Turán–Kubilius inequality

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

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

${\displaystyle A(x)=\sum _{p^{\nu }\leq x}f(p^{\nu })p^{-\nu }(1-p^{-1})}$

and

${\displaystyle B(x)^{2}=\sum _{p^{\nu }\leq x}\left|f(p^{\nu })\right|^{2}p^{-\nu }.}$

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

${\displaystyle {\frac {1}{x}}\sum _{n\leq x}|f(n)-A(x)|^{2}\leq (2+\varepsilon (x))B(x)^{2}.}$

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–Kubilius 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. Vol. 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. Vol. 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.