Distribution of ω(n) over h-free and h-full numbers

2024/09/16 by Das, Sourabhashis, Kuo, Wentang, Liu, Yu-Ru
#11N05 #11N37 #11N56 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2409.10430

Abstract

Let ω(n) denote the number of distinct prime factors of a natural number n. In 1917, Hardy and Ramanujan proved that ω(n) has normal order log log n over naturals. In this work, we establish the first and the second moments of ω(n) over h-free and h-full numbers using a new counting argument and prove that ω(n) has normal order log log n over these subsets.

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