Numerically explicit estimates for the distribution of rough numbers

Abstract

For $x\ge y>1$ and $u:= \log x/\log y$, let $\Phi(x,y)$ denote the number of positive integers up to $x$ free of prime divisors less than or equal to $y$. In 1950 de Bruijn [1] studied the approximation of $\Phi(x,y)$ by the quantity \[\mu_y(u)e^{\gamma}x\log y\prod_{p\leq y}\left(1-\frac{1}{p}\right),\] where $\gamma=0.5772156...$ is Euler's constant and \[\mu_y(u):=\int_{1}^{u}y^{t-u}\omega(t)\,dt.\] He showed that the asymptotic formula \[\Phi(x,y)=\mu_y(u)e^{\gamma}x\log y\prod_{p\leq y}\left(1-\frac{1}{p}\right)+O\left(\frac{xR(y)}{\log y}\right)\] holds uniformly for all $x\ge y\ge2$, 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.