Amplified moments of the Riemann zeta function

Abstract

We establish asymptotic formulae for two-piece amplified second and fourth moments of the Riemann zeta function. As applications, we obtain unconditional effective lower bounds for several joint moments of zeta which are in strong agreement with the conjectures of Keating--Wei and Keating--Snaith. In particular, we prove an unconditional lower bound for the sixth moment of zeta $M_3(T)\geq(34.1+o(1))c_3T(\log T)^9$. We further improve some of the lower bounds obtained by Soundararajan, removing the assumption of the Lindelőf Hypothesis, and we obtain effective lower bounds for all joint integer moments of zeta consistent with the predictions of random matrix theory.