Uniform Titchmarsh divisor problems

Primes in arithmetic progressions to large moduli I: Fixed residue classes

“…The product is (log log x) OA,B (1) , so choosing A large enough we see that this gives the Siegel-Walfisz condition (with implied constants depending on B, C).…”

“…Let a ∈ Z, δ > 0 and Q = x 1/2+δ . Then we have q≤Q (q,a)=1 π(x; q, a) − π(x) φ(q) ≪ a δ 2 x log x + x(log log x) O (1) (log x) 3 .…”

“…We will let z 0 := x 1/(log log x) 3 and y 0 := x 1/ log log x two parameters depending on x, which we will think of as a large quantity. We will let ψ 0 : R → R denote a fixed smooth function supported on [1/2, 5/2] which is identically equal to 1 on the interval [1,2] and satisfies the derivative bounds ψ (j) 0 ∞ ≪ (4 j j!) 2 for all j ≥ 0.…”

“…We now apply Lemma 10.2 to the m 2 sum in T (1) (q 1 q 2 ), this time taking y = y 1 /d 1 . Thus it suffices to show for every 1-bounded sequence α ′ d2 and β ′ d2 that (11.2)…”

“…More recently, the key estimate in the work of Zhang [35] on bounded gaps between primes was a variant of (1.2) valid for moduli slightly larger than x 1/2 provided the moduli only had small prime factors. Zhang's work was extended by the Polymath project [32], and related recent results were obtained by Fouvry and Radziwi l l [21,22], Drappeau [10], Drappeau, Radziwi l l and Pratt [11] and Assing, Blomer and Li [1].…”