The fourth moment of quadratic Dirichlet L-functions over function fields
By Alexandra Florea. Geometric and Functional Analysis, 2016
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We obtain an asymptotic formula for the fourth moment of quadratic Dirichlet L-functions over Fq[x]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{F}_q[x]}$$\end{document}, as the base field Fq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{F}_q}$$\end{document} is fixed and the genus of the family goes to infinity. According to conjectures of Andrade and Keating, we expect the fourth moment to be asymptotic to q2g+1P(2g+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${q^{2g+1} P(2g+1)}$$\end{document} up to an error of size o(q2g+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${o(q^{2g+1})}$$\end{document}, where P is a polynomial of degree 10 with explicit coefficients. We prove an asymptotic formula with the leading three terms, which agrees with the conjectured result.
Subfields: Algebra and Number Theory, Artificial Intelligence, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/065a799ad786619f6894247cf6c5dc7203b4c887
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In Algebra and Number Theory
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Citations in the corpus
- Moments of random multiplicative functions, II: High moments (2018, background): “For some other contexts where a transition from orthogonal/symplectic to unitary behaviour arises, as for large t here, see the papers of Florea [4], Keating and Odgers [13], and Soundararajan and Young [20], for example.”