Asymptotic Expansions of Some Series Involving the Riemann Zeta Function

HARDY and Littlewood (stimulated by a conjecture of Ramanujan) proved that the truth of

Σ (-l)ⁿXⁿ/n! Z(2n+1) = 0(X^-i+∈)

(∈ > 0 arbitrary) is a necessary and sufficient condition for the truth of the Riemann hypothesis. Here Z(s) is Riemann's Zeta Function.