On Value Distribution of Certain Beurling Zeta-Functions

Abstract

In this paper, the approximation of analytic functions by shifts ζP(s+iτ) of Beurling zeta-functions ζP(s) of certain systems P of generalized prime numbers is discussed. It is required that the system of generalized integers NP generated by P satisfies ∑m⩽x,m∈N1=ax+O(xδ), a>0, 0⩽δ<1, and the function ζP(s) in some strip lying in σ^<σ<1, σ^>δ, which has a bounded mean square. Proofs are based on the convergence of probability measures in some spaces.