Sur le nombre des entiers sans grand facteur premier

On some sums involving the largest prime divisor of n, II

“…A stronger form of (i) is due to de Bruijn [2], (ii) follows from the definition (1.15) of (u), and (iii)(a) is the case k = 1 of Lemma 3(viii) of [8]. Part (b) of (iii) follows from (a) on noting that…”

“…We concentrate on the slightly more complicated sum Σ f (x); the corresponding formulae for Σ f (x) then follow by a similar argument. Our proof depends on using a result (Lemma 2.7) due to E. Saias [8] instead of the less precise formula in Lemma 2.5.…”

The median largest prime factor

“…In this section, we deduce the main result in a different way using the work of DeBruijn [1] and Saias [6] on integers without large prime factors. Let ψ(x, y) denote the number of integers n with 1 ≤ n ≤ x, all of whose prime factors are ≤ y.…”

“…To obtain proposition 2, we use the expansion for Λ(x, y) given in Saias' paper [6]. Suppose that x 1 u = y, u ≤ (log y) 3 5 −ǫ , and that u ∈ ∪ 1≤k≤n (k + ǫ, k + 1) ∪ (n + 1, ∞), so that u is not too close to an integer.…”

“…In section 4, we use the work of DeBruijn [1] and Saias [6] on integers without large prime factors to give an alternate derivation of theorem 1.…”