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

${\displaystyle \omega (u)={\frac {1}{u}},\qquad \qquad \qquad 1\leq u\leq 2,}$

${\displaystyle {\frac {d}{du}}(u\omega (u))=\omega (u-1),\qquad u\geq 2.}$

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.