An estimate of the second moment of a sampling of the Riemann zeta function on the critical line

TL;DR

This paper estimates the second moment of a randomly sampled Riemann zeta function on the critical line, showing it grows logarithmically with the sampling parameter under certain conditions.

Contribution

It provides a new probabilistic estimate of the second moment of the zeta function for gamma-distributed random samples on the critical line.

Findings

Expected value of |zeta(1/2 + iX_t)|^2 is approximately log t.

The second moment grows logarithmically with the sampling parameter t.

The estimate holds for large t with gamma-distributed sampling X_t.

Abstract

We investigate the second moment of a random sampling $\zeta(1/2+iX_t)$ of the Riemann zeta function on the critical line. Our main result states that if $X_t$ is an increasing random sampling with gamma distribution, then for all sufficiently large $t$, \[\mathbb{E} |\zeta(1/2+iX_t)|^2 = \log t + O(\sqrt{\log t}\log\log t).\]