Probability density functions attached to random Euler products for automorphic $L$-functions

TL;DR

This paper investigates the value distributions of automorphic L-functions associated with cusp forms, establishing the existence of probability density functions for their random Euler product models and analyzing their distributional properties.

Contribution

It introduces the existence of probability density functions for random Euler products linked to automorphic L-functions and connects these to mean value formulas and distribution discrepancies.

Findings

Existence of probability density functions for the models.

Mean values of L-functions expressed via these densities.

Discrepancy estimates between actual and model distributions.

Abstract

In this paper, we study the value-distributions of $L$-functions of holomorphic primitive cusp forms in the level aspect. We associate such automorphic $L$-functions with probabilistic models called the random Euler products. First, we prove the existence of probability density functions attached to the random Euler products. Then various mean values of automorphic $L$-functions are expressed as integrals involving the density functions. Moreover, we estimate the discrepancies between the distributions of values of automorphic $L$-functions and those of the random Euler products.