A generalisation of an expansion for the Riemann zeta function involving incomplete gamma functions

We derive an expansion for the Riemann zeta function ζ(s) involving incomplete gamma functions with their second argument proportional to n^2p, where n is the summation index and p is a positive integer. The possibility is examined of reducing the number of terms below the value N_t (t/2π)^1/2 in the finite main sum appearing in asymptotic approximations for ζ(s) on the critical line s = 1 2 + it as t → ∞. It is shown that the expansion corresponding to quadratic dependence on n (p = 1) is the best possible representation of this type for ζ(s).