On the primes in floor function sets

TL;DR

This paper investigates the distribution of primes within sets formed by the floor function of real numbers, analyzing their properties and counting functions, and compares these distributions to prime sequences.

Contribution

It introduces new results on the distribution of primes in floor function sets and provides bounds and asymptotic formulas for their counting functions.

Findings

Established bounds for the number of primes in floor function sets

Derived asymptotic formulas for prime counts in these sets

Compared distributions with prime sequences

Abstract

Let [t] be the integral part of the real number t and let 1 P be the characteristic function of the primes. Denote by $\pi$ G (x) the number of primes in the floor function set G(x) := {[ x n ] : 1 n x} and by S 1 P (x) the number of primes in the sequence {[ x n ]} n 1. Very recently, Heyman proves