High pseudomoments of the Riemann zeta function

Abstract

The pseudomoments of the Riemann zeta function, denoted Mk(N), are defined as the 2kth integral moments of the Nth partial sum of ζ(s) on the critical line. We improve the upper and lower bounds for the constants in the estimate Mk(N) ≍k (log N) k 2 as N → ∞ for fixed k ≥ 1, thereby determining the two first terms of the asymptotic expansion. We also investigate uniform ranges of k where this improved estimate holds and when Mk(N) may be lower bounded by the 2kth power of the L ∞ norm of the Nth partial sum of ζ(s) on the critical line.