On Discrete Shifts of Some Beurling Zeta Functions

Antanas Laurinčikas and Darius Šiaučiūnas ()
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Antanas Laurinčikas: Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko Str. 24, LT-03225 Vilnius, Lithuania
Darius Šiaučiūnas: Institute of Regional Development, Šiauliai Academy, Vilnius University, Vytauto Str. 84, LT-76352 Šiauliai, Lithuania

Mathematics, 2024, vol. 13, issue 1, 1-17

Abstract: We consider the Beurling zeta function ζ P ( s ) , s = σ + i t , of the system of generalized prime numbers P with generalized integers m satisfying the condition ∑ m ⩽ x 1 = a x + O ( x δ ) , a > 0 , 0 ⩽ δ < 1 , and suppose that ζ P ( s ) has a bounded mean square for σ > σ P with some σ P < 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 ζ P ( s + i l h ) . This set shifts has a positive density. For the proof, a weak convergence of probability measures in the space of analytic functions is applied.

Keywords: approximation of analytic functions; Beurling zeta function; generalized integers; generalized primes; Haar measure; random element; weak convergence (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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