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The law of large numbers for additive arithmetic functions

Published online by Cambridge University Press:  24 October 2008

P. D. T. A. Elliott
P. D. T. A. Elliott
Affiliation:
Boulder, University of Colorado, U.S.A.
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Extract

Let f(n) be a real-valued additive arithmetic function, that is to say, that f(ab) = f(a) + f(b) for each pair of coprime integers a and b. Let α(x) and β(x) > 0 be real-valued functions, defined for x ≥ 2. In this paper, we study the frequencies

We shall establish necessary and sufficient conditions, subject to rather weak growth conditions upon β(x) alone, in order that these frequencies converge to the improper law, in other words, that f(n) obey a form of the weak law of large numbers.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 78 , Issue 1 , July 1975 , pp. 33 - 71
Copyright
Copyright © Cambridge Philosophical Society 1975

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References

REFERENCES

(1)Elliott, P. D. T. A.On connections between the Turan-Kubilius inequality and the Large Sieve: Some applications. Amer. Math. Soc. Symposia, Vol. xxiv (1973).Google Scholar
(2)Gnedenko, B. V. and Kolmogorov, A. N.Limit distributions for sums of independent random variables (Addison-Wesley, Reading, Massachusetts, 1954).Google Scholar
(3)Halász, G.Über die Mittelwerte multiplikativer zahlentheoretischer Funktionen. Acta Math. Acad. Sci. Hungar. 19 (1968), 365–403.CrossRefGoogle Scholar
(4)Hardy, G. H., Littlewood, J. E. and Pólya, G.Inequalities, 2nd ed. (Cambridge University Press, 1964).Google Scholar
(5)Hardy, G. H. and Wright, E. M.The theory of numbers, 4th ed. (Oxford Press, 1960).Google Scholar
(6)Kubilius, J.Probabilistic methods in the theory of numbers. Amer. Math. Soc. Translations vol. xi.Google Scholar
(7)Titchmarsh, E. C.The theory of the Riemann zeta-function (Oxford, 1951)Google Scholar