Sharp conditional bounds for moments of the Riemann zeta function

By Adam J Harper. 2013

140 citations recorded by the source corpus.

We prove, assuming the Riemann Hypothesis, that \int_{T}^{2T} |\zeta(1/2+it)|^{2k} dt \ll_{k} T log^{k^{2}} T for any fixed k \geq 0 and all large T. This is sharp up to the value of the implicit constant. Our proof builds on well known work of Soundararajan, who showed, assuming the Riemann Hypothesis, that \int_{T}^{2T} |\zeta(1/2+it)|^{2k} dt \ll_{k,\epsilon} T log^{k^{2}+\epsilon} T for any fixed k \geq 0 and \epsilon > 0. Whereas Soundararajan bounded \log|\zeta(1/2+it)| by a single Dirichlet polynomial, and investigated how often it attains large values, we bound \log|\zeta(1/2+it)| by a sum of many Dirichlet polynomials and investigate the joint behaviour of all of them. We also work directly with moments throughout, rather than passing through estimates for large values.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c2dabe12b874ff29a7ff8d6f0104e64e92ee5a2d

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In Algebra and Number Theory

In Geometry and Topology

Citations in the corpus

Moments and distribution of central L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L$$\end{document}-values of quadr (2014, methodology): “Finally, we comment that the method developed here is related to the iterative method of Harper [13] (discovered independently) which yields sharp conditional estimates for moments.”
The fourth moment of quadratic Dirichlet L-functions over function fields (2016, methodology): “[14] Adam Harper.”