The Turán–Kubilius inequality is a

mathematical theorem In mathematics, a theorem is a statement that has been proved, or can be proved. 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 consequence of the ...

probabilistic number theory In mathematics, Probabilistic number theory is a subfield of number theory, which explicitly uses probability to answer questions about the integers and integer-valued functions. One basic idea underlying it is that different prime numbers are, in ...

. 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 or ...

. 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 . A limiting case i ...

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. He had a long collaboration with fellow Hungarian mathematician Paul Erdős, lasting ...

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-functionn see Sigma additivity * Additive category, a preadditive category with f ...

complex-valued

arithmetic function In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function ''f''(''n'') whose domain is the positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their d ...

, 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 In mathematics, the Hardy–Ramanujan theorem, proved by , states that the normal order of the number ω(''n'') of distinct prime factors of a number ''n'' is log(log(''n'')). Roughly speaking, this means that most numbers have about this number ...

about the

normal order In quantum field theory a product of quantum fields, or equivalently their creation and annihilation operators, is usually said to be normal ordered (also called Wick order) when all creation operators are to the left of all annihilation operator ...

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 Theorems in number theory