Continued-fraction expansions for the Riemann zeta function and polylogarithms

Abstract

It appears that the only known representations for the Riemann zeta function ((z) in terms of continued fractions are those for z = 2 and 3. Here we give a rapidly converging continued-fraction expansion of ((n) for any integer n > 2. This is a special case of a more general expansion which we have derived for the polylogarithms of order n, n > 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 ((n), n > 2, give values with an error of less than 10-9.