Hurwitz zeta function

The Hurwitz zeta function (after Adolf Hurwitz ) is one of many well-known zeta functions, which plays an important role in analytic number theory, a branch of mathematics.

The formal definition of complex is

The series converges absolutely and can be extended to a meromorphic function for all

The Riemann zeta function is then

  • 6.1 Zeroing
  • 6.2 Rational arguments
  • 6.3 Other
  • 8.1 Bernoulli polynomials
  • 8.2 Jacobi theta function
  • 8.3 Polygammafunktion

Analytic continuation

The Hurwitz zeta function can be continued to a meromorphic function such that it is defined for all complex. When there is a simple pole with residue 1.

It is then

Using the gamma function, and the Digammafunktion.

Series representations

Helmut Hasse was around 1930 that the series representation:

Laurent expansion

The Laurent expansion is to:

The Generalized Stieltjes constants are:

Fourier series

Integral representation

Hurwitz formula

The formula of Hurwitz is a representation of the function and is:

In which

This means the Polylogarithmus.

Functional equation

For all and is

Values

Zeros

Since it is clear and for the Riemann zeta - function and multiplies this by a simple function of, this leads to the complex zeros of calculation of the Riemann zeta function with the Riemann Hypothesis.

For this q is the Hurwitz zeta function has no zeros with real part greater than or equal 1

For and on the other hand, there are zeros for each stripe with a positive - real. This was proved for rational and non- algebraic - irrational of Davenport and Heilbronn; for algebraic irrational of Cassels.

Rational arguments

The Hurwitz zeta function occurs for instance in connection with the Euler polynomials:

And

Furthermore, applies

With. Here, and defined as the set thresh Chi function:

More

It applies (selection):

( Riemann zeta - function, Catalansche constant)

Derivations

It is

Or

With the Pochhammer symbol.

Relations to other functions

Bernoulli polynomials

The defined in section Hurwitz formula function generalizes the Bernoulli polynomials:

Alternatively, one can say that

For the results

Jacobi theta function

With true, the Jacobi theta function

Is all, this simplifies to

( With one argument stands for the Riemann zeta - function! )

Polygammafunktion

The Hurwitz zeta function generalizes the Polygammafunktion to non- whole orders:

With the Euler - Mascheroni constant.

Occurrence

The Hurwitz zeta functions find application in various places, not only in number theory. It occurs in fractals and dynamic systems as well as in Zipf's law.

In particle physics, it occurs in a formula of Julian Schwinger, which gives an accurate result for the pair formation rate described in the Dirac equation in electron fields.

Special cases and generalizations

Provides a generalization of the Hurwitz zeta function

So that

This function is called Lerchsche zeta function.

The Hurwitz zeta function can be expressed by the hypergeometric function:

Also, applies with the Meijer 's G- function:

Literature and links

  • Jonathan Sondow, Eric W. Weisstein: Hurwitz Zeta Function at MathWorld and functions.wolfram.com (English)
  • Milton Abramowitz, Irene A. Stegun: Handbook of Mathematical Functions. Dover Publications, New York 1964, ISBN 0-486-61272-4. (See Section 6.4.10 )
  • Victor S. Adamchik: Derivatives of the Hurwitz Zeta Function for Rational Arguments. In: Journal of Computational and Applied Mathematics. Volume 100, 1998, pp. 201-206.
  • Necdet batit: New inequalities for the Hurwitz zeta function ( PDF; 115 kB). In: Proc. Indian Acad. Sci. ( Math Sci. ) Volume 118, No. 4, November 2008, pp. 495-503
  • Johan Andersson: Mean Value Properties of the Hurwitz Zeta Function. In: Math Scand. Volume 71, 1992, pp. 295-300
List of zeta functions Absolute convergence Harold Davenport Pochhammer-Symbol Bernoulli number Dynamical system Pair production Meijer G-function Irene Stegun Wolfram Research
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