Definition $\text{PolyLog}_s(z)$ for $z=1$ equals $\zeta(s)$?

Question

In wikipedia and also mathworld I find definitions, that the analytic continuation of the PolyLog() at $z=1$ is existent and equals that of the $\zeta()$, but Pari/GP as well as W|A seem to have implemented this only for $s \gt 1$ . W|A gives for the $\text{PolyLog}_s(1)$ at $s=-3$ the symbol $\infty$ but for $\zeta(-3)$ the well known finite value.

In the definition-parts in mathworld I did not see a restriction, but perhaps I've overlooked some thing (while on a second read in wikipedia I find the restriction of the equality to $s \gt 1$).

What is the definition for $\text{Polylog}_s(1) $ for $s\le 1$ ? Is there an analytic continuation?

Answer 1

According to the cited Wiki article the relation to the Zeta function is valid for $\Re(x) >1.$ For negative integers $n$ the polylog is a rational function (http://functions.wolfram.com/10.08.03.0033.01 , see also the series representations of the Lerch transcendent or http://mathworld.wolfram.com/Polylogarithm.html formulas (6ff)) ) $$\mathrm{Li}_{-n}(x) = \frac{1}{(1-x)^{n + 1}} \sum\limits_{m=1}^{n}\left( \sum\limits_{k=1}^{m} (-1)^{k+1}\binom{n+1}{k-1}(m-k+1)^n \right)x^m \quad (n>0) $$

Here $\mathrm{Li}_{-3}(x)=\frac{x^3+4x^2+x}{(1-x)^4}$, and this has no limit for $x\rightarrow 1.$

On the Mathworld site can additionally find the statement regarding the realation to $\zeta(s)$: Note, however, that the meaning of Li_s(z) for fixed complex s is not completely well-defined, since it depends on how s is approached in four-dimensional (s,z)

Answer 2

$$\text{Li}_s(z) = \sum_{n=1}^\infty z^n n^{-s}, \qquad \{|z| < 1\} \cup \{ |z| \le 1, \Re(s)> 1\} $$

$\lim_{z \to 1} \text{Li}_s(z)$ diverges for every $s, \Re(s) < 1$.

Now for $|z| < 1$, $\text{Li}_s(z)$ is entire in $s$, and summing by parts shows it is (locally uniformly) continuous in $z, |z| \le 1, z \ne 1$. Thus $\displaystyle\lim_{z \,\to\, -1} \text{Li}_s(z)$ is analytic in $s$, ie.

$$-(1-2^{1-s}) \zeta(s) = \lim_{z \,\to \,\color{red}{-1}} \text{Li}_s(z), \qquad s \in \mathbb{C}$$