Hurwitz's zeta function

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 ${\displaystyle s,q}$

${\displaystyle \zeta (s,q)=\sum _{n=0}^{\infty }{\frac {1}{(q+n)^{s}}}\qquad \quad \mathrm {Re} (s)>1{\text{ und Re}}(q)>0}$

The series converges absolutely and can be expanded to a meromorphic function for all ${\displaystyle s\not =1.}$

The Riemann zeta function is then ${\displaystyle \zeta (s,1).}$

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. ${\displaystyle s\not =1}$ ${\displaystyle s=1}$

It then applies

${\displaystyle \lim _{s\to 1}\left[\zeta (s,q)-{\frac {1}{s-1}}\right]={\frac {-\Gamma '(q)}{\Gamma (q)}}=-\psi (q)}$

using the gamma function ${\displaystyle \Gamma (\cdot )}$ and the digamma function ${\displaystyle \psi (\cdot )}$ .

Series representations

Helmut Hasse found the series representation in 1930

${\displaystyle \zeta (s,q)={\frac {1}{s-1}}\sum _{n=0}^{\infty }{\frac {1}{n+1}}\sum _{k=0}^{n}(-1)^{k}{n \choose k}(q+k)^{1-s}}$

for and . ${\displaystyle q>-1}$ ${\displaystyle s\in \mathbb {C} \setminus \{1\}}$

Laurent development

The Laurent development around is: ${\displaystyle s=1}$

${\displaystyle \zeta (s,q)={\frac {1}{s-1}}+\sum _{n=0}^{\infty }{\frac {(-1)^{n}\gamma _{n}(q)}{n!}}(s-1)^{n}}$

with . are the generalized Stieltjes constants : ${\displaystyle 0<q\leq 1}$ ${\displaystyle \gamma _{n}(q)}$

${\displaystyle \gamma _{n}(q):=\lim _{N\to \infty }\left(\sum _{k=0}^{N}{\frac {\log ^{n}(k+q)}{k+q}}-{\frac {\log ^{n+1}(N+q)}{n+1}}\right)}$

For ${\displaystyle n=0,1,2,\dots }$

Fourier series

${\displaystyle \zeta (s,a)=2(2\pi )^{s-1}\Gamma (1-s)\left(\sin \left({\frac {\pi s}{2}}\right)\sum _{k=1}^{\infty }{\frac {\cos(2\pi ak)}{k^{1-s}}}+\cos \left({\frac {\pi s}{2}}\right)\sum _{k=1}^{\infty }{\frac {\sin(2\pi ak)}{k^{1-s}}}\right)}$

with . ${\displaystyle \mathrm {Re} (s)<1{\text{ und }}0<a\leq 1}$

Integral representation

The integral representation is

${\displaystyle \zeta (s,q)={\frac {1}{\Gamma (s)}}\int \limits _{0}^{\infty }{\frac {t^{s-1}e^{-qt}}{1-e^{-t}}}\mathrm {d} t}$

where and ${\displaystyle \mathrm {Re} (s)>1}$ ${\displaystyle \mathrm {Re} (q)>0}$

Hurwitz formula

Hurwitz's formula is a representation of the function for and you reads: ${\displaystyle 0\leq x\leq 1}$ ${\displaystyle s>1.}$

${\displaystyle \zeta (1-s,x)={\frac {1}{2s}}\left[e^{-\mathrm {i} \pi s/2}\beta (x;s)+e^{\mathrm {i} \pi s/2}\beta (1-x;s)\right]}$

in which

${\displaystyle \beta (x;s)=2\Gamma (s+1)\sum _{n=1}^{\infty }{\frac {\exp(2\pi \mathrm {i} nx)}{(2\pi n)^{s}}}={\frac {2\Gamma (s+1)}{(2\pi )^{s}}}{\mbox{Li}}_{s}(e^{2\pi \mathrm {i} x})}$

It denotes the polylogarithm . ${\displaystyle {\mbox{Li}}_{s}(z)}$

Functional equation

For everyone and applies ${\displaystyle s}$ ${\displaystyle 1\leq m\leq n}$

${\displaystyle \zeta \left(1-s,{\frac {m}{n}}\right)={\frac {2\Gamma (s)}{(2\pi n)^{s}}}\sum _{k=1}^{n}\cos \left({\frac {\pi s}{2}}-{\frac {2\pi km}{n}}\right)\;\zeta \left(s,{\frac {k}{n}}\right).}$

values

zeropoint

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 . ${\displaystyle q=1}$ ${\displaystyle q={\tfrac {1}{2}}}$ ${\displaystyle s}$

