Lerch's zeta function

The Lerch zeta function (after Mathias Lerch ) is a very general zeta function . Very many series of reciprocal powers (including the Hurwitz zeta function and the polylogarithm ) can be represented as a special case of this function.

definition

The two functions

${\displaystyle L(\lambda ,\alpha ,s)=\sum _{n=0}^{\infty }{\frac {\exp(2\,\pi \,i\,\lambda \,n)}{(n+\alpha )^{s}}}}$

and

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

are called Lerch's zeta function . The relationship between the two is through

${\displaystyle \,\Phi (\exp(2\,\pi \,\,i\,\lambda ),s,\alpha )=L(\lambda ,\alpha ,s)}$

given.

Special cases and special values

• The Hurwitz zeta function :

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

• The polylogarithm :

${\displaystyle \,{\textrm {Li}}_{s}(z)=z\,\Phi (z,s,1)}$

• The Legendresche Chi function :

${\displaystyle \,\chi _{n}(z)=2^{-n}\,z\,\Phi (z^{2},n,{\tfrac {1}{2}})}$

• The Riemann ζ-function :

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

• The Dirichlet η function :

${\displaystyle \,\eta (s)=\Phi (-1,s,1)}$

• The Dirichlet beta function :

${\displaystyle \beta (s)=2^{-s}\,\Phi (-1,s,{\tfrac {1}{2}})}$

The following special cases also apply (selection):

${\displaystyle \Phi (z,s,1)={\frac {\mathrm {Li} _{s}(z)}{z}}}$

${\displaystyle \Phi (z,0,a)={\frac {1}{1-z}}}$

${\displaystyle \Phi (0,s,a)=\left(a^{2}\right)^{-{\frac {s}{2}}}}$

${\displaystyle \Phi (0,s,a)=a^{-s}\,}$

${\displaystyle \Phi (z,1,1)=-{\frac {\log(1-z)}{z}}}$

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

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

${\displaystyle \Phi (0,1,a)={\frac {1}{\sqrt {a^{2}}}}}$

Furthermore is

${\displaystyle {\begin{aligned}&\Phi (-1,2,{\tfrac {1}{2}})&=&\;4\,G\\&{\frac {\partial \Phi }{\partial s}}(-1,-1,1)&=&\;\log \left({\frac {A^{3}}{{\sqrt[{3}]{2}}\,{\sqrt[{4}]{\mathrm {e} }}}}\right)\\&{\frac {\partial \Phi }{\partial s}}(-1,-2,1)&=&\;{\frac {7\,\zeta (3)}{4\,\pi ^{2}}}\\&{\frac {\partial \Phi }{\partial s}}(-1,-1,{\tfrac {1}{2}})&=&\;{\frac {G}{\pi }}\end{aligned}}}$

with the Catalan constant ${\displaystyle G}$ , the Glaisher-Kinkelin constant ${\displaystyle A}$ and the Apéry constant of ${\displaystyle \zeta (3)}$ the Riemann zeta function.

More formulas

Integral representations

One possible integral representation is

${\displaystyle \Phi (z,s,a)={\frac {1}{\Gamma (s)}}\int \limits _{0}^{\infty }{\frac {t^{s-1}\mathrm {e} ^{-at}}{1-z\,\mathrm {e} ^{-t}}}\,\mathrm {d} t\qquad \quad }$ For ${\displaystyle {\begin{cases}&\mathrm {Re} \;a>0{\text{ und }}\mathrm {Re} \;s>0{\text{ und }}z<1\\{\text{oder }}&\mathrm {Re} \;a>0{\text{ und }}\mathrm {Re} \;s>1{\text{ und }}z=1\end{cases}}}$

The curve integral

${\displaystyle \Phi (z,s,a)=-{\frac {\Gamma (1-s)}{2\,\pi \,i}}\int \limits _{0}^{\infty }{\frac {(-t)^{s-1}\mathrm {e} ^{-a\,t}}{1-z\,\mathrm {e} ^{-t}}}\,\mathrm {d} t}$

with must not contain the points . ${\displaystyle \mathrm {Re} \;a>0,\;\mathrm {Re} \;s<0,\;z<1}$ ${\displaystyle t=\log z+2\,k\,\pi \,i,\;k\in \mathbb {Z} }$

