Buchstab function explained

The Buchstab function (or Buchstab's function) is the unique continuous function

\omega: \R _{\ge} → \R _{>0} defined by the delay differential equation\omega(u) = (1)/(u,) 1 \le u\le 2 , { (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} ≈ 0 . 561 rapidly as u\to inf ty, where \gamma is the Euler–Mascheroni constant. In fact,|\omega(u) - e ^{-\gamma} |\le (\rho(u -1 ))/(u) , 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) (x)/(log x ^{1/u}) , x\to inf ty.

References

• • "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, .
• "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.

Notes and References

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.