On the number of positive integers \(\leq x\) and free of prime factors \(>y\)
Let \(\Psi(x,y)\) denote the number of positive integers \(\leq x\), all whose prime factors are \(\leq y\), and let \(\rho(u)\) be the Dickman function, defined by the conditions \(\rho(u)=1\) \((0\leq u\leq 1)\), \(\rho'(u)=-u\rho(u-1)\) \((u>1)\), \(\rho(u)\) continuous at \(u=1\). The main result of this paper is that the estimate \[ \Psi(y^ u,y)=y^ u \rho(u)(1+O_{\varepsilon}(\log (u+1)/\log y)), \] which has been originally established by \textit{N. G. de Bruijn} [Ned. Akad. Wet. Proc. Ser. A 54, 50--60 (1951; Zbl 0042.04204)] for the range \(1\leq u\leq (\log y)^{3/5-\varepsilon}\), remains valid in the range \[ 1\leq u\leq \exp ((\log y)^{3/5-\varepsilon}).\tag{*} \] The proof proceeds by a careful induction argument, using the identity \[ \Psi (x,y) \log x=\int^{x}_{1}(\Psi (t,y)/t)\,dt+\sum_{p^ m\leq x, p\leq y}\Psi (xp^{-m},y) \log p\quad (x,y\geq 1) \] and the trivial estimate \(\Psi (x,y)=x+O(1)\) (1\(\leq x\leq y)\) to initiate the induction process. The same method is employed to prove two further results on \(\Psi(x,y)\), namely a short interval estimate valid in the range (*) and a lower bound valid uniformly in \(x\geq y\geq 1\).
scientific article; zbMATH DE number 3918482