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Limit Theorems for Empirical Density of Greatest Common Divisors

Published online by Cambridge University Press:  23 May 2016

BEHZAD MEHRDAD  and
LINGJIONG ZHU
BEHZAD MEHRDAD
Affiliation:
Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY-10012, U.S.A. e-mail: [email protected], [email protected]
LINGJIONG ZHU
Affiliation:
Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY-10012, U.S.A. e-mail: [email protected], [email protected]
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Abstract

The law of large numbers for the empirical density for the pairs of uniformly distributed integers with a given greatest common divisor is a classic result in number theory. We study the large deviations of the empirical density. We will also obtain a rate of convergence to the normal distribution for the central limit theorem. Some generalisations are provided.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 161 , Issue 3 , November 2016 , pp. 517 - 533
Copyright
Copyright © Cambridge Philosophical Society 2016 

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References

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