AN ASYMPTOTIC REPRESENTATION FOR THE RIEMANN ZETA-FUNCTION ON THE CRITICAL LINE

A representation for the Riemann zeta function zeta(s) is given as an absolutely convergent expansion involving incomplete gamma functions which is valid for all finite complex values of s (not equal 1). It is then shown how use of the uniform asymptotics of the incomplete gamma function leads to a new asymptotic representation for zeta(s) on the critical line s = 1/2 + it when t --> infinity. This new result involves an error function smoothing of an infinite sum and consequently shares some similarity to, though is quite different from, the recent asymptotic expansion for zeta(1/2 + it) developed by Berry & Keating. Numerical examples suggest that term for term (with a little extra computational effort) the new representation is at least as accurate as the Riemann-Siegel formula.