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 \[μ_y(u)e^γx\log y\prod_{p\leq y}\left(1-\frac{1}{p}\right),\] where $\gamma =\mathrm{0.5772156...}$ is Euler's constant and \[μ_y(u):=\int_{1}^{u}y^{t-u}ω(t)\,dt.\] He showed that the asymptotic formula \[Φ(x,y)=μ_y(u)e^γx\log y\prod_{p\leq y}\left(1-\frac{1}{p}\right)+O\left(\frac{xR(y)}{\log y}\right)\] holds uniformly for all $x\ge y\ge 2$, 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.

arXiv comment24 pages; submitted