Lerch zeta function

In mathematics, the Lerch zeta function, sometimes called the Hurwitz–Lerch zeta-function, is a special function that generalizes the Hurwitz zeta function and the polylogarithm. It is named after the Czech mathematician Mathias Lerch .

Definition

The Lerch zeta function is given by

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

A related function, the Lerch transcendent, is given by

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

The two are related, as

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

Integral representations

An integral representation is given by

${\displaystyle \Phi (z,s,a)={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }{\frac {t^{s-1}e^{-at}}{1-ze^{-t}}}\,dt}$

for

${\displaystyle \Re (a)>0\wedge \Re (s)>0\wedge z<1\vee \Re (a)>0\wedge \Re (s)>1\wedge z=1.}$

A contour integral representation is given by

${\displaystyle \Phi (z,s,a)=-{\frac {\Gamma (1-s)}{2\pi i}}\int _{0}^{(+\infty )}{\frac {(-t)^{s-1}e^{-at}}{1-ze^{-t}}}\,dt}$

for

${\displaystyle \Re (a)>0\wedge \Re (s)<0\wedge z<1}$

where the contour must not enclose any of the points ${\displaystyle t=\log(z)+2k\pi i,k\in Z.}$

A Hermite-like integral representation is given by

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

for

${\displaystyle \Re (a)>0\wedge |z|<1}$

and

${\displaystyle \Phi (z,s,a)={\frac {1}{2a^{s}}}+{\frac {\log ^{s-1}(1/z)}{z^{a}}}\Gamma (1-s,a\log(1/z))+{\frac {2}{a^{s-1}}}\int _{0}^{\infty }{\frac {\sin(s\arctan(t)-ta\log(z))}{(1+t^{2})^{s/2}(e^{2\pi at}-1)}}\,dt}$

for

${\displaystyle \Re (a)>0.}$

Similar representations include

${\displaystyle \Phi (z,s,a)={\frac {1}{2a^{s}}}+\int _{0}^{\infty }{\frac {\cos(t\log z)\sin {\Big (}s\arctan {\tfrac {t}{a}}{\Big )}-\sin(t\log z)\cos {\Big (}s\arctan {\tfrac {t}{a}}{\Big )}}{{\big (}a^{2}+t^{2}{\big )}^{\frac {s}{2}}\tanh \pi t}}\,dt,}$

and

${\displaystyle \Phi (-z,s,a)={\frac {1}{2a^{s}}}+\int _{0}^{\infty }{\frac {\cos(t\log z)\sin {\Big (}s\arctan {\tfrac {t}{a}}{\Big )}-\sin(t\log z)\cos {\Big (}s\arctan {\tfrac {t}{a}}{\Big )}}{{\big (}a^{2}+t^{2}{\big )}^{\frac {s}{2}}\sinh \pi t}}\,dt,}$

holding for positive z (and more generally wherever the integrals converge). Furthermore,

${\displaystyle \Phi (e^{i\varphi },s,a)=L{\big (}{\tfrac {\varphi }{2\pi }},a,s{\big )}={\frac {1}{a^{s}}}+{\frac {1}{2\Gamma (s)}}\int _{0}^{\infty }{\frac {t^{s-1}e^{-at}{\big (}e^{i\varphi }-e^{-t}{\big )}}{\cosh {t}-\cos {\varphi }}}\,dt,}$

The last formula is also known as Lipschitz formula.

Special cases

The Hurwitz zeta function is a special case, given by

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

The polylogarithm is a special case of the Lerch Zeta, given by

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

The Legendre chi function is a special case, given by

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

The Riemann zeta function is given by

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

The Dirichlet eta function is given by

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

Identities

For λ rational, the summand is a root of unity, and thus ${\displaystyle L(\lambda ,\alpha ,s)}$ may be expressed as a finite sum over the Hurwitz zeta-function. Suppose ${\displaystyle \lambda ={\frac {p}{q}}}$ with ${\displaystyle p,q\in \mathbb {Z} }$ and ${\displaystyle q>0}$. Then ${\displaystyle z=\omega =e^{2\pi i{\frac {p}{q}}}}$ and ${\displaystyle \omega ^{q}=1}$.

