Limit theorem for the hybrid joint universality theorem on zeta and $L$-functions

TL;DR

This paper proves a limit theorem for the hybrid joint universality theorem concerning zeta and L-functions, advancing the understanding of their joint value distribution.

Contribution

It provides the first probabilistic proof of the hybrid joint universality theorem using Bagchi's approach, filling a gap in the existing literature.

Findings

Established the limit theorem for the hybrid joint universality of zeta and L-functions

Developed a probabilistic proof method based on Bagchi's approach

Enhanced understanding of the joint value distribution of zeta and L-functions

Abstract

In 1979, Gonek presented the hybrid joint universality theorem for Dirichlet $L$-functions and proved the universality theorem for Hurwitz zeta-functions with rational parameter as an application. Following the introduction of the hybrid universality theorem, several generalizations, refinements, and applications have been developed. Despite these advancements, no probabilistic proof based on Bagchi's approach has been formulated due to the complexities of adapting his method to the hybrid joint universality theorem. In this paper, we prove the limit theorem for the hybrid joint universality theorem.