Turán–Kubilius inequality

Template:Short description 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.Template:Rp The theorem was proved in a special case in 1934 by Pál Turán and generalized in 1956 and 1964 by Jonas Kubilius.Template:Rp

Statement of the theorem

This formulation is from Tenenbaum.Template:Rp Other formulations are in NarkiewiczTemplate:Rp and in Cojocaru & Murty.Template:Rp

Suppose f is an additive complex-valued arithmetic function, and write p for an arbitrary prime and Template:Math for an arbitrary positive integer. Write

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

and

$B(x{)}^2=\sum _{p^{\nu }\le x}{|f(p^{\nu })|}^2{p}^{-\nu }.$

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

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

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.Template:Rp There is an exposition of Turán's proof in Hardy & Wright, §22.11. TenenbaumTemplate:Rp gives a proof of the Hardy–Ramanujan theorem using the Turán–Kubilius inequality and states without proof several other applications.