Sums of greatest prime factors

Abstract

Suppose that k ≥ 2 and m ≥ 1 are fixed positive integers. Let B(n, p) be the number of positive integers not exceeding n such that the prime p is their greatest prime factor. In this article we obtain asymptotic formulae for C k,m (n) = n k <p≤n p m B(n, p) and D k,m (n) = 2≤p≤ n k p m B(n, p) Let b m (n) be the m-th power of the greatest prime factor in the prime factorization of n. In this article we proved the following asymptotic formula n i=2 b m (i) ∼ ζ(m + 1) m + 1 n m+1 log n , where ζ(s) is the Riemann's Zeta Function.

On the quotient between the greatest prime factor and the least prime factor

“…Let b(n) be the greatest prime factor in the prime factorization of n. In previous articles [1] [2], we proved the following asymptotic formula…”

“…where ζ(s) is the Riemann's Zeta Function. In articles [1] [2] we use the notation b m (i) = b(i) m . Let a(n) be the least prime factor in the prime factorization of n. In a previous article [2], we proved the following asymptotic formula n i=2 a(i) m ∼ 1 m + 1 n m+1 log n ,…”