Define $$ \operatorname{Li}_2(z) \coloneqq \sum_{n\geq 1} \frac{z^n}{n^2},\qquad |z|\lt 1 $$ More generally, the polylogarithm $m=1,2,\dots$ $$ \operatorname{Li}_m(z) \coloneqq \sum_{n\geq 1} \frac{z^n}{n^m},\qquad |z|\lt 1 $$ Note that $$ \operatorname{Li}_1(z) = -\log(1-z) $$ and $$ \frac{d}{d z} \operatorname{Li}_m(z) = \operatorname{Li}_{m-1}(z) $$ So we get an analytic continuation $$ \operatorname{Li}_2(z) = -\int_0^z \log(1-u) \frac{d u}{u} $$ where the path from $0$ to $z$ is in $\mathbb{C}\setminus [1,\infty)$ Functional equations: $$ \begin{gathered} \operatorname{Li}_1(1-x y) = \operatorname{Li}_1(1-x) + \operatorname{Li}_1(1-y)\\ \operatorname{Li}_2 = \text{5 terms (Spence 1809, Abel 1828, ...)} \end{gathered} $$ Monodromy (on $\operatorname{Li}_2(x),\log(x),1$) $$ \gamma_0=\begin{pmatrix}1&0&0\\0&1&2\pi i\\0&0&1\end{pmatrix}, \gamma_1=\begin{pmatrix}1&-2\pi i&0\\0&1&0\\0&0&1\end{pmatrix} $$ generate a Heisenberg group $$ \begin{pmatrix}1&\mathbb{Z}(1)&\mathbb{Z}(2)\\ 0&1&\mathbb{Z}(1)\\0&0&1\end{pmatrix} $$ ##Bloch-Wigner Dilogarithm ## {:#Bloch-Wigner} $$ D(z) \coloneqq \operatorname{Im} \operatorname{Li}_2(z) + \arg(1-z)\log|z| $$ is real-analytic in $\mathbb{C}\setminus\{0,1\}$ and continuous in $\mathbb{C}$. $$ \begin{gathered} D\left(e^{i\theta}\right) = \sum_{n\geq 1} \frac{\sin n\theta}{n^2}\\ D(\overline{z}) = - D(z) \end{gathered} $$ hence vanishes on $\mathbb{R}$. $$ \begin{split} D(z)&= D\left(1-z^{-1}\right)= D\left({(1-z)}^{-1}\right)\\ & - D\left(z^{-1}\right) = - D(1-z) = -D\left(-\frac{z}{1-z}\right) \end{split} $$ So we have a continuous real-vaued function on $\mathbb{P}^1(\mathbb{C})$ with a maximum at $z=(1+\sqrt{-3})/2$: $D(1+\sqrt{-3})/2)=1.0149\dots$. Define recursively $$ z_{n+1}z_{n-1} = 1-z_n $$ then $z_{n+5}=z_n$. If we call $z_0=x$, $z_1=y$, then we find $$ x,y,\frac{1-y}{x},\frac{x+y-1}{xy},\frac{1-x}{y} $$ (Laurent phenomenon). (Cremona transformation of order 5 on $\mathbb{P}^2(\mathbb{C})$ is $(x,y)\mapsto\left(y,\tfrac{1-y}{x}\right)$.) The 5-term recursion relation is $$ \sum_{j=0}^4 D(z_j)=0 $$