math lessons - Hurwitz 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 = k/Q and k = 1, 2, ..., Q. 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 Q. Then we can write the Dirichlet L-function as

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

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 is generalizes the polygamma function:

ψ (m) (z) = ( - ) m + 1 m!ζ(m + 1, z)

Relation to the Lerch transcendant

The Lerch transcendant generalizes the Hurwitz zeta:

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

and thus

ζ(s, q) = Φ(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

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

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. This distinction based on z accounts for the fact that the Jacobi theta function converges to the Dirac delta function in z as $t\rightarrow 0$ .

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. 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.
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

Categories: Zeta functions

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