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.^[1]

Definition

The Lerch zeta function is given by

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

A related function, the Lerch transcendent, is given by

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

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

${\displaystyle |z|<1}$, or

${\displaystyle {\mathfrak {R}}(s)>1}$, and ${\displaystyle |z|=1}$.^[2]

The two are related, as

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

Integral representations

The Lerch transcendent has an integral representation:

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

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

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

and then interchanging the sum and integral. The resulting integral representation converges for ${\displaystyle z\in \mathbb {C} \setminus [1,\infty ),}$ Re(s) > 0, and Re(a) > 0. This analytically continues ${\displaystyle \Phi (z,s,a)}$ 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.^[3][4]

A contour integral representation is given by

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

where C is a Hankel contour counterclockwise around the positive real axis, not enclosing any of the points ${\displaystyle t=\log(z)+2k\pi i}$ (for integer k) which are poles of the integrand. The integral assumes Re(a) > 0.^[5]

Other integral representations

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,

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

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^[6]

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

The polylogarithm is another special case:^[6]

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

The Riemann zeta function is a special case of both of the above:^[6]

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

Other special cases include:

• The Dirichlet eta function:^[6]

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

• The Dirichlet beta function:^[6]

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

• The Legendre chi function:^[6]

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

• The polygamma function:^{[citation needed]}

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

Identities

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

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

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.^[7]

A Taylor series in the first parameter was given by Arthur Erdélyi. It may be written as the following series, which is valid for^[8]

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

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

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

The representation as a generalized hypergeometric function is^[9]

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

Asymptotic expansion

The polylogarithm function ${\displaystyle \mathrm {Li} _{n}(z)}$ 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.^[10]

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 ${\displaystyle \Re a\to \infty }$.^[11]

Software

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

References

1. 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
2. https://arxiv.org/pdf/math/0506319.pdf
3. Bateman & Erdélyi 1953, p. 27
4. Guillera & Sondow 2008, Lemma 2.1 and 2.2
5. Bateman & Erdélyi 1953, p. 28
6. Guillera & Sondow 2008, p. 248–249
7. "The Analytic Continuation of the Lerch Transcendent and the Riemann Zeta Function". 27 April 2020. Retrieved 28 April 2020.
8. B. R. Johnson (1974). "Generalized Lerch zeta function". Pacific J. Math. 53 (1): 189–193. doi:10.2140/pjm.1974.53.189.
9. Gottschalk, J. E.; Maslen, E. N. (1988). "Reduction formulae for generalized hypergeometric functions of one variable". J. Phys. A. 21 (9): 1983–1998. Bibcode:1988JPhA...21.1983G. doi:10.1088/0305-4470/21/9/015.
10. 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.
11. 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.

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.