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

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.