Buchstab function

The Buchstab function approaches ${\displaystyle e^{-\gamma }\approx 0.561}$ rapidly as ${\displaystyle u\to \infty ,}$ where ${\displaystyle \gamma }$ is the Euler–Mascheroni constant. In fact,

${\displaystyle |\omega (u)-e^{-\gamma }|\leq {\frac {\rho (u-1)}{u}},\qquad u\geq 1,}$

where ρ is the Dickman function.^[1] Also, ${\displaystyle \omega (u)-e^{-\gamma }}$ oscillates in a regular way, alternating between extrema and zeroes; the extrema alternate between positive maxima and negative minima. The interval between consecutive extrema approaches 1 as u approaches infinity, as does the interval between consecutive zeroes.^[2]

The Buchstab function is used to count rough numbers. If Φ(x, y) is the number of positive integers less than or equal to x with no prime factor less than y, then for any fixed u > 1,

${\displaystyle \Phi (x,x^{1/u})\sim \omega (u){\frac {x}{\log x^{1/u}}},\qquad x\to \infty .}$