AN EXTENSION TO THE BRUN–TITCHMARSH THEOREM

The Siegel–Walfisz theorem states that for any B > 0, we have $$\sum_{p\leq x\ p\equiv a\ {\rm mod}\ k)}\quad1\sim\frac {x}{\varphi(k)\ {\rm log}\ x}$$ 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, $$\sum_{p\leq x\ p\equiv a\ {\rm mod}\ k)}\quad1\ll\frac {x}{\varphi(k)\ {\rm log}(x/k)}.$$ 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 $$\sum_{y\lt n\leq x+y\ n\equiv a({\rm mod}\ k)\omega(n)\leq s}\quad 1\ll \frac {x}{\varphi(k){\rm log}(x/k)}\sum^{s-1}_{\ell=0}\frac {{\rm log\ log}(x/k)+K)^\ell}{\ell !}$$ for any x, y > 0, s ≥ 1 and 1 ≤ k < x. In particular, for s ≤ log log (x/k), we have $$\sum_{y\lt n\leq x+y\ n\equiv a({\rm mod}\ k)\omega(n)\leq s}\quad 1\ll \frac {x}{\varphi(k){\rm log}(x/k)}\frac {({\rm log\ log}(x/k)+K)^{s-1}}{(s-1)!}\sqrt {{\rm log\ log}(x/k)+K}$$ and for any ε∈(0, 1) and s ≤ (1−ε) log log (x/k), we have $$\sum_{y\lt n\leq x+y\ n\equiv a({\rm mod}\ k)\omega(n)\leq s}\quad 1\ll \frac {\varepsilon^{-1}x}{\varphi(k){\rm log}(x/k)}\frac {({\rm log\ log}(x/k)+K)^{s-1}}{(s-1)!}.$$