Hurwitz zeta function
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In mathematics, the Hurwitz zeta function is one of the many zeta functions. It defined as
- <math>\zeta(s,q) = \sum_{k=0}^\infty (k+q)^{-s}.<math>
When q = 1, this coincides with Riemann's zeta function.
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Relation to Dirichlet L-functions
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 <math>\chi<math> be a character mod Q. Then we can write the Dirichlet L-function as
- <math>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}) <math>.
Hurwitz's formula
Hurwitz's formula is the theorem that
- <math>\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]<math>
where
- <math>\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}) <math> is a representation of the zeta that is valid for <math>0\le x\le 1<math> and <math>s>1<math>. Here, <math>\mbox{Li}_s (z)<math> is the polylogarithm.
Relation to Bernoulli polynomials
The function <math>\beta<math> defined above 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. Alternately,
- <math>\zeta(-n,x)=-{B_{n+1}(x) \over n+1}<math>
Relation to the polygamma function
The Hurwitz zeta is generalizes the polygamma function:
- <math>\psi^{(m)}(z)= (-1)^{m+1} m! \zeta (m+1,z)\,<math>
Relation to the Lerch transcendant
The Lerch transcendant generalizes the Hurwitz zeta:
- <math>\Phi(z, s, q) = \sum_{k=0}^\infty
\frac { z^k} {(k+q)^s}<math> and thus
- <math>\zeta (s,q)=\Phi(1, s, q)\,<math>
Functional equation
The functional equation relates values of the zeta on the left- and right-hand sides of the complex plane. For integers <math>1\leq m \leq n <math>,
- <math>\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) <math> holds for all values of s.
Taylor series
The derivative of the zeta in the second argument is a shift:
- <math>\frac {\partial} {\partial q} \zeta (s,q) = -s\zeta(s+1,q)<math>
Thus, the Taylor series can be written as
- <math>\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) <math>
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 <math>\vartheta (z,\tau)<math> is the Jacobi theta function, then
- <math>\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]<math> holds for <math>\Re s > 0<math> and z complex, but not an integer. For z=n an integer, this simplifies to
- <math>\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)<math> 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 <math>t\rightarrow 0<math>.
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) (http://www.ams.org/journal-getitem?pii=S0025-5718-99-01091-1)
- Linas Vepstas, The Bernoulli Operator, the Gauss-Kuzmin-Wirsing Operator, and the Riemann Zeta (http://www.linas.org/math/chap-gkw/gkw.html)
