Buchstab function

The continuous solution of the system

This function occurs in number theory as the limit

where denotes the number of positive integers not exceeding that are free of prime factors smaller than ; see [a1].

The function is positive-valued and converges to the constant as , where is the Euler constant. The difference behaves asymptotically like a trigonometric function with period and decaying amplitudes of size . These and similar results have been exploited in the study of irregularities in the distribution of prime numbers; see [a2], [a3].

References

[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