On the connection between the Beurling-Malliavin density and the asymptotic density

2023/11/08 by Giuliano, Rita, Grekos, Georges
#11B05 (Secondary) #11B25 (Primary) 14F05 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2311.04762

Abstract

We study the notion of Beurling-Malliavin density from the point of view of Number Theory. We prove a general relation between the Beurling-Malliavin density and the upper asymptotic density; we identify a class of sequences for which the two densities coincide; this class contains the arithmetic progressions. Last, by means of an alternative definition of Beurling-Malliavin density, we study the connection with the asymptotic density for another kind of sequences that again generalizes the arithmentic progressions.

Related

• The Beurling-Malliavin density, the Polya density and their connection
• Conditions for Beurling's integers to have a density
• Asymptotic ψ-densities of subsets of natural numbers
• Determination of the asymptotic behavior of probabilistic characteristics of arithmetic functions and some other questions of probabilistic number theory
• Gaussian lower bounds for the density via Malliavin calculus