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].


[a1] A.A. Bukhstab, "Asymptotic estimates of a general number-theoretic function" Mat. Sb. , 44 (1937) pp. 1239–1246 (In Russian)
[a2] J. Friedlander, A. Granville, A. Hildebrand, H. Maier, "Oscillation theorems for primes in arithmetic progressions and for sifting functions" J. Amer. Math. Soc. , 4 (1991) pp. 25–86
[a3] H. Maier, "Primes in short intervals" Michigan Math. J. , 32 (1985) pp. 221–225
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