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

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.