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.