A1 Refereed original research article in a scientific journal
Products of primes in arithmetic progressions
Authors: Matomäki Kaisa, Teräväinen Joni
Publisher: De Gruyter
Publishing place: Berlin
Publication year: 2024
Journal: Journal fur die reine und angewandte mathematik
Journal name in source: JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK
Journal acronym: J REINE ANGEW MATH
Volume: 2024
Issue: 808
Number of pages: 48
ISSN: 0075-4102
eISSN: 1435-5345
DOI: https://doi.org/10.1515/crelle-2023-0096
Web address : https://doi.org/10.1515/crelle-2023-0096
Self-archived copy’s web address: https://research.utu.fi/converis/portal/detail/Publication/386942683
Preprint address: https://arxiv.org/abs/2301.07679
A conjecture of Erdos states that, for any large prime q, every reduced residue class (mod q) can be represented as a product p(1) p(2) of two primes p(1) , p(2) <= q. We establish a ternary version of this conjecture, showing that, for any sufficiently large cube-free integer q, every reduced residue class ( mod q) can be written as p(1) p(2) p(3) with p 1 , p 2 , p 3 <= q primes. We also show that, for any epsilon > 0 and any sufficiently large integer q, at least (2/3 - epsilon) phi (q) reduced residue classes (mod q) can be represented as a product p(1)p(2) of two primes p 1 , p 2 <= q. The problems naturally reduce to studying character sums. The main innovation in the paper is the establishment of a multiplicative dense model theorem for character sums over primes in the spirit of the transference principle. In order to deal with possible local obstructions we establish bounds for the logarithmic density of primes in certain unions of cosets of subgroups of DOUBLE-STRUCK CAPITAL Z(q)(x) of small index and study in detail the exceptional case that there exists a quadratic character psi (mod q) such that psi (p) = - 1 for very many primes p <= q.
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