2024/06/09 by Petr Kucheriaviy, Kucheriaviy, Petr
Mathematics · #Advanced Harmonic Analysis Research #FOS: Mathematics #Mathematical Approximation and Integration #Number Theory (math.NT) #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2406.05896
openalex publication_date 2024/06/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A set of natural numbers A is called primitive if no element of A divides any other. Let Ω(n) be the number of prime divisors of n counted with multiplicity. Let fz(A) = ∑a ∈ A\fraczΩ(a)a (log a)z, where z ∈ ℝ> 0. Erdős proved in 1935 that f1(A) = ∑a ∈ A(1)/(a log a) is uniformly bounded over all choices of primitive sets A. We prove the same fact for fz(A), when z ∈ (0, 2). Also we discuss the limz → 0 fz(A). Some other results about primitive sets are generalized. In particular we study the asymptotic of fz(ℙk), where ℙk = \ n : Ω(n) = k \. In case of z = 1 we find the next term in asymptotic expansion of f1(ℙk) compared to the recent result of Gorodetsky, Lichtman, Wong.