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Erdős-Kac Theorem

A deeper result than the Hardy-Ramanujan theorem. Let N(x,a,b) be the number of integers in [n,x] such that inequality

a<=(omega(n)-lnlnn)/(sqrt(lnlnn))<=b
(1)

holds, where omega(n) is the number of distinct prime factors of n. Then

lim_(x->infty)N(x,a,b)=((x+o(x)))/(sqrt(2pi))int_a^be^(-t^2/2)dt
(2)
=((x+o(x)))/2[erf(b/(sqrt(2)))-erf(a/(sqrt(2)))],
(3)

where o(x) is a Landau symbol.

The theorem is discussed in Kac (1959).

See also

Distinct Prime Factors, Hardy-Ramanujan Theorem

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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.

Referenced on Wolfram|Alpha

Erdős-Kac Theorem

Cite this as:

Weisstein, Eric W. "Erdős-Kac Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Erdos-KacTheorem.html

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