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Hurwitz Zeta FunctionIn 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. 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 be a character mod Q. Then we can write the Dirichlet L-function as -
\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 -
where -
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 and . Here, is the polylogarithm. Relation to Bernoulli polynomials The function defined above generalizes the Bernoulli polynomials: -
where denotes the real part of z. Alternately, -
Relation to the polygamma function The Hurwitz zeta is generalizes the polygamma function: -
Relation to the Lerch transcendant The Lerch transcendant generalizes the Hurwitz zeta: -
\frac { z^k} {(k+q)^s} and thus -
Functional equation The functional equation relates values of the zeta on the left- and right-hand sides of the complex plane. For integers , -
\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: -
Thus, the Taylor series can be written as -
\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 is the Jacobi theta function, then -
\pi^{-(1-s)/2} \Gamma \left( \frac {1-s}{2} \right) \left+ \zeta(1-s,1-z) \right holds for and z complex, but not an integer. For z=n an integer, this simplifies to -
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 . 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
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