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.

On Universality of Some Beurling Zeta-Functions

“…In [24], the following statement has been obtained. Denote by Ω P,1 a subset of Ω P such that a product…”

“…Our purpose is to prove the universality of the function ζ P (s) with a certain system P. We began studying the approximation of analytic functions by shifts ζ P (s + iτ) in [24]. Suppose that the estimate (1) is valid.…”

“…Here, and in the sequel, the notation a ≪ c b, a ∈ C, b > 0, shows that there exists a constant c = c(ε) > 0 such that |a| ⩽ cb. Denote by H(D) the space of analytic on D functions equipped with the topology of uniform convergence on compacta, and by measA the Lebesgue measure of a measurable set A ⊂ R. The main result of [24] is the following theorem.…”