Arakawa–Kaneko zeta function


In mathematics, the Arakawa–Kaneko zeta function is a generalisation of the Riemann zeta function which generates special values of the polylogarithm function.

Definition

The zeta function is defined by
where Lik is the k-th polylogarithm

Properties

The integral converges for and has analytic continuation to the whole complex plane as an entire function.
The special case k = 1 gives where is the Riemann zeta-function.
The special case s = 1 remarkably also gives where is the Riemann zeta-function.
The values at integers are related to multiple zeta function values by
where