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Fan, Kai
Journal of number theory, July 2024, 2024-07-00, Volume: 260Journal Article
For x≥y>1 and u:=logx/logy, 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)eγxlogy∏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)eγxlogy∏p≤y(1−1p)+O(xR(y)logy) 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.
