Continued-fraction expansions for the Riemann zeta function and polylogarithms

It appears that the only known representations for the Riemann zeta function $\zeta (z)$ in terms of continued fractions are those for $z=2$ and 3. Here we give a rapidly converging continued-fraction expansion of $\zeta (n)$ for any integer $n\geq 2$. This is a special case of a more general expansion which we have derived for the polylogarithms of order $n$, $n\geq 1$, by using the classical Stieltjes technique. Our result is a generalisation of the Lambert-Lagrange continued fraction, since for $n=1$ we arrive at their well-known expansion for $\log (1+z)$. Computation demonstrates rapid convergence. For example, the 11th approximants for all $\zeta (n)$, $n\geq 2$, give values with an error of less than 10$^{-9}$.