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. [1]: 316
This formulation is from Tenenbaum. [1]: 302 Other formulations are in Narkiewicz [2]: 243 and in Cojocaru & Murty. [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. [4] Tenenbaum [1]: 305–308 gives a proof of the Hardy–Ramanujan theorem using the Turán–Kubilius inequality and states without proof several other applications.