Buchstab function

Graph of the Buchstab function ω(u) from u = 1 to u = 4.

The Buchstab function (or Buchstab's function) is the unique continuous function $\omega: \R_{\ge 1}\rightarrow \R_{>0}$ defined by the delay differential equation

$\omega(u)=\frac 1 u, \qquad\qquad\qquad 1\le u\le 2,$

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

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.

Asymptotics

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

$|\omega(u)-e^{-\gamma}|\le \frac{\rho(u-1)}{u}, \qquad u\ge 1,$

where ρ is the Dickman function.^[1] Also, $\omega(u)-e^{-\gamma}$ oscillates in a regular way, alternating between extrema and zeroes; the extrema alternate between positive maxima and negative minima. The interval between consecutive extrema approaches 1 as u approaches infinity, as does the interval between consecutive zeroes.^[2]

Applications

The Buchstab function is used to count rough numbers. If Φ(x, y) is the number of positive integers less than or equal to x with no prime factor less than y, then for any fixed u > 1,

$\Phi(x,x^{1/u}) \sim \omega(u)\frac{x}{\log x^{1/u}}, \qquad x\to\infty.$

Notes

1. (5.13), Jurkat and Richert 1965. In this paper the argument of ρ has been shifted by 1 from the usual definition.
2. p. 131, Cheer and Goldston 1990.

References

• Бухштаб, А. А. (1937), "Асимптотическая оценка одной общей теоретикочисловой функции" [Asymptotic estimation of a general number-theoretic function], Matematicheskii Sbornik (in Russian), 2(44) (6): 1239–1246, Zbl 0018.24504
• "Buchstab Function", Wolfram MathWorld. Accessed on line Feb. 11, 2015.
• §IV.32, "On Φ(x,y) and Buchstab's function", Handbook of Number Theory I, József Sándor, Dragoslav S. Mitrinović, and Borislav Crstici, Springer, 2006, ISBN 978-1-4020-4215-7.
• "A differential delay equation arising from the sieve of Eratosthenes", A. Y. Cheer and D. A. Goldston, Mathematics of Computation 55 (1990), pp. 129–141.
• "An improvement of Selberg’s sieve method", W. B. Jurkat and H.-E. Richert, Acta Arithmetica 11 (1965), pp. 217–240.
• Hildebrand, A. (2001), "Bukhstab function", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4