On Discrete Shifts of Some Beurling Zeta Functions

We consider the Beurling zeta function ζP(s), s=σ+it, of the system of generalized prime numbers P with generalized integers m satisfying the condition ∑m⩽x1=ax+O(xδ), a>0, 0⩽δ<1, and suppose that ζP(s) has a bounded mean square for σ>σP with some σP<1. Then, we prove that, for every h>0, there exists a closed non-empty set of analytic functions that are approximated by discrete shifts ζP(s+ilh). This set shifts has a positive density. For the proof, a weak convergence of probability measures in the space of analytic functions is applied.