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- Title
On Universality of Some Beurling Zeta-Functions.
- Authors
Geštautas, Andrius; Laurinčikas, Antanas
- Abstract
Let P be the set of generalized prime numbers, and ζ P (s) , s = σ + i t , denote the Beurling zeta-function associated with P. In the paper, we consider the approximation of analytic functions by using shifts ζ P (s + i τ) , τ ∈ R . We assume the classical axioms for the number of generalized integers and the mean of the generalized von Mangoldt function, the linear independence of the set { log p : p ∈ P } , and the existence of a bounded mean square for ζ P (s) . Under the above hypotheses, we obtain the universality of the function ζ P (s) . This means that the set of shifts ζ P (s + i τ) approximating a given analytic function defined on a certain strip σ ^ < σ < 1 has a positive lower density. This result opens a new chapter in the theory of Beurling zeta functions. Moreover, it supports the Linnik–Ibragimov conjecture on the universality of Dirichlet series. For the proof, a probabilistic approach is applied.
- Subjects
ANALYTIC functions; DIRICHLET series; ZETA functions; AXIOMS; HAAR integral; INTEGERS; RIEMANN hypothesis
- Publication
Axioms (2075-1680), 2024, Vol 13, Issue 3, p145
- ISSN
2075-1680
- Publication type
Article
- DOI
10.3390/axioms13030145