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The celebrated theorem of Bombieri and A. I. Vinogradov states that
for any ε > 0 and A ε 0, the implied constant in the symbol «ε depending at most on ε and A (see [1] and [14]). The original proofs of Bombieri and Vinogradov were greatly simplified by P. X. Gallagher [4]. An elegant proof has been given recently by R. C. Vaughan [13]. For other references see H. L. Montgomery [10] and H. -E. Richert [12]. Estimates of type (1) are required in various applications of sieve methods. Having this in mind distinct generalizations have been investigated (see for example [15] and [2]). Y. Motohashi established a general theorem which, roughly speaking, says that if (1) holds for two arithmetic functions then it also holds for their Dirichlet convolution; for precise assumptions and statement see [11]. So far, all methods depend on the large sieve inequality (see [10])
which sets the limit x1/2 for the modulus q in (1) and in its generalizations.
