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THE PRIMES ARE NOT METRIC POISSONIAN

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

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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