Lerch Transcendent

A generalization of the Hurwitz Zeta Function and Polylogarithm function. Many sums of reciprocal Powers can be expressed in terms of it. It is defined by

$$\Phi(z,s,a)\equiv \sum_{k=0}^\infty {z^k\over (a+k)^s},$$ (1)

where any term with $a+k=0$ is excluded.

The Lerch transcendent can be used to express the Dirichlet Beta Function

$$\beta(s)\equiv \sum_{k=0}^\infty (-1)^k(2k+1)^{-s} 2^{-s}\Phi(-1,s,{\textstyle{1\over 2}}),$$ (2)

the integral of the Fermi-Dirac Distribution

$$\int_0^\infty {k^s\over e^{k-\mu}+1}\,dk = e^\mu\Gamma(s+1)\Phi(-e^\mu,s+1,1),$$ (3)

where $\Gamma(z)$ is the Gamma Function, and to evaluate the Dirichlet L-Series.

See also Dirichlet Beta Function, Dirichlet L-Series, Fermi-Dirac Distribution, Hurwitz Zeta Function, Polylogarithm