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 [1].

Definition

The Lerch zeta function is given by

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

A related function, the Lerch transcendent, is given by

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

The two are related, as

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

Integral representations

An integral representation is given by

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

for

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

A contour integral representation is given by

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

for

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

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

A Hermite-like integral representation is given by

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

for

$\mathrm{\Re }(a)>0\wedge \left|z\right|<1$

and

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

for

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

Similar representations include

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

and

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

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

$\mathrm{\Phi }(e^{i\phi },s,a)=L(\frac{\phi }{2\pi },a,s)=\frac{1}{a^s}+\frac{1}{2\mathrm{\Gamma }(s)}\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{t^{s-1}{e}^{-at}(e^{i\phi }-e^{-t})}{\cosh t-\cos \phi }dt,$

The last formula is also known as Lipschitz formula.

Special cases

The Hurwitz zeta function is a special case, given by

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

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

${\text{<mrow data-mjx-texclass="ORD">Li</mrow>}}_s(z)=z\mathrm{\Phi }(z,s,1).$

The Legendre chi function is a special case, given by

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

The Riemann zeta function is given by

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

The Dirichlet eta function is given by

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

Identities

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

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

Various identities include:

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

and

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

and

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

Series representations

A series representation for the Lerch transcendent is given by

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

(Note that $(\genfrac{}{}{0ex}{}{n}{k})$ 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 Erdélyi. It may be written as the following series, which is valid for

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

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

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

$\mathrm{\Phi }(z,n,a)=z^{-a}\{\sum _{\genfrac{}{}{0ex}{}{k=0}{k\ne n-1}}^{\mathrm{\infty }}\zeta (n-k,a)\frac{{\log }^k(z)}{k!}+[\psi (n)-\psi (a)-\log (-\log (z))]\frac{{\log }^{n-1}(z)}{(n-1)!}\},$

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

A Taylor series in the third variable is given by

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

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

Series at a = -n is given by

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

A special case for n = 0 has the following series

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

where ${\mathrm{Li}}_s(z)$ is the polylogarithm.

An asymptotic series for $s\to -\mathrm{\infty }$

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

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

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

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

An asymptotic series in the incomplete gamma function

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

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

Asymptotic expansion

The polylogarithm function ${\mathrm{L}\mathrm{i}}_n(z)$ is defined as

${\mathrm{L}\mathrm{i}}_0(z)=\frac{z}{1-z},{\mathrm{L}\mathrm{i}}_{-n}(z)=z\frac{d}{dz}{\mathrm{L}\mathrm{i}}_{1-n}(z).$

Let

${\mathrm{\Omega }}_a\equiv \{\begin{array}{ll}\mathbb{ℂ}\setminus [1,\mathrm{\infty })& \text{if }\mathrm{\Re }a>0,\\ z\in \mathbb{ℂ},\left|z\right|<1& \text{if }\mathrm{\Re }a\le 0.\end{array}$

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

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

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

Let

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

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

$\mathrm{\Phi }(z,s,a)-\frac{{\mathrm{L}\mathrm{i}}_s(z)}{z^a}=\sum _{n=0}^{N-1}{C}_n(z,a)\frac{(s{)}_n}{a^{n+s}}+O((\mathrm{\Re }a{)}^{1-N-s}+a{z}^{-\mathrm{\Re }a}),$

as $\mathrm{\Re }a\to \mathrm{\infty }$.

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.
• Lerch, Mathias (1887), "Note sur la fonction $\mathfrak{K}(w,x,s)=\sum _{k=0}^{\mathrm{\infty }}\frac{e^{2k\pi ix}}{(w+k{)}^s}$", Acta Mathematica (in French), 11 (1–4): 19–24, doi:10.1007/BF02612318, JFM 19.0438.01, MR 1554747, S2CID 121885446CS1 maint: unrecognized language (link).

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.