Moments and distribution of central L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs}…

By Maksym Radziwill, Kannan Soundararajan. Inventiones Mathematicae, 2014

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We show that if one can compute a little more than a particular moment for some family of L-functions, then one has upper bounds of the conjectured order of magnitude for all smaller (positive, real) moments and a one-sided central limit theorem holds. We illustrate our method for the family of quadratic twists of an elliptic curve, obtaining sharp upper bounds for all moments below the first. We also establish a one sided central limit theorem supporting a conjecture of Keating and Snaith. Our work leads to a conjecture on the distribution of the order of the Tate-Shafarevich group for rank zero quadratic twists of an elliptic curve, and establishes the upper bound part of this conjecture (assuming the Birch-Swinnerton-Dyer conjecture).

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d705c9fa08823e75588646f45dfc04d6a58a4e18

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Citations in the corpus

Lower bounds for moments of zeta and $L$-functions revisited (2020, background): “For example, [8] enunciates the principle that an upper bound for a particular moment (with a little flexibility) may be used to establish upper bounds of the correct order of magnitude for all smaller moments.”
SHARP UPPER BOUNDS FOR FRACTIONAL MOMENTS OF THE RIEMANN ZETA FUNCTION (2019, methodology): “This proposition is an analogue of Lemma 2 of [10], and is proved similarly.”