# Buchstab function

The Buchstab function (or Buchstab's function) is the unique continuous function $\omega :\mathbb {R} _{\geq 1}\rightarrow \mathbb {R} _{>0}$ defined by the delay differential equation

$\omega (u)={\frac {1}{u}},\qquad \qquad \qquad 1\leq u\leq 2,$ ${\frac {d}{du}}(u\omega (u))=\omega (u-1),\qquad u\geq 2.$ In the second equation, the derivative at u = 2 should be taken as u approaches 2 from the right. It is named after Alexander Buchstab, who wrote about it in 1937.

## Asymptotics

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

$|\omega (u)-e^{-\gamma }|\leq {\frac {\rho (u-1)}{u}},\qquad u\geq 1,$ where ρ is the Dickman function. Also, $\omega (u)-e^{-\gamma }$ 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.

## Applications

The Buchstab function is used to count rough numbers. If Φ(xy) 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,

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