On the upper bound of the number of primes in arithmetic progression (Q1098878)

As to the Brun-Titchmarsh problem \textit{H. Iwaniec} [J. Math. Soc. Japan 34, 95-123 (1982; Zbl 0486.10033)] proved the deep result \[ (1)\quad \pi (x;q,a)\quad <\quad (2+\epsilon)x/\phi (q)\log (xq^{-3/8}) \] valid for any \(\epsilon >0\), \(x>x_ 0(\epsilon)\) and all \(q\leq x^{9/20- \epsilon}\). By using a finer subdivision in Iwaniec's character sums approach, the present author claims the improved range \(q\leq x^{5/11- \epsilon}\). Although very concise, the proof seems to follow Iwaniec's closely.