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Two new proofs of the Erdös–Kac Theorem, with bound on the rate of convergence, by Stein's method for distributional approximations

Published online by Cambridge University Press:  01 July 2009

ADAM J. HARPER
ADAM J. HARPER*
Affiliation:
King's College, Cambridge, CB2 1ST e-mail: A.J.Harper@dpmms.cam.ac.uk
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Abstract

In this paper, we apply Stein's method for distributional approximations to prove a quantitative form of the Erdös–Kac Theorem. We obtain our best bound on the rate of convergence, on the order of log log log n (log log n)−1/2, by making an intermediate Poisson approximation; we believe that this approach is simpler and more probabilistic than others, and we also obtain an explicit numerical value for the constant implicit in the bound. Different ways of applying Stein's method to prove the Erdös–Kac Theorem are discussed, including a Normal approximation argument via exchangeable pairs, where the suitability of a Poisson approximation naturally suggests itself.

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Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 147 , Issue 1 , July 2009 , pp. 95 - 114
Copyright
Copyright © Cambridge Philosophical Society 2009

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References

REFERENCES

[1]Billingsley, P.On the central limit theorem for the prime divisor function. The American Mathematical Monthly 76, no. 2 (1969), pp 132–139.CrossRefGoogle Scholar
[2]Brun, V.Le crible d'Eratosthène et le théoreme de Goldbach. Skr. Norske Vid. Akad, I, no. 3 (1920), pp 1–36.Google Scholar
[3]Chen, L. Poisson approximation for dependent trials. Ann. Probab., no. 3 (1975), pp 534–545.CrossRefGoogle Scholar
[4]Chen, L. and Shao, Q-M. Stein's method for normal approximation. Tutorial Notes for the Workshop on Stein's Method and Applications. In An Introduction to Stein's Method, eds. Barbour, A. and Chen, L., Lecture Notes Series, Volume 4, pp 1–59 (Institute of Mathematical Sciences, National University of Singapore, 2005).Google Scholar
[5]Cojocaru, A. and Murty, M. RamAn Introduction to Sieve Methods and their Applications. LMS Student Text 66 (Cambridge University Press, 2005).CrossRefGoogle Scholar
[6]Erdös, P.Note on the number of prime divisors of integers. J. London Math. Soc. 12 (1937), pp 308–314.CrossRefGoogle Scholar
[7]Erdös, P. and Kac, M.On the Gaussian law of errors in the theory of additive functions. Proc. Natl. Acad. Sci. USA, 25, no. 4 (1939), pp 206–207.CrossRefGoogle ScholarPubMed
[8]Erdös, P. and Kac, M.The Gaussian law of errors in the theory of additive number theoretic functions. Amer. J. Math. 62, no. 1 (1940), pp 738–742.CrossRefGoogle Scholar
[9]Erdös, P. and Sárközy, A.On the number of prime factors of integers. Acta Sci. Math. 42 (1980), pp 237–246.Google Scholar
[10]Erhardsson, T. Stein's method for Poisson and compound Poisson approximation. Tutorial Notes for the Workshop on Stein's Method and Applications. In An Introduction to Stein's Method, eds. Barbour, A. and Chen, L., Lecture Notes Series, Volume 4, pp 61–113, (Institute of Mathematical Sciences, National University of Singapore, 2005).CrossRefGoogle Scholar
[11]Hardy, G. H. and Wright, E. M.An Introduction to the Theory of Numbers (Oxford University Press, 1979).Google Scholar
[12]Kubilyus, I. P.Probabilistic methods in the theory of numbers. Amer. Math. Soc. Trans. ser. 2 19 (1962), pp 47–85 (Translated by Bateman, P. T.) (Originally published in Russian, 1956).Google Scholar
[13]Kubilius, J. Probabilistic methods in the theory of numbers. AMS Trans. Math. Monogr. 11 (Translated by Burgie, G. and Schuur, S.) 1964 (Originally published in Russian, 1962).Google Scholar
[14]LeVeque, W. J.On the size of certain number-theoretic functions. Trans. Amer. Math. Soc. 66, no. 2 (1949), pp 440–463.CrossRefGoogle Scholar
[15]Rényi, A. and Turán, P.On a theorem of Erdös–Kac. Acta Arith. 4 (1958), pp 71–84.CrossRefGoogle Scholar
[16]Rinott, Y. and Rotar, V.On coupling constructions and rates in the CLT for dependent summands with applications to the antivoter model and weighted U-statistics. Ann. Appl. Probab. 7 (1997), pp 1080–1105.CrossRefGoogle Scholar
[17]Rinott, Y. and Rotar, V.Normal approximations by Stein's method. Decis. Econ. Finance 23 (2000), pp 15–29.CrossRefGoogle Scholar
[18]Rosser, J. B. and Schoenfeld, L.Approximate formulas for some functions of prime numbers. Illinois J. Math. 6 (1962), pp 64–94.CrossRefGoogle Scholar
[19]Stein, C.Approximate Computation of Expectations. Lecture Notes, Monograph Series, Volume 7 (Institute of Mathematical Statistics, Hayward, California, 1986).Google Scholar