On a generalization of the Selberg formula

In this paper, we prove a theorem related to the asymptotic formula for ψk(x;q,a) which is used to count numbers up to x with at most k distinct prime factors (or k-almost primes) in a given arithmetic progression . This theorem not only gives the asymptotic formula for ψk(x;q,a) (or Selberg formula), but has played an essential role, recently, in obtaining a lower bound for the variance of distribution of almost primes in arithmetic progressions.