ON PRIMES IN ARITHMETIC PROGRESSIONS

with some absolute constant L, vide [16, Kap. X]. Many works have been done to obtain an explicit value of this Linnik constant. The best known result is L = 5.5 due to D. R. Heath-Brown [14]. The Bombieri-Vinogradov theorem, see [7,§28], has the same power as the extended Riemann hypothesis in some sense. Indeed, it yields (1) for any given a t^ 0 and almost all q. In 1980 E. Fouvry and H. Iwaniec [10, 11] made a significantstep beyond the extended Riemann hypothesis. Their ideas have been surprisinglydeveloped by E. Fouvry [8, 9] and E. Bombieri, J. B. Friedlander and H. Iwaniec [4, 5].In particular,it follows from [5] that, for any fixed a ^ 0 and almost all q,