A note on sums of greatest (least) prime factors

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 ,…”

Sums of greatest prime factors

“…Let b m (n) be the m-th power of the greatest prime factor in the prime factorization of n. In a previous article [2], we proved the following asymptotic formula…”

“…Now, if k is large then D k,m is very small (see (27)) in comparation with C k,m . Therefore the contribution to n i=2 b m (i) of the smooth numbers whose density is 1 (see (2)) is insignificant in comparation with the contribution to n i=2 b m (i) of the rest of numbers whose density is zero (see (1)).…”

On certain sums extended over prime factors

“…However, we shall see that formula (3) can be obtained from formula (2). The case s = 0 was studied by R. Jakimczuk [2]. This author obtained the following asymptotic formula…”