Hurwitz's zeta function

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The Hurwitz zeta function (after Adolf Hurwitz ) is one of the many known zeta functions that 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 expanded to a meromorphic function for all

The Riemann zeta function is then

Analytical continuation

The Hurwitz zeta function may be a function meromorphic continued to be so for all complex is defined. At is a simple pole with residual 1.

It then applies

using the gamma function and the digamma function .

Series representations

Helmut Hasse found the series representation in 1930

for and .

Laurent development

The Laurent development around is:

with . are the generalized Stieltjes constants :


Fourier series

with .

Integral representation

The integral representation is

where and

Hurwitz formula

Hurwitz's formula is a representation of the function for and you reads:

in which

It denotes the polylogarithm .

Functional equation

For everyone and applies



Since for and results in the Riemann zeta function or this multiplied by a simple function of , this leads to the complicated zero point calculation of the Riemann zeta function with the Riemann conjecture .

For these , the Hurwitz zeta function has no zeros with a real part greater than or equal to 1.

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

Rational arguments

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


Furthermore applies

with . And are defined with the legendary Chi function as follows :



The following applies (selection):

( Riemann zeta function , Catalan's constant )


It applies

with as well as and .

The derivatives after result in

for and using the Pochhammer symbol .

Relationships with other functions

Bernoulli polynomials

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

Alternatively, you can say that

For that results

Jacobian theta function

With , the Jacobian theta function applies

where and .

Is whole, this simplifies itself too

( with one argument stands for the Riemann zeta function )

Polygamma function

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

with the Euler-Mascheroni constant .


Hurwitz's zeta functions are used in various places, not just 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 by Julian Schwinger , which gives an exact result for the pair formation rate of electrons in fields described in the Dirac equation .

Special cases and generalizations

A generalization of the Hurwitz zeta function offers


so that

This function is called Lerch's zeta function .

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


In addition, with Meijer's G function :

with .

Literature and web links

Individual evidence

  1. Helmut Hasse: A summation method for the Riemann ζ series In: Mathematische Zeitschrift. Volume 32, 1930, pp. 458-464.
  3. Eric W. Weisstein : Hurwitz's Formula . In: MathWorld (English).
  4. ^ H. Davenport and H. Heilbronn: On the zeros of certain Dirichlet series . In: Journal of the London Mathematical Society. Volume 11, 1936, pp. 181-185
  5. ^ JWS Cassels: Footnote to a note of Davenport and Heilbronn . In: Journal of the London Mathematical Society. Volume 36, 1961, pp. 177-184
  6. Đurđe Cvijovic and Jacek Klinowski: Values of the Legendre chi and Hurwitz zeta functions at rational arguments . In: Mathematics of Computation . Volume 68, 1999, pp. 1623-1630.
  10. Oliver Espinosa and Victor H. Moll: A Generalized Polygamma Function on e-Print archive 2003.
  11. ^ J. Schwinger: On gauge invariance and vacuum polarization . In: Physical Review . Volume 82, 1951, pp. 664-679.