Numerically explicit estimates for the distribution of rough numbers

Abstract

For x≥y>1 and u:=log⁡x/log⁡y, let Φ(x,y) denote the number of positive integers up to x free of prime divisors less than or equal to y. In 1950 de Bruijn [4] studied the approximation of Φ(x,y) by the quantityμy(u)eγxlog⁡y∏p≤y(1−1p), where γ=0.5772156... is Euler's constant andμy(u):=∫1uyt−uω(t)dt. He showed that the asymptotic formulaΦ(x,y)=μy(u)eγxlog⁡y∏p≤y(1−1p)+O(xR(y)log⁡y) holds uniformly for all x≥y≥2, where R(y) is a positive decreasing function related to the error estimates in the Prime Number Theorem. In this paper we obtain numerically explicit versions of de Bruijn's result.