Polynomial moments with a weighted zeta square measure on the critical line

We prove closed-form identities for the sequence of moments $\int t^{2n}|\Gamma (s)\zeta (s)|^2\,dt$ on the whole critical line $s=1/2+it$. They are finite sums involving binomial coefficients, Bernoulli numbers, Stirling numbers and $\pi $, in particular featuring the numbers $\zeta (n)B_n/n$ appearing in work of Bettin and Conrey (2013). Their main power series identity, together with a remark (2021) of the present authors, 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 by the present 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 (1915). The sequence of moments studied here, not to be confused with the moments of the Riemann zeta function, entirely characterizes $|\zeta |$ on the critical line. They arise in some generalizations of the Nyman–Beurling criterion, but might be of independent interest.