Value of polylogarithms $\mathrm{Li}_s(1)$ for $s<1$.

Question

For real $s>1$ the polylogarithm for $z=1$ reduces to the Riemann zeta function $\mathrm{Li}_s(1) = \zeta(s)$. For real $s<1$ Wolfram Alpha and the Wolfram function site give $\mathrm{Li}_s(1) = \infty, s<1$ whereas the mpmath package returns $\mathrm{Li}_s(1) = \zeta(s)$ for all $s\ne 1$. Is this just a convention or is it a kind of continuation for $s?$

Answer 1

If you define the polylogarithm and zeta functions with the series :$$\operatorname{Li}_s(z):=\sum_{k=1}^\infty \frac {z^k}{k^s}\\\zeta(s):=\sum_{k=1}^\infty \frac 1{k^s}$$ then you'll be in the first case and have to suppose $\;\Re(s)>1\;$ for convergence but if you consider the analytic continuation of these two series as being the actual functions then things become much more interesting and considering every complex $s$ such that $\;s\ne 1\;$ should be the fine choice (see the discussions in the Wikipedia links or here concerning zeta).