A generalization of the Montgomery-Hooley theorem

Let Ω(n) be the total number of prime factors of n andΘk(x;q,a)=∑n≤xn≡a(modq)Ω(n)=k1, where k is allowed to tend to infinity with respect to x. Combining the circle method with the Selberg-Delange method, together with the result of Bombieri-type sum for exponential sums, we investigate the behavior of the error term of Θk(x;q,a) in the “mean” and obtain its upper bound.