Year: 2023
Type: preprint
Abstract: For $x\ge y>1$ and $u:= \log x/\log y$, let $Φ(x,y)$ 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 $Φ(x,y)$ by the quantity \[μ_y(u)e^γx\log y\prod_{p\leq y}\left(1-\frac{1}{p}\right),\] where $γ=0.5772156...$ 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 $x\ge y\ge2$, 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.
Source: arXiv (Cornell University)
Author: Fan, Steve
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4380092762