Pseudomoments of the Riemann zeta function

Abstract

The 2kth pseudomoments of the Riemann zeta function ζ ( s ) are, following Conrey and Gamburd, the 2 k th integral moments of the partial sums of ζ ( s ) on the critical line. For fixed k > 1 / 2 , these moments are known to grow like ( log N ) k 2 , where N is the length of the partial sum, but the true order of magnitude remains unknown when k ⩽ 1 / 2 . We deduce new Hardy–Littlewood inequalities and apply one of them to improve on an earlier asymptotic estimate when k → ∞ . In the case k < 1 / 2 , we consider pseudomoments of ζ α ( s ) for α > 1 and the question of whether the lower bound ( log N) k 2 α 2 known from earlier work yields the true growth rate. Using ideas from recent work of Harper, Nikeghbali and Radziwiłł and some probabilistic estimates of Harper, we obtain the somewhat unexpected result that these pseudomements are bounded below by log N to a power larger than k 2 α 2 when k < 1 / e and N is sufficiently large.