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THE PRIMES ARE NOT METRIC POISSONIAN

Part of: Multiplicative number theory Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  14 February 2018

Aled Walker
Aled Walker*
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, U.K. email walker@maths.ox.ac.uk
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Abstract

It has been known since Vinogradov that, for irrational $\unicode[STIX]{x1D6FC}$, the sequence of fractional parts $\{\unicode[STIX]{x1D6FC}p\}$ is equidistributed in $\mathbb{R}/\mathbb{Z}$ as $p$ ranges over primes. There is a natural second-order equidistribution property, a pair correlation of such fractional parts, which has recently received renewed interest, in particular regarding its relation to additive combinatorics. In this paper we show that the primes do not enjoy this stronger equidistribution property.

MSC classification

Primary: 11K38: Irregularities of distribution, discrepancy
Secondary: 11N05: Distribution of primes
Type
Research Article
Information
Mathematika , Volume 64 , Issue 1 , 2018 , pp. 230 - 236
Copyright
Copyright © University College London 2018 

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References

Aistleitner, C., Lachmann, T. and Pausinger, F., Pair correlations and equidistribution. J. Number Theory 182 2018, 206–220.CrossRefGoogle Scholar
Aistleitner, C., Larcher, G. and Lewko, M., Additive Energy and the Hausdorff dimension of the exceptional set in metric pair correlation problems. Israel J. Math. 222(1) 2017, 463–485 ; with an appendix by Jean Bourgain.Google Scholar
Baker, R. C., Metric number theory and the large sieve. J. Lond. Math. Soc. (2) 24(1) 1981, 34–40.Google Scholar
Bloom, T., Chow, S., Gafni, A. and Walker, A., Additive energy and the metric poissonian property. Preprint, 2017, arXiv:1709.02634.CrossRefGoogle Scholar
Grepstad, S. and Larcher, G., On pair correlation and discrepancy. Arch. Math. (Basel) 109(2) 2017, 143–149.Google Scholar
Harman, G., Metric Diophantine approximation with two restricted variables. I. Two square-free integers, or integers in arithmetic progressions. Math. Proc. Cambridge Philos. Soc. 103(2) 1988, 197–206.CrossRefGoogle Scholar
Harman, G., Metric Number Theory (London Mathematical Society Monographs, New Series 18 ), The Clarendon Press, Oxford University Press (New York, 1998).Google Scholar
Heath-Brown, D. R., Pair correlation for fractional parts of 𝛼n 2 . Math. Proc. Cambridge Philos. Soc. 148(3) 2010, 385–407.CrossRefGoogle Scholar
Iwaniec, H. and Kowalski, E., Analytic Number Theory (American Mathematical Society Colloquium Publications 53 ), American Mathematical Society (Providence, RI, 2004).Google Scholar
Lachmann, T. and Technau, N., On exceptional sets in the metric poissonian pair correlations problem. Preprint, 2017, arXiv:1708.08599.Google Scholar
Rudnick, Z. and Sarnak, P., The pair correlation function of fractional parts of polynomials. Comm. Math. Phys. 194(1) 1998, 61–70.CrossRefGoogle Scholar
Rudnick, Z., Sarnak, P. and Zaharescu, A., The distribution of spacings between the fractional parts of n 2𝛼. Invent. Math. 145(1) 2001, 37–57.Google Scholar
Rudnick, Z. and Zaharescu, A., A metric result on the pair correlation of fractional parts of sequences. Acta Arith. 89(3) 1999, 283–293.CrossRefGoogle Scholar
Steinerberger, S., Localized quantitative criteria for equidistribution. Acta Arith. 180(2) 2017, 183–199.CrossRefGoogle Scholar
Vaughan, R. C., The Hardy–Littlewood Method, 2nd edn., (Cambridge Tracts in Mathematics 125 ), Cambridge University Press (Cambridge, 1997).CrossRefGoogle Scholar