We prove closed-form identities for the sequence of moments ∫t2n∣Γ(s)ζ(s)∣2dt on the whole critical line s=1/2+it. They are
finite sums involving binomial coefficients, Bernoulli numbers, Stirling
numbers and π, especially featuring the numbers ζ(n)Bn/n unveiled by
Bettin and Conrey in 2013.
Their main power series identity allows for a short proof of an auxiliary
result: the computation of the k-th derivatives at 1 of the "exponential
auto-correlation" function studied in a recent paper by the authors. We also
provide an elementary and self-contained proof of this secondary result. The
starting point of our work is a remarkable identity proven by Ramanujan in
1915.
The sequence of moments studied here, not to be confused with the moments of
the Riemann zeta function, entirely characterizes ∣ζ∣ on the critical
line. They arise in some generalizations of the Nyman-Beurling criterion, but
might be of independent interest for the numerous connections concerning the
above mentioned numbers