# Arakawa–Kaneko zeta function

A generalisation of the Riemann zeta function which generates special values of the polylogarithm function.

## Definition

The zeta function $\xi_k(s)$ is defined by $$\xi_k(s) = \frac{1}{\Gamma(s)} \int_0^{+\infty} \frac{t^s-1}{e^t-1}\mathrm{Li}_k(1-e^{-t}) dt$$ where $\mathrm{Li}_k$ is the$k$-th polylogarithm $$\mathrm{Li}_k(z) = \sum_{n=1}^{\infty} \frac{z^n}{n^k} \ .$$

## Properties

The integral converges for $\Re(s) > 0$ and $\xi_k(s)$ has analytic continuation to the whole complex plane as an entire function.

The special case$k=1$ gives $\xi_1(s) = s \zeta(s+1)$ where $\zeta$ is the Riemann zeta-function.

The values at integers are related to multiple zeta function values by

$$\xi_k(m) = \zeta_m^*(k,1,\ldots,1)$$ where $$\zeta_n^*(k_1,\dots,k_{n-1},k_n)=\sum_{0<m_1<m_2<\cdots<m_n}\frac{1}{m_1^{k_1}\cdots m_{n-1}^{k_{n-1}}m_n^{k_n}} \ .$$

How to Cite This Entry:
Richard Pinch/sandbox-WP2. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox-WP2&oldid=39626