta function in z as $t\rightarrow 0$ .

Applications

Hurwitz's zeta function occurs in a variety of disciplines. Most commonly, it occurs in number theory, where its theory is the deepest and most developed. However, it also occurs in the study of fractals and dynamical systems. In applied statistics, it occurs in Zipf's law and the Zipf-Mandelbrot law. In particle physics, it occurs in a formula by Julian Schwinger, given in 1951, giving an exact result for the pair production rate in a uniform electric field.

References

Tom M. Apostol Introduction to Analytic Number Theory, (1976) Springer-Verlag, New York. ISBN 0-387-90163-9 (See Chapter 12)
Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 486-61272-4 . (See Paragraph 6.4.10 for relationship to polygamma function.)
Helmut Hasse, Ein Summierungsverfahren fur die Riemannsche ζ-Reihe, (1930) Math. Z. 32 pp 458-464. (Globally convergent series expression.)
Djurdje Cvijović and Jacek Klinowski, Values of the Legendre chi and Hurwitz zeta functions at rational arguments, Math. Comp. 68 (1999), 1623-1630, 1999.
Djurdje Cvijović and Jacek Klinowski, A Note on the Hurwitz Zeta Function, Matematichki Vesnik 52 (2000) pp 47-54.
Victor S. Adamchik, Derivatives of the Hurwitz Zeta Function for Rational Arguments, Journal of Computational and Applied Mathematics, 100 (1998), pp 201--206.
Linas Vepstas, The Bernoulli Operator, the Gauss-Kuzmin-Wirsing Operator, and the Riemann Zeta

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Category: Zeta and L-functions