Buchstab function
The Buchstab function (or Buchstab's function) is the unique continuous function formula_1 defined by the delay differential equation 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 formula_4 rapidly as formula_5 where formula_6 is the Euler–Mascheroni constant. In fact, where "ρ" is the Dickman function. Also, formula_8 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. 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,