ABSTRACT. A representation for the Riemann zeta function £(s) is given as an absolutely convergent expansion involving incomplete gamma functions which is valid for all finite complex values of s (^ 1). It is then shown how use of the uniform asymptotics of the incomplete gamma function leads to an asymptotic representation for £(s) on the critical line s = ^ + it when t ->• oo. This result, which represents an improvement on an earlier treatment [10], involves an error function smoothing of the original Dirichlet series together with a correction term whose coefficients can be given explicitly to any desired order in terms of coefficients arising in the asymptotics of the incomplete gamma function. By examination of the higher-order coefficients in the correction term, it is shown that the expansion diverges like the familiar 'factorial divided by a power' dependence, decorated by a slowly varying multiplier function, as has recently been demonstrated by M. V. Berry for the Riemann-Siegel formula.