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

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

and

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

{\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

^ ^a ^b ^c ^d ^e 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.
^ Narkiewicz, Władysław (1983). Number Theory. Singapore: World Scientific. ISBN 978-9971-950-13-2.
^ 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.
^ 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.