Moments for primes in arithmetic progressions, I

Abstract

The second moment ∑q≤Q ∑a=1q (ψ(x; q, a) - ρ (x; q, a))2 is investigated with the novel approximation ρ(x; q, a) = ∑n≤x n≡a (mod q) F R(n), where FR(n) = ∑r≤R μ(r)/φ(r) ∑b=1 (b,r)=1r e(bn/r), and it is shown that when R ≤ logA x, this leads to more precise conclusions than in the classical Montgomery-Hooley case.