Products of primes in arithmetic progressions: a footnote in parity breaking
Olivier Ramaré1; Aled Walker2
1 CNRS / Institut de Mathématiques de Marseille Aix Marseille Université, U.M.R. 7373 Site Sud, Campus de Luminy, Case 907 13288 MARSEILLE Cedex 9, France
2 Mathematical Institute University of Oxford Andrew Wiles Building Radcliffe Observatory Quarter Woodstock Road Oxford OX2 6GG, United Kingdom
Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 1, pp. 219-225.

We prove that, if x and q⩽x 1/16 are two parameters, then for any invertible residue class a modulo q there exists a product of exactly three primes, each one below x 1/3 , that is congruent to a modulo q.

Nous montrons que, étant donnés x et q⩽x 1/16 , toute classe inversible a modulo q contient au moins un produit d’exactement trois nombres premiers, chacun étant inférieur ou égal à x 1/3 .

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1024
Classification: 11N13, 11A41, 11N37, 11B13
Keywords: Primes in arithmetic progressions, Least prime quadratic residue, Linnik’s Theorem
Olivier Ramaré 1; Aled Walker 2

1 CNRS / Institut de Mathématiques de Marseille Aix Marseille Université, U.M.R. 7373 Site Sud, Campus de Luminy, Case 907 13288 MARSEILLE Cedex 9, France
2 Mathematical Institute University of Oxford Andrew Wiles Building Radcliffe Observatory Quarter Woodstock Road Oxford OX2 6GG, United Kingdom
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Olivier Ramaré; Aled Walker. Products of primes in arithmetic progressions: a footnote in parity breaking. Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 1, pp. 219-225. doi : 10.5802/jtnb.1024. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1024/

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