Polylogarithm

Summary

The polylogarithm $\operatorname{Li}_s(z)$ is the power series $\sum_{k \ge 1} \frac{z^k}{k^s}$ and its analytic continuation. For $s = 1$ it is $-\ln(1 - z)$, which explains the name, and for $\operatorname{Re} s > 1$ its value at $z = 1$ is the Riemann zeta function $\zeta(s)$. The case $s = 2$ is the dilogarithm. Polylogarithms appear in number theory, in algebraic K-theory, in quantum field theory and in the statistics of Fermi and Bose gases.