Asymptotic density of k-almost primes

2014/01/13 by Belton, Martin
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1401.2694

Abstract

Landau's well known asymptotic formula N_k(x):= |\n≤ x : Ω(n)=k\| ∼ ( (x)/(log x) ) \frac(loglog x)^k-1(k - 1)! (x → ∞), which also holds for π_k(x):= |\n≤ x : ω(n)=k\|, is known to be fairly poor for k > 1, and when k is allowed to tend to infinity with x, the study of N_k(x) and π_k(x) becomes very technical [1, Chapter II.6, § 6.1, p.200]. I hope to show that the method described below provides not only a more accurate approach, but rather increases in its asymptotic accuracy as k tends to infinity.

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