Turán Kubilius Inequality

High Quality Content by WIKIPEDIA articles! 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. :305 308 The theorem was proved in a special case in 1934 by Paul Turán and generalized in 1956 and 1964 by Jonas Kubilius. Turán developed the inequality to create a simpler proof of the Hardy Ramanujan theorem about the normal order of the number (n) of distinct prime divisors of an integer n.:316 There is an exposition of Turán's proof in Hardy & Wright,
22.11. Tenenbaum:305 308 gives a proof of the Hardy Ramanujan theorem using the Turán Kubilus inequality and states without proof several other applications.