Oscillation of the Buchstab function
= Oscillation of the Buchstab function
The <Buchstab function> takes values both above and below $e^{-\gamma}$ on every interval of length one in $[1,\infty)$. The equation $\frac{d}{du}(u(w(u)-e^{-\gamma}))=w(u-1)-e^{-\gamma}$ propagates any fixed sign forwards; the rapid decay to zero then forces an impossible identically zero tail.