${\displaystyle \Phi (\omega ,s,\alpha )=\sum _{n=0}^{\infty }{\frac {\omega ^{n}}{(n+\alpha )^{s}}}=\sum _{m=0}^{q-1}\sum _{n=0}^{\infty }{\frac {\omega ^{qn+m}}{(qn+m+\alpha )^{s}}}=\sum _{m=0}^{q-1}\omega ^{m}q^{-s}\zeta (s,{\frac {m+\alpha }{q}})}$

Various identities include:

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

and

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

and

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

Series representations

A series representation for the Lerch transcendent is given by

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

(Note that ${\displaystyle {\tbinom {n}{k}}}$ is a binomial coefficient.)

The series is valid for all s, and for complex z with Re(z)<1/2. Note a general resemblance to a similar series representation for the Hurwitz zeta function. [1]

A Taylor series in the first parameter was given by Erdélyi. It may be written as the following series, which is valid for

${\displaystyle |\log(z)|<2\pi ;s\neq 1,2,3,\dots ;a\neq 0,-1,-2,\dots }$

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

B. R. Johnson (1974). "Generalized Lerch zeta-function". Pacific J. Math. 53 (1): 189–193. doi:10.2140/pjm.1974.53.189.

If n is a positive integer, then

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

where ${\displaystyle \psi (n)}$ is the digamma function.

A Taylor series in the third variable is given by

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

where ${\displaystyle (s)_{k}}$ is the Pochhammer symbol.

Series at a = -n is given by

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

A special case for n = 0 has the following series

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

where ${\displaystyle \operatorname {Li} _{s}(z)}$ is the polylogarithm.

An asymptotic series for ${\displaystyle s\rightarrow -\infty }$

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

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

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

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

An asymptotic series in the incomplete gamma function

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

for ${\displaystyle |a|<1;\Re (s)<0.}$

Asymptotic expansion

The polylogarithm function ${\displaystyle \mathrm {Li} _{n}(z)}$ is defined as

${\displaystyle \mathrm {Li} _{0}(z)={\frac {z}{1-z}},\qquad \mathrm {Li} _{-n}(z)=z{\frac {d}{dz}}\mathrm {Li} _{1-n}(z).}$

Let

${\displaystyle \Omega _{a}\equiv {\begin{cases}\mathbb {C} \setminus [1,\infty )&{\text{if }}\Re a>0,\\{z\in \mathbb {C} ,|z|<1}&{\text{if }}\Re a\leq 0.\end{cases}}}$

For ${\displaystyle |\mathrm {Arg} (a)|<\pi ,s\in \mathbb {C} }$ and ${\displaystyle z\in \Omega _{a}}$, an asymptotic expansion of ${\displaystyle \Phi (z,s,a)}$ for large ${\displaystyle a}$ and fixed ${\displaystyle s}$ and ${\displaystyle z}$ is given by

${\displaystyle \Phi (z,s,a)={\frac {1}{1-z}}{\frac {1}{a^{s}}}+\sum _{n=1}^{N-1}{\frac {(-1)^{n}\mathrm {Li} _{-n}(z)}{n!}}{\frac {(s)_{n}}{a^{n+s}}}+O(a^{-N-s})}$

for ${\displaystyle N\in \mathbb {N} }$, where ${\displaystyle (s)_{n}=s(s+1)\cdots (s+n-1)}$ is the Pochhammer symbol.[2]

Let

${\displaystyle f(z,x,a)\equiv {\frac {1-(ze^{-x})^{1-a}}{1-ze^{-x}}}.}$

Let ${\displaystyle C_{n}(z,a)}$ be its Taylor coefficients at ${\displaystyle x=0}$. Then for fixed ${\displaystyle N\in \mathbb {N} ,\Re a>1}$ and ${\displaystyle \Re s>0}$,

${\displaystyle \Phi (z,s,a)-{\frac {\mathrm {Li} _{s}(z)}{z^{a}}}=\sum _{n=0}^{N-1}C_{n}(z,a){\frac {(s)_{n}}{a^{n+s}}}+O\left((\Re a)^{1-N-s}+az^{-\Re a}\right),}$

as ${\displaystyle \Re a\to \infty }$.[3]

Software

The Lerch transcendent is implemented as LerchPhi in Maple and Mathematica, and as lerchphi in mpmath and SymPy.

