Erdős-Kac Theorem

A deeper result than the Hardy-Ramanujan theorem. Let  be the number of integers in  such that inequality

(1)

holds, where  is the number of distinct prime factors of . Then

			(2)
			(3)

where  is a Landau symbol.

The theorem is discussed in Kac (1959).

REFERENCES:

Kac, M. Statistical Independence in Probability, Analysis and Number Theory. New York: Wiley, 1959.

Riesel, H. "The Erdős-Kac Theorem." Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser, pp. 158-159, 1994.