Numerically explicit estimates for the distribution of rough numbers

For $x\ge y>1$ and $u:=\log x\mathrm{/}\log y$, let $\mathrm{\Phi }(x,y)$ 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 $\mathrm{\Phi }(x,y)$ by the quantity $${\mu }_y(u)e^{\gamma }x\log y\prod _{p\le y}(1-\frac{1}{p}),$$ where $\gamma =\mathrm{0.5772156...}$ is Euler’s constant and $${\mu }_y(u):={\int }_1^u{y}^{t-u}\omega (t)dt\mathrm{.}$$ He showed that the asymptotic formula $$\mathrm{\Phi }(x,y)={\mu }_y(u)e^{\gamma }x\log y\prod _{p\le y}(1-\frac{1}{p})+O\left(\frac{xR(y)}{\log y}\right)$$ holds uniformly for all $x\ge y\ge 2$, 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.