On the Barban‐Davenport‐Halberstam Theorem: VI

we continue by investigating a topic that may be seen as a synthesis of those studied in I, II, and XII (as before, we refer to former papers with the present title by the Roman number indicating their position in the series, details of those cited being given in the list of references at the end). In the first two of those papers we were mainly interested in Barban–Montgomery asymptotic formulae (with remainder terms) of the type (3) S(x,Q) ∼ Qx logQ that were valid unconditionally for (4) x log−A x < Q ≤ x and also conditionally for (5) x1/2+ε < Q ≤ x on the extended Riemann hypothesis, while in the last paper, improving on the work of Friedlander and Goldston [1], we shewed unconditionally that (6) G(x, k) > ( 1 2 − ε ) x log k (x > x0(ε))