Moments of the Riemann zeta function

Abstract

Assuming the Riemann hypothesis, we obtain an upper bound for the moments of the Riemann zeta function on the critical line. Our bound is nearly as sharp as the conjectured asymptotic formulae for these moments. The method extends to moments in other families of L-functions.

Upper bounds for the moments of derivatives of Dirichlet L-functions

“…On the other hand, no upper bounds have been known for the moments of L ( ) (1/2 χ), although upper bounds for the moments of derivatives of the Riemann zeta function are already obtained by Milinovich [11]. In this paper, we extend the results by Heath-Brown [7] and Soundararajan [17], and give certain upper bounds for these moments when ≥ 1/2. The first result is a generalization of Heath-Brown's estimate (2).…”

“…By Cauchy's integral formula, the -th derivative of L( χ) at = 1/2 is given by an integral in the neighborhood of = 1/2, and this integral is bounded by the weighted moment J(γ) of L( χ) introduced by Heath-Brown [7]. Thus the estimate (3) [17] (see also the undergraduate thesis of Koltes [8]). Then, by using the method of Milinovich [10] again, the estimate (4) is obtained.…”

“…Then, the former is simply evaluated by ≤log 8 1 log 3 By straightforward exercises in partial summation, the latter is evaluated by O(1), by (17). Therefore, we obtain (21).…”

“…holds for all ∈ (0 2) under the assumption of the Generalized Riemann Hypothesis (or simply GRH), and that (2) holds unconditionally for = 1/ , ∈ N. More recently, Soundararajan [17] showed that the upper bound * χ (mod )…”

“…On the other hand, by extending Soundararajan's method for the moments of the Riemann zeta function [17] and using Milinovich's ideas introduced in [11], the following upper bound is obtained in the case of ≥ 2. …”