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
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 .
Helmut Hasse found the series representation in 1930
for and .
The Laurent development around is:
with . are the generalized Stieltjes constants :
The integral representation is
Hurwitz's formula is a representation of the function for and you reads:
It denotes the polylogarithm .
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 .
The Hurwitz zeta function occurs in connection with the Euler polynomials :
with . And are defined with the legendary Chi function as follows :
The following applies (selection):
( Riemann zeta function , Catalan's constant )
with as well as and .
The derivatives after result in
for and using the Pochhammer symbol .
Relationships with other functions
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 )
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
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 :
Literature and web links
- Jonathan Sondow, Eric W. Weisstein: Hurwitz Zeta Function on MathWorld and in functions.wolfram.com (English)
Milton Abramowitz , Irene A. Stegun : Handbook of Mathematical Functions . Dover Publications, New York 1964, ISBN 0-486-61272-4 . (See paragraph 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.) Vol. 118, No. 4, Nov. 2008, pp. 495-503
- Johan Andersson: Mean Value Properties of the Hurwitz Zeta Function . In: Math. Scand. Volume 71, 1992, pp. 295-300
↑ Helmut Hasse: A summation method for the Riemann ζ series In: Mathematische Zeitschrift. Volume 32, 1930, pp. 458-464.
↑ Eric W. Weisstein : Hurwitz's Formula . In: MathWorld (English).
^ 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
^ JWS Cassels: Footnote to a note of Davenport and Heilbronn . In: Journal of the London Mathematical Society. Volume 36, 1961, pp. 177-184
↑ 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.
↑ Oliver Espinosa and Victor H. Moll: A Generalized Polygamma Function on arXiv.org e-Print archive 2003.
^ J. Schwinger: On gauge invariance and vacuum polarization . In: Physical Review . Volume 82, 1951, pp. 664-679.