Furthermore is

${\displaystyle \Phi (z,s,a)={\frac {1}{2\,a^{s}}}+\int \limits _{0}^{\infty }{\frac {z^{t}}{(a+t)^{s}}}\,\mathrm {d} t+{\frac {2}{a^{s-1}}}\int \limits _{0}^{\infty }{\frac {\sin(s\arctan t-a\,t\log z)}{(1+t^{2})^{s/2}\cdot (\mathrm {e} ^{2\,\pi \,a\,t}-1)}}\,\mathrm {d} t}$

for and . ${\displaystyle \mathrm {Re} \;a>0}$ ${\displaystyle |z|<1}$

Likewise is

${\displaystyle \Phi (z,s,a)={\frac {1}{2\,a^{s}}}+{\frac {\log ^{s-1}{\dfrac {1}{z}}}{z^{a}}}\,\Gamma (1-s,a\log {\dfrac {1}{z}})+{\frac {2}{a^{s-1}}}\int \limits _{0}^{\infty }{\frac {\sin(s\arctan t-a\,t\log z)}{(1+t^{2})^{s/2}\cdot (\mathrm {e} ^{2\,\pi \,a\,t}-1)}}\,\mathrm {d} t}$

for . ${\displaystyle \mathrm {Re} \,a>0}$

Series representations

A series representation for the transcendent lark is

${\displaystyle \Phi (z,s,q)={\frac {1}{1-z}}\sum _{n=0}^{\infty }\left({\frac {-z}{1-z}}\right)^{n}\sum _{k=0}^{n}(-1)^{k}{n \choose k}(q+k)^{-s}.}$

It applies to all and complex with ; compare the series representation of the Hurwitz zeta function . ${\displaystyle s}$ ${\displaystyle z}$ ${\displaystyle \mathrm {Re} \,z<{\tfrac {1}{2}}}$

If is positive and whole, then applies ${\displaystyle s}$

${\displaystyle \Phi (z,n,a)=z^{-a}\left\{\sum _{{k=0} \atop k\neq n-1}^{\infty }\zeta (n-k,a){\frac {\log ^{k}z}{k!}}+\left[\Psi (n)-\Psi (a)-\log(-\log z)\right]{\frac {\log ^{n-1}z}{(n-1)!}}\right\}.}$

A Taylor series of the third variable is through

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

given for using the Pochhammer symbol . ${\displaystyle |x|<\mathrm {Re} \,a}$ ${\displaystyle (s,k)}$

The following applies in the limit value ${\displaystyle a\rightarrow -n}$

${\displaystyle \Phi (z,s,a)=\sum _{k=0}^{n}{\frac {z^{k}}{(a+k)^{s}}}+z^{n}\sum _{m=0}^{\infty }(1-m-s,m)\,\mathrm {Li} _{s+m}(z){\frac {(a+n)^{m}}{m!}}}$.

The special case has the following series: ${\displaystyle n=0}$

${\displaystyle \Phi (z,s,a)={\frac {1}{a^{s}}}+\sum _{m=0}^{\infty }(1-m-s,m)\,\mathrm {Li} _{s+m}(z){\frac {a^{m}}{m!}}}$

for . ${\displaystyle |a|<1}$

The asymptotic expansion for is given by ${\displaystyle s\rightarrow -\infty }$

${\displaystyle \Phi (z,s,a)=z^{-a}\,\Gamma (1-s)\sum _{k=-\infty }^{\infty }\left[2\,k\,\pi \,i-\log z\right]^{s-1}\mathrm {e} ^{2\,k\,\pi \,a\,i}}$

for and ${\displaystyle |a|<1,\;\mathrm {Re} \;s<0,\;z\notin (-\infty ,0)}$

${\displaystyle \Phi (-z,s,a)=z^{-a}\,\Gamma (1-s)\sum _{k=-\infty }^{\infty }\left[(2\,k+1)\pi \,i-\log z\right]^{s-1}\mathrm {e} ^{(2\,k+1)\pi \,a\,i}}$

