On the number of positive integers ≦𝑥 all of whose prime factors are ≦𝑦

Abstract

where e is an arbitrarily small positive number, p denotes primes, h> 2, and the constants implied in the 0-symbols depend on e and h. Vinogradoff [4] in his proof of the theorexm that every large odd number is the sum of three primes, gives a much cruder estimate than (1'). Recently Ankeny2 proved that if p-3 (mod 4), then n(p) Po(e), where n(p) is the least positive quadratic nonresidue of the prime p. This proof uses de Bruijn's estimates for J (x, y) and g(x, y) in a sharper form, however, than those given above. We follow de Bruijn in proving (1') and (1) which are listed as Theorems 1 and 2 below. Although these results are only special cases of de Bruijn's work, they seem important enough to merit simple proofs. Although our proof follows de Bruijn's very closely, we believe it is considerably simpler. Writing