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

TL;DR

This paper investigates the distribution of extreme values of various families of $L$-functions within the critical strip $1/2< ext{Re}(s)<1$, using probabilistic models and conditional results to understand their behavior.

Contribution

It introduces a unified approach to model large and small $L$-values across different families using random Euler products and establishes new $ ext{Omega}$-results under GRH.

Findings

$L$-values are well modeled by random Euler products

Established new $ ext{Omega}$-results for quadratic Dirichlet $L$-functions

Provided insights into large moments of $ ext{zeta}(\sigma+it)$

Abstract

We study the distribution of large (and small) values of several families of $L$-functions on a line $\text{Re(s)}=\sigma$ where $1/2<\sigma<1$. We consider the Riemann zeta function $\zeta(s)$ 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 $\Omega$-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 $\zeta(\sigma+it)$.