if . ${\displaystyle |a|<1,\;\mathrm {Re} \,s<0,\;z\notin (0,\infty )}$

Using the incomplete gamma function holds

${\displaystyle \Phi (z,s,a)={\frac {1}{2\,a^{s}}}+{\frac {1}{z^{a}}}\sum _{k=1}^{\infty }{\frac {\mathrm {e} ^{-2\,\pi \,i\,(k-1)a}\,\Gamma (1-s,a\,(-2\,\pi \,i\,(k-1)-\log z))}{(-2\,\pi \,i\,(k-1)-\log z)^{1-s}}}+{\frac {\mathrm {e} ^{2\,\pi \,i\,k\,a}\,\Gamma (1-s,a\,(2\,\pi \,i\,k-\log z))}{(2\,\pi \,i\,k-\log z)^{1-s}}}}$

with and . ${\displaystyle |a|<1}$ ${\displaystyle \mathrm {Re} \,s<0}$

Identities and other formulas

${\displaystyle \Phi (z,s,a)=z^{n}\,\Phi (z,s,a+n)+\sum _{k=0}^{n-1}{\frac {z^{k}}{(k+a)^{s}}}}$

${\displaystyle \Phi (z,s-1,a)=\left(a+z{\frac {\partial }{\partial z}}\right)\Phi (z,s,a)}$

${\displaystyle \Phi (z,s+1,a)=-\,{\frac {1}{s}}\,{\frac {\partial }{\partial a}}\Phi (z,s,a)}$

Furthermore applies to the integral representation with or ${\displaystyle \{z\in \mathbb {C} \,\setminus [1,\infty ){\text{ und }}\mathrm {Re} \;s>-2\}}$ ${\displaystyle \{z=1{\text{ und Re}}\;s>-1\}}$

${\displaystyle \int \limits _{0}^{1}\int \limits _{0}^{1}{\frac {x^{u-1}\cdot y^{v-1}}{1-x\,y\,z}}(-\log(x\,y))^{s}\,\mathrm {d} x\,\mathrm {d} y=\Gamma (s+1)\,{\frac {\Phi (z,s+1,v)-\Phi (z,s+1,u)}{u-v}}}$

and

${\displaystyle \int \limits _{0}^{1}\int \limits _{0}^{1}{\frac {(x\,y)^{u-1}}{1-x\,y\,z}}(-\log(x\,y))^{s}\,\mathrm {d} x\,\mathrm {d} y=\Gamma (s)\,\Phi (z,s+2,u)}$.

literature

• Mathias Lerch : Démonstration élémentaire de la formule :, L'Enseignement Mathématique 5 (1903): pp. 450–453 ${\displaystyle \textstyle {\frac {\pi ^{2}}{\sin ^{2}{\pi x}}}=\sum _{\nu =-\infty }^{\infty }{\frac {1}{(x+\nu )^{2}}}}$
• M. Jackson: On Lerch's transcendent and the basic bilateral hypergeometric series ${\displaystyle {}_{2}\psi _{2}}$ , J. London Math. Soc. 25 (3), 1950: pp. 189-196
• Jesús Guillera, Jonathan Sondow: Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent . In: Ramanujan J. Volume 16, Number 3, 2008, pages 247-270; see. in arxiv
• Antanas Laurinčikas and Ramūnas Garunkštis: The Lerch zeta-function , Dordrecht: Kluwer Academic Publishers, 2002, ISBN 978-1-4020-1014-9 online

Web links

• Ramunas Garunkstis: Home Page (reference collection)
• Ramunas Garunkstis, Approximation of the Lerch Zeta Function (PDF; 112 kB)
• S. Kanemitsu, Y. Tanigawa and H. Tsukada: A generalization of Bochner's formula , (undated, 2005 or earlier)
• Eric W. Weisstein : Lerch Transcendent . In: MathWorld (English).

Individual evidence

1. ↑ http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/LerchPhi/03/ShowAll.html
2. ↑ Guillera, Sondow 2008, Theorem 3.1 (see Ref.)