Sur le nombre des entiers sans grand facteur premier. (On the number of integers without large prime factor)
An estimate for \(\psi(x,y)\), the number of positive integers not exceeding \(x\) with no prime factor exceeding \(y\), is required in various problems in number theory. \textit{N. G. de Bruijn} [Indagationes Math. 13, 50-60 (1951; Zbl 0042.042)] showed that for all \(\epsilon >0\) and \(x>1\), \(y\geq 2\), \[ (1)\quad \psi(x,y)= \Lambda (x,y)+O_{\epsilon}(x \exp (-(\log y)^{3/5-\epsilon})), \] where \[ \Lambda(x,y)= x\int^{\infty}_{0}\rho (\frac{\log x-\log t}{\log y})d([t]/t),\quad for\quad x\not\in {\mathbb{N}} \] with \(\rho\) denoting the Dickman function, and he deduced the well known formula \[ \psi(x,y)= x \rho (u)(1+O_{\epsilon}(\log (u+1)/(\log y))),\quad u=(\log x)/(\log y), \] for a range which was eventually extended to \[ (2)\quad \exp ((\log \log x)^{5/3+\epsilon})\leq y\leq x,\quad x\geq x_ 0(\epsilon) \] by \textit{A. Hildebrand} [J. Number Theory 22, 289-307 (1986; Zbl 0575.10038)]. In Trans. Am. Math. Soc. 296, 265-290 (1986; Zbl 0601.10028) \textit{A. Hildebrand} and \textit{G. Tenenbaum} obtained another formula for \(\psi\) (x,y), in terms of \(\zeta(s,y)= \prod_{p\leq y}(1-p^{-s})^{-1}\) and other functions, which is valid whenever \(x\geq y\geq 2\). By building on these developments, the present author derives the following improvement of (1) in the range (2): \[ \psi(x,y)= \Lambda (x,y)(1+O_{\epsilon}(\exp (-(\log y)^{3/5-\epsilon}))). \]
scientific article; zbMATH DE number 4106961
scientific article