A Joint Limit Theorem for Epstein and Hurwitz Zeta-Functions

Abstract

In the paper, we prove a joint limit theorem in terms of the weak convergence of probability measures on C2 defined by means of the Epstein ζ(s;Q) and Hurwitz ζ(s,α) zeta-functions. The limit measure in the theorem is explicitly given. For this, some restrictions on the matrix Q and the parameter α are required. The theorem obtained extends and generalizes the Bohr-Jessen results characterising the asymptotic behaviour of the Riemann zeta-function.

Keywords

• Dirichlet L-function
• Epstein zeta-function
• Hurwitz zeta-function
• limit theorem
• Haar probability measure
• weak convergence