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AN $L$ -FUNCTION-FREE PROOF OF VINOGRADOV’S THREE PRIMES THEOREM

Part of: Additive number theory; partitions Multiplicative number theory

Published online by Cambridge University Press:  04 November 2014

XUANCHENG SHAO
XUANCHENG SHAO*
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK; Xuancheng.Shao@maths.ox.ac.uk

Abstract

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We give a new proof of Vinogradov’s three primes theorem, which asserts that all sufficiently large odd positive integers can be written as the sum of three primes. Existing proofs rely on the theory of $L$ -functions, either explicitly or implicitly. Our proof is sieve theoretical and uses a transference principle, the idea of which was first developed by Green [Ann. of Math. (2) 161 (3) (2005), 1609–1636] and used in the proof of Green and Tao’s theorem [Ann. of Math. (2) 167 (2) (2008), 481–547]. To make our argument work, we also develop an additive combinatorial result concerning popular sums, which may be of independent interest.

MSC classification

Primary: 11P32: Goldbach-type theorems; other additive questions involving primes 11N35: Sieves
Secondary: 11P70: Inverse problems of additive number theory, including sumsets

Information

Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 2 , 01 July 2014 , e27
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/3.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author 2014

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