Irregularities in the distribution of primes in an arithmetic progression

Abstract

(1) |∆(x; q, a)| e (q/x)1/2−e log x uniformly for q ≤ x, for any given e > 0. Recently, Friedlander and Granville [1] disproved Montgomery’s conjecture (1). They showed that for any A > 0 there exist arbitrarily large values of x and integers q ≤ x/(log x) and a with (a, q) = 1 for which |∆(x; q, a)| 1. Then Friedlander, Granville, Hildebrand and Maier [2] further showed that (1) fails to hold for almost all moduli q as small as x exp{−(log x)1/3−δ}, for any fixed δ > 0, if the parameter e in (1) is sufficiently small. They also showed the following

Locations

• Acta Arithmetica - View - PDF