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 Czech mathematician Mathias Lerch, who published a paper about the function in 1887.

Definition

The Lerch zeta function is given by

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

A related function, the Lerch transcendent, is given by

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

The transcendent only converges for any real number $\alpha >0$, where:

$|z|<1$, or

${\mathfrak {R}}(s)>1$, and $|z|=1$.

The two are related, as

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

Integral representations

The Lerch transcendent has an integral representation:

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

The proof is based on using the integral definition of the Gamma function to write

$\Phi (z,s,a)\Gamma (s)=\sum _{n=0}^{\infty }{\frac {z^{n}}{(n+a)^{s}}}\int _{0}^{\infty }x^{s}e^{-x}{\frac {dx}{x}}=\sum _{n=0}^{\infty }\int _{0}^{\infty }t^{s}z^{n}e^{-(n+a)t}{\frac {dt}{t}}$

and then interchanging the sum and integral. The resulting integral representation converges for $z\in \mathbb {C} \setminus [1,\infty ),$ Re(s) > 0, and Re(a) > 0. This analytically continues $\Phi (z,s,a)$ to z outside the unit disk. The integral formula also holds if z = 1, Re(s) > 1, and Re(a) > 0; see Hurwitz zeta function.

A contour integral representation is given by

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

where C is a Hankel contour counterclockwise around the positive real axis, not enclosing any of the points $t=\log(z)+2k\pi i$ (for integer k) which are poles of the integrand. The integral assumes Re(a) > 0.

Other integral representations

A Hermite-like integral representation is given by

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

for

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

and

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

for

$\Re (a)>0.$

Similar representations include

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

and

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

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

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

The last formula is also known as Lipschitz formula.

Special cases

The Lerch zeta function and Lerch transcendent generalize various special functions.

The Hurwitz zeta function is the special case

$\zeta (s,\alpha )=L(0,s,\alpha )=\Phi (1,s,\alpha )=\sum _{n=0}^{\infty }{\frac {1}{(n+\alpha )^{s}}}.$

The polylogarithm is another special case:

${\textrm {Li}}_{s}(z)=z\Phi (z,s,1)=\sum _{n=1}^{\infty }{\frac {z^{n}}{n^{s}}}.$

The Riemann zeta function is a special case of both of the above:

$\zeta (s)=\Phi (1,s,1)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}$

Other special cases include:

• The Dirichlet eta function:

$\eta (s)=\Phi (-1,s,1)=\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}}{n^{s}}}$

• The Dirichlet beta function:

$\beta (s)=2^{-s}\Phi (-1,s,1/2)=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{(2k+1)^{s}}}$

• The Legendre chi function:

$\chi _{s}(z)=2^{-s}z\Phi (z^{2},s,1/2)=\sum _{k=0}^{\infty }{\frac {z^{2k+1}}{(2k+1)^{s}}}$

• The polygamma function:

$\psi ^{(n)}(\alpha )=(-1)^{n+1}n!\Phi (1,n+1,\alpha )$

Identities

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

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

Various identities include:

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

and

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

and

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

Series representations

A series representation for the Lerch transcendent is given by

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

(Note that ${\tbinom {n}{k}}$ 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.

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

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

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

If n is a positive integer, then

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

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

A Taylor series in the third variable is given by

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

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

Series at a = −n is given by

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

A special case for n = 0 has the following series

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

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

An asymptotic series for $s\rightarrow -\infty$

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

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

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

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

An asymptotic series in the incomplete gamma function

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

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

The representation as a generalized hypergeometric function is

$\Phi (z,s,\alpha )={\frac {1}{\alpha ^{s}}}{}_{s+1}F_{s}\left({\begin{array}{c}1,\alpha ,\alpha ,\alpha ,\cdots \\1+\alpha ,1+\alpha ,1+\alpha ,\cdots \\\end{array}}\mid z\right).$

Asymptotic expansion

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

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

Let

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

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

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

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

Let

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

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

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

as $\Re a\to \infty$.

Software

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

References

• 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.

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.)
• Garunkstis, Ramunas (2004). "Approximation of the Lerch Zeta Function" (PDF). Lithuanian Mathematical Journal. 44 (2): 140–144. doi:10.1023/B:LIMA.0000033779.41365.a5. S2CID 123059665.
• Kanemitsu, S.; Tanigawa, Y.; Tsukada, H. (2015). "A generalization of Bochner's formula". Kanemitsu, S.; Tanigawa, Y.; Tsukada, H. (2004). "A generalization of Bochner's formula". Hardy-Ramanujan Journal. 27. doi:10.46298/hrj.2004.150.
• Weisstein, Eric W. "Lerch Transcendent". MathWorld.
• Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., eds. (2010), "Lerch's Transcendent", NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.