AN EXTENSION TO THE BRUN-TITCHMARSH THEOREM

The Siegel-Walfisz theorem states that for any B > 0, we have Sigma(p <= x p equivalent to a (mod k)) 1 similar to x/phi(k)logx for k <= log(B) x and (k, a) = 1. This only gives an asymptotic formula for the number of primes over an arithmetic progression for quite small moduli k compared with x. However, if we are only concerned about upper bound, we have the Brun-Titchmarsh theorem, namely for any 1 <= k < x, Sigma(p <= x p equivalent to a (mod k)) 1 << x/phi(k)log(x/k). In this article, we prove an extension to the Brun-Titchmarsh theorem on the number of integers, with at most s prime factors, in an arithmetic progression, namely Sigma(y < n <= x+y n equivalent to a(mod k) omega(n)<= s) 1 <= x/phi(k)log(x/k) Sigma(s-1)(l=0) (log log(x/k) +K)(l)/l! for any x, y > 0, s >= 1 and 1 <= k < x. In particular, for s <= log log( x/ k), we have Sigma(y < n <= x+y n equivalent to a(mod k) omega(n)<= s) 1 <= x/phi(k)log(x/k) (log log(x/k) + K)(s-1)/(s-1)! root log log(x/k) + K and for any epsilon is an element of (0, 1) and s <= (1 - epsilon) log log(x/ k), we have Sigma(y < n <= x+y n equivalent to a(mod k) omega(n)<= s) 1 << epsilon(-1)x/phi(k) log(x/k) (log log(x/k) + K)s(-1)/(s-1)!