On the distribution of extreme values of zeta and L-functions in the strip 1/2<σ<1

2010/05/25 by Youness Lamzouri, Lamzouri, Youness · 1 citation
Mathematics · #11F66 #11M06 #60F10 #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Probability (math.PR) #advanced mathematical theories #math.NT #math.PR #msc:11F66 #msc:11M06 #msc:60F10

paper · pdf · doi:10.48550/arxiv.1005.4640

45 pages. To appear in Int. Math. Res. Not

openalex publication_date 2010/05/25 · arxiv created 2011/01/09 · arxiv updated 2011/01/11 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28

Abstract

We study the distribution of large (and small) values of several families of L-functions on a line Re(s)=σ where 1/2<σ<1. We consider the Riemann zeta function ζ(s) in the t-aspect, Dirichlet L-functions in the q-aspect, and L-functions attached to primitive holomorphic cusp forms of weight 2 in the level aspect. For each family we show that the L-values can be very well modeled by an adequate random Euler product, uniformly in a wide range. We also prove new Ω-results for quadratic Dirichlet L-functions (predicted to be best possible by the probabilistic model) conditionally on GRH, and other results related to large moments of ζ(σ+it).

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