Buchstab function


The Buchstab function is the unique continuous function defined by the delay differential equation
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 rapidly as where is the Euler–Mascheroni constant. In fact,
where ρ is the Dickman function. Also, 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 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,