Polynomial Moments with a weighted Zeta Square measure on the critical line

We prove closed-form identities for the sequence of moments $\int t^{2n}\mathrm{\mid }\mathrm{\Gamma }(s)\zeta (s){\mathrm{\mid }}^{2}dt$ on the whole critical line $s=1\mathrm{/}2+it$. They are finite sums involving binomial coefficients, Bernoulli numbers, Stirling numbers and $\pi$, especially featuring the numbers $\zeta (n)B_n\mathrm{/}n$ unveiled by Bettin and Conrey. Their main power series identity, together with our previous work, 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 \cite{DH21a}. 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. %today interpreted as a Mellin-Plancherel isometry involving the $\mathrm{\Gamma }$ and $\zeta$ functions. The sequence of moments studied here, not to be confused with the moments of the Riemann zeta function, entirely characterizes $\mathrm{\mid }\zeta \mathrm{\mid }$ on the critical line. They arise in some generalizations of the Nyman-Beurling criterion, but might be of independent interest for %various other applications, as well as for the numerous connections concerning the above mentioned numbers.