Full asymptotics and log-periodicity for coefficients in representations of the Riemann zeta function

For each integer $b\ge2$, we obtain representations of the Riemann zeta function in the complex plane as geometrically convergent series. Mellin transform methods and asymptotics for ratios of Gamma functions from the author's recent work allow us to prove, for the coefficients of these series, asymptotic expansions to all orders in inverse powers of the index, with numerators which are periodic functions of the base $b$ logarithm of the index. Such complete asymptotics were first established in the author's recent work for quantities which arose decades ago in theoretical computer science.