# Buchstab function

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The continuous solution $\omega(u)$ of the differential-delay equation $$(u\,\omega(u))' = \omega(u-1)$$ for $u > 2$ with initial values $$\omega(u) = \frac{1}{u}\ ,\ \ (1 \le u \le 2) \ .$$
This function occurs in number theory as the limit $$\omega(u) = \lim_{x\rightarrow\infty} \frac{ \Phi(x,x^{1/u}) \log(x^{1/u}) }{ x }$$ where $\Phi(x,y)$ denotes the number of positive integers not exceeding $x$ that are free of prime factors smaller than $y$; see [a1].
The function $\omega(u)$ is positive-valued and converges to the constant $e^{-\gamma}$ as $u\rightarrow\infty$, where $\gamma$ is the Euler constant. The difference $\omega(u)-e^{-\gamma}$ behaves asymptotically like a trigonometric function with period $2$ and decaying amplitudes of size $\exp((1+o(1))\log u)$. These and similar results have been exploited in the study of irregularities in the distribution of prime numbers; see [a2], [a3].