Primes in arithmetic progressions to large moduli

On the greatest prime factor of ab + 1

“…It is also worth noticing that in [21, §4] the author expresses the presentiment of the importance of the work of Bombieri, Friedlander and Iwaniec [4] to improve her results. The present paper confirms this intuition.…”

“…It is a challenge to improve the value of Q in (9), even by modifying the way of summing the error terms π(y; q, a) − π(y) ϕ(q) or even by approaching the characteristic function of the set of primes by the characteristic function of another set of the same, but easier, combinatorial structure. The first breakthrough in that direction is due to Fouvry and Iwaniec [14] (see also [9]) and it was followed by several papers of Bombieri, Fouvry, Friedlander and Iwaniec ([10], [11], [15], [13], [3], [4], [5]...) Also see [2, §12 p.89-103] for an introduction to these techniques, based on Linnik's dispersion method and on several types of bounds for Kloosterman sums.…”

“…The proof we shall give is highly based on the work [4]. We shall use the same tools and, as far as possible the same notations, but our proof is more than a paraphrase of the original proof of [4]: the intrusion of the integer variable m creates a new combinatorial situation that we cannot ignore.…”

“…We shall use the same tools and, as far as possible the same notations, but our proof is more than a paraphrase of the original proof of [4]: the intrusion of the integer variable m creates a new combinatorial situation that we cannot ignore. In the same order of ideas, the variable m brings unsuspected difficulty linked with coprimality conditions (we shall work a lot to circumvent the condition (A 3 (x)) below).…”

“…The purpose of the present paper is to improve the following It is important to note that, in the particular case where A and B are dense subsets of [1, N ], (which means that they satisfy |A|, |B| δN , for some fixed positive δ and N → +∞), we then have Γ + (A, B, N ) ≫ δ N 1+δ1 , where δ 1 is a positive function of δ. However the relation (1) never produces a lower bound better than N 4 3 . Actually, much more is conjectured since in [24,Conj.1], the authors propose the following Conjecture 1.…”