Products of primes in arithmetic progressions: a footnote in parity breaking

Ramaré, Olivier; Walker, Aled

Journal de théorie des nombres de Bordeaux, Tome 30 (2018) no. 1, pp. 219-225.

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 $x^{1/3}$.

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 $q$.

Classification : 11N13, 11A41, 11N37, 11B13
Mots clés : Primes in arithmetic progressions, Least prime quadratic residue, Linnik’s Theorem

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