Buchstab function

It is named after Alexander Buchstab, who wrote about it in 1937.

In fact, where ρ is the Dickman function.

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,