On the density function for the value-distribution of automorphic \(L\)-functions (Q1710759)

The paper is devoted to the study of value distribution of autormorphic \(L\)-functions. More precisely, it is proved an analogue of the Bohr-Jessen type limit theorem in the case of automorphic \(L\)-functions of normalized Hecke-eigen newform \(f\) of weight \(\kappa\) with respect to the congruence subgroup \(\Gamma_0(N)\) (denote it by \(L_f(s)\)). The main result says that, for \(T>0\) and any \(\sigma > 1/2\), the limit \[ W_\sigma (R;L_f ) = \lim\limits_{T \to \infty} \frac{1}{2T}\mu_1\big\{t \in [-T,T]: \log L_f(\sigma+it)\in R\big\} \] exists, and can be written as \[ W_\sigma(R;L_f) = \int_{R}{\mathcal{M}}_\sigma(z,L_f )|dz| \] with explicitly given certain continuous non-negative function \({\mathcal{M}}_\sigma(z,L_f)\) (note that \({\mathcal{M}}_\sigma(z,L_f )\) is called the density function for the value-distribution of \(L_f(s)\)). Here, \(R\) is a fixed rectangle in \(\mathbb{C}\) with the edges parallel to the axes, and \(\mu_1\) is one-dimensional Lebesgue measure.