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]:{{{1}}} Other formulations are in Narkiewicz[2]:{{{1}}} and in Cojocaru & Murty.[3]:{{{1}}}
Suppose f is an additive complex-valued arithmetic function, and write p for an arbitrary prime and ν for an arbitrary positive integer. Write
- [math]\displaystyle{ A(x)=\sum_{p^\nu \le x} f(p^\nu) p^{-\nu}(1-p^{-1}) }[/math]
and
- [math]\displaystyle{ B(x)^2 = \sum_{p^\nu \le x} \left| f(p^\nu) \right| ^2 p^{-\nu}. }[/math]
Then there is a function ε(x) that goes to zero when x goes to infinity, and such that for x ≥ 2 we have
- [math]\displaystyle{ \frac{1}{x} \sum_{n \le x} |f(n) - A(x)|^2 \le (2 + \varepsilon(x)) B(x)^2. }[/math]
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]:{{{1}}} There is an exposition of Turán's proof in Hardy & Wright, §22.11.[4] Tenenbaum[1]:{{{1}}} gives a proof of the Hardy–Ramanujan theorem using the Turán–Kubilius inequality and states without proof several other applications.
Notes
- ↑ 1.0 1.1 1.2 1.3 1.4 Tenenbaum, Gérald (1995). Introduction to Analytic and Probabilistic Number Theory. Cambridge studies in advanced mathematics. 46. Cambridge University Press. ISBN 0-521-41261-7.
- ↑ Narkiewicz, Władysław (1983). Number Theory. Singapore: World Scientific. ISBN 978-9971-950-13-2. https://books.google.com/books?id=4CUUmYrem2YC.
- ↑ Cojocaru, Alina Carmen; Murty, M. Ram (2005). An Introduction to Sieve Methods and Their Applications. London Mathematical Society Student Texts. 66. Cambridge University Press. ISBN 0-521-61275-6.
- ↑ Hardy, G. H.; Wright, E. M. (2008). 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.
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