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Vinogradov’s three primes theorem with primes having given primitive roots

Part of: Algebraic number theory: global fields Additive number theory; partitions

Published online by Cambridge University Press:  05 November 2019

C. FREI ,
P. KOYMANS  and
E. SOFOS
C. FREI
Affiliation:
University of Manchester, School of Mathematics, Oxford Road, Manchester M13 9PL, UK, e-mail: christopher.frei@manchester.ac.uk
P. KOYMANS
Affiliation:
Universiteit Leiden, Mathematisch Instituut, Niels Bohrweg 1, Leiden, 2333 CA, Netherlands. e-mail: p.h.koymans@math.leidenuniv.nl
E. SOFOS
Affiliation:
Max-Planck–Institut für Mathematik, Vivatsgasse 7, Bonn, 53072, Germany. e-mail: sofos@mpim-bonn.mpg.de
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Abstract

The first purpose of our paper is to show how Hooley’s celebrated method leading to his conditional proof of the Artin conjecture on primitive roots can be combined with the Hardy–Littlewood circle method. We do so by studying the number of representations of an odd integer as a sum of three primes, all of which have prescribed primitive roots. The second purpose is to analyse the singular series. In particular, using results of Lenstra, Stevenhagen and Moree, we provide a partial factorisation as an Euler product and prove that this does not extend to a complete factorisation.

MSC classification

Primary: 11P32: Goldbach-type theorems; other additive questions involving primes
Secondary: 11P55: Applications of the Hardy-Littlewood method 11R45: Density theorems
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 170 , Issue 1 , January 2021 , pp. 75 - 110
Copyright
© Cambridge Philosophical Society 2019

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References

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