For these , the Hurwitz zeta function has no zeros with a real part greater than or equal to 1. ${\displaystyle q}$

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 . ${\displaystyle 0<q<1}$ ${\displaystyle q\not ={\tfrac {1}{2}}}$ ${\displaystyle 1<\mathrm {Re} (s)<1+\epsilon }$ ${\displaystyle \epsilon }$ ${\displaystyle q}$ ${\displaystyle q}$

Rational arguments

The Hurwitz zeta function occurs in connection with the Euler polynomials : ${\displaystyle E_{n}(x)}$

${\displaystyle E_{2n-1}\left({\frac {p}{q}}\right)=(-1)^{n}{\frac {4(2n-1)!}{(2\pi q)^{2n}}}\sum _{k=1}^{q}\zeta \left(2n,{\frac {2k-1}{2q}}\right)\cos {\frac {(2k-1)\pi p}{q}}}$

and

${\displaystyle E_{2n}\left({\frac {p}{q}}\right)=(-1)^{n}{\frac {4(2n)!}{(2\pi q)^{2n+1}}}\sum _{k=1}^{q}\zeta \left(2n+1,{\frac {2k-1}{2q}}\right)\sin {\frac {(2k-1)\pi p}{q}}.}$

Furthermore applies

${\displaystyle \zeta \left(s,{\frac {2p-1}{2q}}\right)=2(2q)^{s-1}\sum _{k=1}^{q}\left[C_{s}\left({\frac {k}{q}}\right)\cos \left({\frac {(2p-1)\pi k}{q}}\right)+S_{s}\left({\frac {k}{q}}\right)\sin \left({\frac {(2p-1)\pi k}{q}}\right)\right]}$

with . And are defined with the legendary Chi function as follows : ${\displaystyle 1\leq p\leq q}$ ${\displaystyle C_{\nu }(x)}$ ${\displaystyle S_{\nu }(x)}$ ${\displaystyle \chi _{\nu }}$

${\displaystyle C_{\nu }(x)=\operatorname {Re} \,\chi _{\nu }(e^{\mathrm {i} x})}$

or.

${\displaystyle S_{\nu }(x)=\operatorname {Im} \,\chi _{\nu }(e^{\mathrm {i} x}).}$

Further

The following applies (selection):

${\displaystyle \zeta (s,-1)=\zeta (s)+1\,}$

${\displaystyle \zeta (s,2)=\zeta (s)-1\,}$

${\displaystyle \zeta (s,0)=\zeta (s,1)\,}$

${\displaystyle \zeta \left(s,{\frac {m}{n}}\right)={\frac {1}{n}}\sum _{k=1}^{n}n^{s}\cdot \mathrm {Li} _{s}\left(e^{\frac {2\pi \mathrm {i} k}{n}}\right)e^{-{\frac {2\pi \mathrm {i} km}{n}}}\qquad \qquad m,n\in \mathbb {N} ^{+}{\text{ und }}m\leq n}$

${\displaystyle \zeta (0,a)={\frac {1}{2}}-a}$

${\displaystyle \zeta (2,{\tfrac {1}{4}})=\pi ^{2}+8G}$

${\displaystyle \zeta (2,{\tfrac {1}{2}}+{\tfrac {x}{\pi }})+\zeta (2,{\tfrac {1}{2}}-{\tfrac {x}{\pi }})={\frac {\pi ^{2}}{\cos ^{2}x}}}$

( Riemann zeta function , Catalan's constant )

Derivatives

It applies

${\displaystyle {\frac {\partial ^{n}\zeta (s,a)}{\partial s^{n}}}={\frac {(-1)^{n}}{2^{n}}}\sum _{k=0}^{\infty }{\frac {\log ^{n}\left((a+k)^{2}\right)}{\left((a+k)^{2}\right)^{s/2}}}}$

with as well as and . ${\displaystyle -a\notin \mathbb {N} }$ ${\displaystyle \mathrm {Re} (s)>1}$ ${\displaystyle n\in \mathbb {N} }$

The derivatives after result in ${\displaystyle a}$

${\displaystyle {\frac {\partial ^{n}\zeta (s,a)}{\partial a^{n}}}=(1-n-s,n)\sum _{k=0}^{\infty }{\frac {1}{(a+k)^{n}\left((a+k)^{2}\right)^{s/2}}}}$

for and using the Pochhammer symbol . ${\displaystyle a\notin \mathbb {N} }$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle (x,n)}$

