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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 Mathias Lerch [1].

Definition

The Lerch zeta-function is given by

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

A related function, the Lerch transcendent, is given by

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

The two are related, as

$$\,\Phi(\exp (2\pi i\lambda), s,\alpha)=L(\lambda, \alpha,s).$$

Integral representations

An integral representation is given by

$$\Phi(z,s,a)=\frac{1}{\Gamma(s)}\int_0^\infty \frac{t^{s-1}e^{-at}}{1-ze^{-t}}\,dt$$

for

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

A contour integral representation is given by

$$\Phi(z,s,a)=-\frac{\Gamma(1-s)}{2\pi i}\int_0^{(+\infty)} \frac{(-t)^{s-1}e^{-at}}{1-ze^{-t}}\,dt$$

for

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

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

$$\Phi(z,s,a)= \frac{1}{2a^s}+ \int_0^\infty \frac{z^t}{(a+t)^s}\,dt+ \frac{2}{a^{s-1}} \int_0^\infty \frac{\sin(s\arctan(t)-ta\log(z))}{(1+t^2)^{s/2}(e^{2\pi at}-1)}\,dt$$

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

$$\Re(a)>0.$$

Special cases

The Hurwitz zeta-function is a special case, given by

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

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

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

The Legendre chi function is a special case, given by

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

The Riemann zeta-function is given by

$$\,\zeta(s)=\Phi (1,s,1).$$

The Dirichlet eta-function is given by

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

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.

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

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

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

$$|\log(z)|<2 \pi;s\neq 1,2,3,\dots; a\neq 0,-1,-2,\dots \Phi(z,s,a)=z^{-a}\left[\Gamma(1-s)\left(-\log (z)\right)^{s-1} +\sum_{k=0}^\infty \zeta(s-k,a)\frac{\log^k (z)}{k!}\right]$$

(the correctness of this formula is disputed, please see the talk page)

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

If s is a positive integer, then

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

A Taylor series in the third variable is given by

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

Series at a = -n is given by

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

A special case for n = 0 has the following series

$$\Phi(z,s,a)=\frac{1}{a^s} +\sum_{m=0}^\infty (1-m-s)_m Li_{s+m}(z)\frac{a^m}{m!}; |a|<1.$$

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.
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 978-0521192255, MR2723248.
Bateman, H.; Erdélyi, A. (1953), Higher Transcendental Functions, Vol. I, New York: McGraw-Hill. (See § 1.11, "The function Ψ(z,s,v)", p. 27)
Gradshteyn, I.S.; Ryzhik, I.M. (1980), Tables of Integrals, Series, and Products (4th ed.), New York: Academic Press, ISBN 0-12-294760-6. (see Chapter 9.55)
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, MR2429900. (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, MR0036882.
Laurinčikas, Antanas; Garunkštis, Ramūnas (2002), The Lerch zeta-function, Dordrecht: Kluwer Academic Publishers, ISBN 9781402010149, MR1979048.
Lerch, Mathias (1887), "Note sur la fonction $$\scriptstyle{\mathfrak K}(w,x,s) = \sum_{k=0}^\infty {e^{2k\pi ix} \over (w+k)^s}$$" (in French), Acta Mathematica 11 (1–4): 19–24, doi:10.1007/BF02612318, MR1554747.

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" from MathWorld.
"§25.14, Lerch’s Transcendent". NIST Digital Library of Mathematical Functions. National Institute of Standards and Technology. 2010. Retrieved 28 January 2012.

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