Distribution of values of $L$-functions at the edge of the critical strip

TL;DR

This paper investigates the distribution of $L$-function values near the critical line by analyzing random Euler products, with applications to symmetric power $L$-functions, the Selberg class, and quadratic twists.

Contribution

It introduces a new framework using random Euler products to study $L$-function value distributions at the edge of the critical strip, including several novel applications.

Findings

Distribution results for symmetric power $L$-functions at $s=1$

Distribution analysis for functions in the Selberg class

Results on quadratic twists of automorphic cusp forms

Abstract

We prove several results on the distribution of values of $L$-functions at the edge of the critical strip, by constructing and studying a large class of random Euler products. Among new applications, we study families of symmetric power $L$-functions of holomorphic cusp forms in the level aspect (assuming the automorphy of these $L$-functions) at $s=1$, functions in the Selberg class (in the height aspect), and quadratic twists of a fixed $GL(m)/{\Bbb Q}$-automorphic cusp form at $s=1$.