Buchstab function

The Buchstab function (or Buchstab's function) is the unique continuous function ${\displaystyle \omega :\mathbb {R} _{\geq 1}\rightarrow \mathbb {R} _{>0}}$ 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.

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

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