# Turán–Kubilius inequality

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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 Paul Turán and generalized in 1956 and 1964 by Jonas Kubilius.[1]:316

## Statement of the theorem

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

$A(x)=\sum_{p^\nu \le x} f(p^\nu) p^{-\nu}(1-p^{-1})$

and

$B(x)^2 = \sum_{p^\nu \le x} \left| f(p^\nu) \right| ^2 p^{-\nu}.$

Then there is a function ε(x) that goes to zero when x goes to infinity, and such that for x ≥ 2 we have

$\frac{1}{x} \sum_{n \le x} |f(n) - A(x)|^2 \le (2 + \varepsilon(x)) B(x)^2.$

## Applications of the theorem

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–Kubilus inequality and states without proof several other applications.

## Notes

1. Tenenbaum, Gérald (1995). Introduction to Analytic and Probabilistic Number Theory. Cambridge studies in advanced mathematics 46. Cambridge University Press. ISBN 0-521-41261-7.
2. ^ Narkiewicz, Władysław (1983). Number Theory. Singapore: World Scientific. ISBN 978-9971-950-13-2.
3. ^ Cojocaru, Alina Carmen; Murty, M. Ram (2005). An Introduction to Sieve Methods and Their Applications. London Mathematical Society Student Texts 66. Cambridge University Press. ISBN 0-521-61275-6.
4. ^ Hardy, G. H.; Wright, E. M. (2008) [First edition 1938]. An Introduction to the Theory of Numbers. Revised by D. R. Heath-Brown and Joseph H. Silverman (Sixth ed.). Oxford, Oxfordshire: Oxford University Press. ISBN 978-0-19-921986-5.