Mean values of L-functions and symmetry

1999/12/13 by J. Brian Conrey, David W. Farmer, Conrey, J. Brian +1 · 4 citations
Mathematics · Physics and Astronomy · #11F66 #11M06 #81Q50 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Number Theory (math.NT) #math-ph #math.MP #math.NT #msc:11F66 #msc:11M06 #msc:81Q50

paper · pdf · doi:10.48550/arxiv.math/9912107

23 pages, 6 figures

arxiv created 1999/12/13 · openalex publication_date 1999/12/13 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Recently Katz and Sarnak introduced the idea of a symmetry group attached to a family of L-functions, and they gave strong evidence that the symmetry group governs many properties of the distribution of zeros of the L-functions. We consider the mean-values of the L-functions and the mollified mean-square of the L-functions and find evidence that these are also governed by the symmetry group. We use recent work of Keating and Snaith to give a complete description of these mean values. We find a connection to the Barnes-Vignéras Γ₂-function and to a family of self-similar functions.

Citations

• Nonvanishing of Quadratic Dirichlet L-Functions at S = &#189

Cited by

• Decorrelation of SO(5)× SO(2) Bessel periods for symmetric cubes of GL(2)
• Selberg orthogonality for half-integral weight modular forms
• The maximum size of L-functions
• Upper tail distributions of central L-values of quadratic twists of elliptic curves at the variance scale

Related

• Families of L-functions and their Symmetry
• Statistics for low-lying zeros of symmetric power L-functions in the level aspect
• Small first zeros of L-functions
• A note on the mean value of the zeta and L-functions. XIV
• A note on the mean value of the zeta and L-functions. XV