On the Distribution of Extreme Values of Zeta and L-Functions in the Strip 1/2 < < 1

Abstract

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).