On the Brun-Titchmarsh theorem (Q5905403)

It is shown that the inequality \(\pi(x,q,a)\leq(18+\epsilon)x/\varphi(q)\log(x^ 6/q)\) is true for almost all reduced residue classes \(a\bmod q\) if \(x^{6/7}\leq q\leq x(\log x)^{-A}\). This improves slightly a result of \textit{Y. Motohashi} [J. Math. Soc. Japan 26, 306-323 (1974; Zbl 0273.10040)]. In the proof the Rosser-Iwaniec sieve is used.