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

[math]\displaystyle{ L(\lambda, s, \alpha) = \sum_{n=0}^\infty \frac { e^{2\pi i\lambda n}} {(n+\alpha)^s}. }[/math]

A related function, the Lerch transcendent, is given by

[math]\displaystyle{ \Phi(z, s, \alpha) = \sum_{n=0}^\infty \frac { z^n} {(n+\alpha)^s} }[/math].

The transcendent only converges for any real number [math]\displaystyle{ \alpha \gt 0 }[/math], where:

[math]\displaystyle{ |z| \lt 1 }[/math], or

[math]\displaystyle{ \mathfrak{R}(s) \gt 1 }[/math], and [math]\displaystyle{ |z| = 1 }[/math].

The two are related, as

[math]\displaystyle{ \,\Phi(e^{2\pi i\lambda}, s,\alpha)=L(\lambda, s, \alpha). }[/math]

Integral representations

The Lerch transcendent has an integral representation:

[math]\displaystyle{ \Phi(z,s,a)=\frac{1}{\Gamma(s)}\int_0^\infty \frac{t^{s-1}e^{-at}}{1-ze^{-t}}\,dt }[/math]

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

[math]\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} }[/math]

and then interchanging the sum and integral. The resulting integral representation converges for [math]\displaystyle{ z \in \Complex \setminus [1,\infty), }[/math] Re(s) > 0, and Re(a) > 0. This analytically continues [math]\displaystyle{ \Phi(z,s,a) }[/math] 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

[math]\displaystyle{ \Phi(z,s,a)=-\frac{\Gamma(1-s)}{2\pi i} \int_C \frac{(-t)^{s-1}e^{-at}}{1-ze^{-t}}\,dt }[/math]

where C is a Hankel contour counterclockwise around the positive real axis, not enclosing any of the points [math]\displaystyle{ t = \log(z) + 2k\pi i }[/math] (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

[math]\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 }[/math]

for

[math]\displaystyle{ \Re(a)\gt 0\wedge |z|\lt 1 }[/math]

and

[math]\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 }[/math]

for

[math]\displaystyle{ \Re(a)\gt 0. }[/math]

Similar representations include

[math]\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, }[/math]

and

[math]\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, }[/math]

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

[math]\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, }[/math]

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

[math]\displaystyle{ \zeta(s,\alpha) = L(0, s, \alpha) = \Phi(1,s,\alpha) = \sum_{n=0}^\infty \frac{1}{(n+\alpha)^s}. }[/math]

The polylogarithm is another special case:

[math]\displaystyle{ \textrm{Li}_s(z)=z\Phi(z,s,1) = \sum_{n=1}^\infty \frac{z^n}{n^s}. }[/math]

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

[math]\displaystyle{ \zeta(s) = \Phi(1,s,1) = \sum_{n=1}^\infty \frac{1}{n^s} }[/math]

Other special cases include:

• The Dirichlet eta function:

[math]\displaystyle{ \eta(s) = \Phi(-1,s,1) = \sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^s} }[/math]

• The Dirichlet beta function:

[math]\displaystyle{ \beta(s) = 2^{-s} \Phi(-1,s,1/2) = \sum_{k=0}^\infty \frac{(-1)^{k}}{(2k+1)^s} }[/math]

• The Legendre chi function:

[math]\displaystyle{ \chi_s(z)=2^{-s}z \Phi(z^2,s,1/2) = \sum_{k=0}^\infty \frac{z^{2k+1}}{(2k+1)^s} }[/math]

• The polygamma function:

[math]\displaystyle{ \psi^{(n)}(\alpha)= (-1)^{n+1} n!\Phi (1,n+1,\alpha) }[/math]

Identities

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

[math]\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) }[/math]

Various identities include:

[math]\displaystyle{ \Phi(z,s,a)=z^n \Phi(z,s,a+n) + \sum_{k=0}^{n-1} \frac {z^k}{(k+a)^s} }[/math]

and

[math]\displaystyle{ \Phi(z,s-1,a)=\left(a+z\frac{\partial}{\partial z}\right) \Phi(z,s,a) }[/math]

and

[math]\displaystyle{ \Phi(z,s+1,a)=-\frac{1}{s}\frac{\partial}{\partial a} \Phi(z,s,a). }[/math]

Series representations

A series representation for the Lerch transcendent is given by

[math]\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}. }[/math]

(Note that [math]\displaystyle{ \tbinom{n}{k} }[/math] 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

[math]\displaystyle{ \left|\log(z)\right| \lt 2 \pi;s\neq 1,2,3,\dots; a\neq 0,-1,-2,\dots }[/math]

[math]\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] }[/math]

If n is a positive integer, then

[math]\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\}, }[/math]

where [math]\displaystyle{ \psi(n) }[/math] is the digamma function.

