The Buchstab function (or Buchstab's function) is the unique continuous function <math>\omega: \R_{\ge 1}\rightarrow \R_{>0}</math> defined by the delay differential equation
<math>\omega(u)=\frac 1 u, \qquad\qquad\qquad 1\le u\le 2,</math>
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.
The Buchstab function approaches <math>e^{-\gamma} \approx 0.561</math> rapidly as <math>u\to\infty,</math> where <math>\gamma</math> is the Euler–Mascheroni constant. In fact,
where ρ is the Dickman function.[1] Also, <math>\omega(u)-e^{-\gamma}</math> 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]
Applications
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,
§IV.32, "On Φ(x,y) and Buchstab's function", Handbook of Number Theory I, József Sándor, Dragoslav S. Mitrinović, and Borislav Crstici, Springer, 2006, Шаблон:ISBN.
