An extension to the Brun-Titchmarsh theorem

The Siegel-Walfisz theorem states that for any B>0, we have Sp=x, p?d (mod v) 1 ~ x/f(v) log(x) for v = logB(x) and (v,d)=1. This only gives an asymptotic formula for the number of primes in an arithmetic progression for quite a small modulus v compared to x. However, if we are concerned only with an upper bound, the Brun-Titchmarsh theorem says that for any 1=v=x, we have Sp=x, p?d (mod v) 1 << x/f(v) log(x). In this talk, we will discuss an extension to the Brun-Titichmarsh theorem that concerns the number of integers with exactly s distinct prime factors in an arithmetic progression. This is joint work with Kai Man Tsang and Tsz Ho Chan.