The Turán–Kubilius inequality is a mathematical theorem in probabilistic number theory.
It is useful for proving results about the normal order of an arithmetic function.
[1]: 305–308 The theorem was proved in a special case in 1934 by Pál Turán and generalized in 1956 and 1964 by Jonas Kubilius.
[3]: 45–46 Suppose f is an additive complex-valued arithmetic function, and write p for an arbitrary prime and ν for an arbitrary positive integer.
Write and Then there is a function ε(x) that goes to zero when x goes to infinity, and such that for x ≥ 2 we have Turán developed the inequality to create a simpler proof of the Hardy–Ramanujan theorem about the normal order of the number ω(n) of distinct prime divisors of an integer n.[1]: 316 There is an exposition of Turán's proof in Hardy & Wright, §22.11.