An estimate for the number of integers without large prime factors

Abstract:

$\Psi (x,y)$ denotes the number of positive integers $\leq x$ and free of prime factors $>y$. Hildebrand and Tenenbaum provided a good approximation of $\Psi (x,y)$. However, their method requires the solution $\alpha =\alpha (x,y)$ to the equation $\sum _{p \leq y} \log p/(p^{\alpha } - 1) = \log x$, and therefore it needs a large amount of time for the numerical solution of the above equation for large $y$. Hildebrand also showed $1-\xi _u/\log y$ approximates $\alpha$ for $1 \leq u \leq y/(2 \log y)$, where $u=(\log x)/\log y$ and $\xi _u$ is the unique solution to $e^{\xi _u}=1+ u\xi _u$. Let $E(i)$ be defined by $E(0)=\log u; E(i)=\log u+\log ( E(i-1)+1/u )$ for $i>0$. We show $E(m)$ approximates $\xi _u$, and $1-E(m)/\log y$ also approximates $\alpha$, where $m=\lceil (\log u+\log \log y)/\log \log u \rceil +1$. Using these approximations, we give a simple method which approximates $\Psi (x,y)$ within a factor $1+O(1/u+1/\log y)$ in the range $(\log \log x)^{5/3+\epsilon }<\log y<(\log x)/e$, where $\epsilon$ is any positive constant.