References

1. "The Analytic Continuation of the Lerch Transcendent and the Riemann Zeta Function". Retrieved 28 April 2020.
2. Ferreira, Chelo; López, José L. (October 2004). "Asymptotic expansions of the Hurwitz–Lerch zeta function". Journal of Mathematical Analysis and Applications. 298 (1): 210–224. doi:10.1016/j.jmaa.2004.05.040.
3. Cai, Xing Shi; López, José L. (10 June 2019). "A note on the asymptotic expansion of the Lerch's transcendent". Integral Transforms and Special Functions. 30 (10): 844–855. arXiv:1806.01122. doi:10.1080/10652469.2019.1627530. S2CID 119619877.

• Apostol, T. M. (2010), "Lerch's Transcendent", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
• Bateman, H.; Erdélyi, A. (1953), Higher Transcendental Functions, Vol. I (PDF), New York: McGraw-Hill. (See § 1.11, "The function Ψ(z,s,v)", p. 27)
• Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich; Jeffrey, Alan (2015) [October 2014]. "9.55.". In Zwillinger, Daniel; Moll, Victor Hugo (eds.). Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (8 ed.). Academic Press. ISBN 978-0-12-384933-5. LCCN 2014010276.
• Guillera, Jesus; Sondow, Jonathan (2008), "Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent", The Ramanujan Journal, 16 (3): 247–270, arXiv:math.NT/0506319, doi:10.1007/s11139-007-9102-0, MR 2429900, S2CID 119131640. (Includes various basic identities in the introduction.)
• Jackson, M. (1950), "On Lerch's transcendent and the basic bilateral hypergeometric series ₂ψ₂", J. London Math. Soc., 25 (3): 189–196, doi:10.1112/jlms/s1-25.3.189, MR 0036882.
• Johansson, F.; Blagouchine, Ia. (2019), "Computing Stieltjes constants using complex integration", Mathematics of Computation, 88 (318): 1829–1850, arXiv:1804.01679, doi:10.1090/mcom/3401, MR 3925487, S2CID 4619883.
• Laurinčikas, Antanas; Garunkštis, Ramūnas (2002), The Lerch zeta-function, Dordrecht: Kluwer Academic Publishers, ISBN 978-1-4020-1014-9, MR 1979048.
• Lerch, Mathias (1887), "Note sur la fonction ${\displaystyle \scriptstyle {\mathfrak {K}}(w,x,s)=\sum _{k=0}^{\infty }{e^{2k\pi ix} \over (w+k)^{s}}}$", Acta Mathematica (in French), 11 (1–4): 19–24, doi:10.1007/BF02612318, JFM 19.0438.01, MR 1554747, S2CID 121885446.

External links

• Aksenov, Sergej V.; Jentschura, Ulrich D. (2002), C and Mathematica Programs for Calculation of Lerch's Transcendent.
• Ramunas Garunkstis, Home Page (2005) (Provides numerous references and preprints.)
• Ramunas Garunkstis, Approximation of the Lerch Zeta Function (PDF)
• S. Kanemitsu, Y. Tanigawa and H. Tsukada, A generalization of Bochner's formula, (undated, 2005 or earlier)
• Weisstein, Eric W. "Lerch Transcendent". MathWorld.
• "§25.14, Lerch's Transcendent". NIST Digital Library of Mathematical Functions. National Institute of Standards and Technology. 2010. Retrieved 28 January 2012.