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

Abstract

We prove closed-form identities for the sequence of moments $\int t^{2n}|\Gamma(s)\zeta(s)|^2dt$ on the whole critical line $s=1/2+it$. They are finite sums involving binomial coefficients, Bernoulli numbers, Stirling numbers and $\pi$, especially featuring the numbers $\zeta(n)B_n/n$ 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 $\Gamma$ and $\zeta$ functions. 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 for %various other applications, as well as for the numerous connections concerning the above mentioned numbers.