The Turán–Kubilius inequality is a

mathematical theorem In mathematics and formal logic, a theorem is a statement that has been proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical c ...

in probabilistic number theory. It is useful for proving results about the

normal order of an arithmetic function In number theory, a normal order of an arithmetic function is some simpler or better-understood function which "usually" takes the same or closely approximate values. Let ''f'' be a function on the natural numbers. We say that ''g'' is a normal o ...

. The theorem was proved in a

special case In logic, especially as applied in mathematics, concept is a special case or specialization of concept precisely if every instance of is also an instance of but not vice versa, or equivalently, if is a generalization of .Brown, James Robert.� ...

in 1934 by

Pál Turán Pál Turán (; 18 August 1910 – 26 September 1976) also known as Paul Turán, was a Hungarian mathematician who worked primarily in extremal combinatorics. In 1940, because of his Jewish origins, he was arrested by History of the Jews in Hun ...

and generalized in 1956 and 1964 by Jonas Kubilius.

Statement of the theorem

This formulation is from Tenenbaum. Other formulations are in Narkiewicz and in Cojocaru & Murty. Suppose ''f'' is an

additive Additive may refer to: Mathematics * Additive function, a function in number theory * Additive map, a function that preserves the addition operation * Additive set-function see Sigma additivity * Additive category, a preadditive category with fin ...

complex-valued

arithmetic function In number theory, an arithmetic, arithmetical, or number-theoretic function is generally any function whose domain is the set of positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their definition th ...

, and write ''p'' for an arbitrary prime and for an arbitrary positive integer. Write :

A(x)=\sum_ f(p^\nu) p^(1-p^)

and :

B(x)^2 = \sum_ \left,  f(p^\nu) \ ^2 p^.

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

Notes

{{DEFAULTSORT:Turan-Kubilius inequality Inequalities (mathematics) Theorems in number theory