A Taylor series in the third variable is given by

[math]\displaystyle{ \Phi(z,s,a+x)=\sum_{k=0}^\infty \Phi(z,s+k,a)(s)_{k}\frac{(-x)^k}{k!};|x|\lt \Re(a), }[/math]

where [math]\displaystyle{ (s)_{k} }[/math] is the Pochhammer symbol.

Series at a = −n is given by

[math]\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 }[/math]

A special case for n = 0 has the following series

[math]\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|\lt 1, }[/math]

where [math]\displaystyle{ \operatorname{Li}_s(z) }[/math] is the polylogarithm.

An asymptotic series for [math]\displaystyle{ s\rightarrow-\infty }[/math]

[math]\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} }[/math]

for [math]\displaystyle{ |a|\lt 1;\Re(s)\lt 0 ;z\notin (-\infty,0) }[/math] and

[math]\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} }[/math]

for [math]\displaystyle{ |a|\lt 1;\Re(s)\lt 0 ;z\notin (0,\infty). }[/math]

An asymptotic series in the incomplete gamma function

[math]\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}} }[/math]

for [math]\displaystyle{ |a|\lt 1;\Re(s)\lt 0. }[/math]

The representation as a generalized hypergeometric function is

[math]\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). }[/math]

Asymptotic expansion

The polylogarithm function [math]\displaystyle{ \mathrm{Li}_n(z) }[/math] is defined as

[math]\displaystyle{ \mathrm{Li}_0(z)=\frac{z}{1-z}, \qquad \mathrm{Li}_{-n}(z)=z \frac{d}{dz} \mathrm{Li}_{1-n}(z). }[/math]

Let

[math]\displaystyle{ \Omega_{a} \equiv\begin{cases} \mathbb{C}\setminus[1,\infty) & \text{if } \Re a \gt 0, \\ {z \in \mathbb{C}, |z|\lt 1} & \text{if } \Re a \le 0. \end{cases} }[/math]

For [math]\displaystyle{ |\mathrm{Arg}(a)|\lt \pi, s \in \mathbb{C} }[/math] and [math]\displaystyle{ z \in \Omega_{a} }[/math], an asymptotic expansion of [math]\displaystyle{ \Phi(z,s,a) }[/math] for large [math]\displaystyle{ a }[/math] and fixed [math]\displaystyle{ s }[/math] and [math]\displaystyle{ z }[/math] is given by

[math]\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}) }[/math]

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

Let

[math]\displaystyle{ f(z,x,a) \equiv \frac{1-(z e^{-x})^{1-a}}{1-z e^{-x}}. }[/math]

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

[math]\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}+a z^{-\Re a} \right), }[/math]

as [math]\displaystyle{ \Re a \to \infty }[/math].

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. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, http://dlmf.nist.gov/25.14 .
• Bateman, H.; Erdélyi, A. (1953), Higher Transcendental Functions, Vol. I, New York: McGraw-Hill, http://apps.nrbook.com/bateman/Vol1.pdf . (See § 1.11, "The function Ψ(z,s,v)", p. 27)
• "9.55." (in English). Table of Integrals, Series, and Products (8 ed.). Academic Press. 2015. ISBN 978-0-12-384933-5. 
• 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, doi:10.1007/s11139-007-9102-0 . (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 .
• Johansson, F.; Blagouchine, Ia. (2019), "Computing Stieltjes constants using complex integration", Mathematics of Computation 88 (318): 1829–1850, doi:10.1090/mcom/3401 .
• Laurinčikas, Antanas; Garunkštis, Ramūnas (2002), The Lerch zeta-function, Dordrecht: Kluwer Academic Publishers, ISBN 978-1-4020-1014-9 .

External links

• Aksenov, Sergej V.; Jentschura, Ulrich D. (2002), C and Mathematica Programs for Calculation of Lerch's Transcendent, http://aksenov.freeshell.org/lerchphi.html .
• Ramunas Garunkstis, Home Page (2005) (Provides numerous references and preprints.)
• Garunkstis, Ramunas (2004). "Approximation of the Lerch Zeta Function". Lithuanian Mathematical Journal 44 (2): 140–144. doi:10.1023/B:LIMA.0000033779.41365.a5. http://www.mif.vu.lt/~garunkstis/preprintai/approx.pdf. 
• Kanemitsu, S.; Tanigawa, Y.; Tsukada, H. (2015). "A generalization of Bochner's formula". https://hal.archives-ouvertes.fr/hal-02220916.  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". http://mathworld.wolfram.com/LerchTranscendent.html. 
• Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., eds. (2010), "Lerch's Transcendent", NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, http://dlmf.nist.gov/25.14