Lerch zeta function

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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 the Czech mathematician Mathias Lerch [1].


The Lerch zeta function is given by

A related function, the Lerch transcendent, is given by

The two are related, as

Integral representations[edit]

An integral representation is given by


A contour integral representation is given by


where the contour must not enclose any of the points

A Hermite-like integral representation is given by




Similar representations include


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

The last formula is also known as Lipschitz formula.

Special cases[edit]

The Hurwitz zeta function is a special case, given by

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

The Legendre chi function is a special case, given by

The Riemann zeta function is given by

The Dirichlet eta function is given by


For λ rational, the summand is a root of unity, and thus may be expressed as a finite sum over the Hurwitz zeta-function. Suppose with and . Then and .

Various identities include:



Series representations[edit]

A series representation for the Lerch transcendent is given by

(Note that 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

B. R. Johnson (1974). "Generalized Lerch zeta-function". Pacific J. Math. 53 (1): 189–193. doi:10.2140/pjm.1974.53.189.[permanent dead link]

If s is a positive integer, then

where is the digamma function.

A Taylor series in the third variable is given by

where is the Pochhammer symbol.

Series at a = -n is given by

A special case for n = 0 has the following series

where is the polylogarithm.

An asymptotic series for

for and


An asymptotic series in the incomplete gamma function


Asymptotic expansion[edit]

The polylogarithm function is defined as


For and , an asymptotic expansion of for large and fixed and is given by

for .[1]


Let be its Taylor coefficients at . Then for fixed and ,

as .[2]


The Lerch transcendent is implemented as LerchPhi in Maple.


  1. ^ 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.
  2. ^ 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: 1–12. arXiv:1806.01122. doi:10.1080/10652469.2019.1627530.

External links[edit]