A joint discrete limit theorem for Epstein and Hurwitz zeta-functions

In the paper, we obtain a joint limit theorem on weak convergence for probability measure defined by discrete shifts of the Epstein and Hurwitz zeta-functions. The limit measure is explicitly given. For the proof, some linear independence restriction is required. The proved theorem extends and continues Bohr–Jessen’s classical results on probabilistic characterization of value distribution for the Riemann zeta-function.