LAUSR

Abstract The pseudomoments of the Riemann zeta function, denoted M k ( 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 M k ( 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 M k ( 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.