Paper detail

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

We study the distribution of large (and small) values of several families of $L$-functions on a line $\text{Re(s)}=σ$ 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 $ζ(σ+it)$.