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

Olivier Ramare (1) , Aled Walker (2)

1 I2M - Institut de Mathématiques de Marseille
2 Mathematical Institute, University of Oxford

We prove that, if $x$ and $q\leqslant x^{1/16}$ 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 $q$.

Mots clés: Prime numbers; Mathematical theorems; Arithmetic progressions; Mathematical inequalities; Combinatorics; Cardinality; Linnik's Theorem

Domaines: Théorie des nombres [math.NT]

Olivier Ramare, Aled Walker. Products of primes in arithmetic progressions: a footnote in parity breaking. Journal de Théorie des Nombres de Bordeaux, 2018, 30 (1), pp.219-225. ⟨hal-02025094⟩