Identities for the Rogers dilogarithm function connected with simple Lie algebras (Q1826004)

The function \(L(x)=-\int^{x}_{0}[\frac{\log (1-x)}{x}+\frac{\log x}{1-x}]dx\), where \(0\leq x\leq 1\), is called Rogers dilogarithmic function. Let G be a simple Lie algebra of rank \(\ell\), g the dual Coxeter number and \(\phi =\pi /(r+g)\). The author conjectures that \[ \sum^{\ell}_{k=0}\sum^{r}_{m=1}L(f_ m^{(k)}(\phi))=(r \dim G/(r+g)\quad)(\pi^ 2/6). \] It is proved this formula for the Lie algebra \(A_ n\) and for classical Lie algebras of rank \(\ell\), whre \(\ell \leq 4\).