Relationships with other functions

Bernoulli polynomials

The function defined in the Hurwitz formula section generalizes the Bernoulli polynomials : ${\displaystyle \beta }$ ${\displaystyle B_{n}(x)}$

${\displaystyle B_{n}(x)=-\mathrm {Re} \left[(-\mathrm {i} )^{n}\beta (x;n)\right]}$

Alternatively, you can say that

${\displaystyle \zeta (-n,x)=-{\frac {B_{n+1}(x)}{n+1}}.}$

For that results ${\displaystyle n=0}$

${\displaystyle \zeta (0,x)={\frac {1}{2}}-x.}$

Jacobian theta function

With , the Jacobian theta function applies ${\displaystyle \vartheta (z,\tau )}$

${\displaystyle \int \limits _{0}^{\infty }\left[\vartheta (z,\mathrm {i} t)-1\right]t^{s/2}{\frac {\mathrm {d} t}{t}}=\pi ^{-(1-s)/2}\Gamma \left({\frac {1-s}{2}}\right)\left[\zeta (1-s,z)+\zeta (1-s,1-z)\right]}$

where and . ${\displaystyle \mathrm {Re} (s)>0}$ ${\displaystyle z\in \mathbb {C} \,\setminus \,\mathbb {Z} }$

Is whole, this simplifies itself too ${\displaystyle z=n}$

${\displaystyle \int \limits _{0}^{\infty }\left[\vartheta (n,\mathrm {i} t)-1\right]t^{s/2}{\frac {\mathrm {d} t}{t}}=2\ \pi ^{-(1-s)/2}\ \Gamma \left({\frac {1-s}{2}}\right)\zeta (1-s)=2\ \pi ^{-s/2}\ \Gamma \left({\frac {s}{2}}\right)\zeta (s).}$

( with one argument stands for the Riemann zeta function ) ${\displaystyle \zeta }$

Polygamma function

The Hurwitz zeta function generalizes the polygamma function to non-whole orders : ${\displaystyle s}$

${\displaystyle \psi _{s}(z)={\frac {1}{\Gamma (-s)}}\left({\frac {\partial }{\partial s}}+\psi (-s)+\gamma \right)\zeta (s+1,z)}$

with the Euler-Mascheroni constant . ${\displaystyle \gamma }$

Occur

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

${\displaystyle \Phi (z,s,q)=\sum _{k=0}^{\infty }{\frac {z^{k}}{(k+q)^{s}}}}$,

so that

${\displaystyle \zeta (s,q)=\Phi (1,s,q).}$

This function is called Lerch's zeta function .

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

${\displaystyle \zeta (s,a)=a^{-s}\cdot {}_{s+1}F_{s}(1,a_{1},a_{2},\ldots a_{s};a_{1}+1,a_{2}+1,\ldots a_{s}+1;1)}$

With ${\displaystyle a_{1}=a_{2}=\ldots =a_{s}=a{\text{ und }}a\notin \mathbb {N} {\text{ und }}s\in \mathbb {N} ^{+}.}$

In addition, with Meijer's G function :

${\displaystyle \zeta (s,a)=G\,_{s+1,\,s+1}^{\,1,\,s+1}\left(-1\;\left|\;{\begin{matrix}0,1-a,\ldots ,1-a\\0,-a,\ldots ,-a\end{matrix}}\right)\right.}$

with . ${\displaystyle s\in \mathbb {N} ^{+}}$

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

Individual evidence

1. ↑ Helmut Hasse: A summation method for the Riemann ζ series In: Mathematische Zeitschrift. Volume 32, 1930, pp. 458-464.
2. ↑ http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta2/06/03/01/01/0001/
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.
7. ↑ http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta2/03/ShowAll.html
8. ↑ http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta2/20/02/01/01/0001/
9. ↑ http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta2/20/02/02/01/0001/
10. ↑ Oliver Espinosa and Victor H. Moll: A Generalized Polygamma Function on arXiv.org e-Print archive 2003.
11. ^ J. Schwinger: On gauge invariance and vacuum polarization . In: Physical Review . Volume 82, 1951, pp. 664-679.
12. ↑ http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta2/26/01/02/01/
13. ↑ http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta2/26/02/01/01/