On the Brun-Titchmarsh theorem
Hiroshi Mikawa · Tsukuba Journal of Mathematics · 1991
MIKAWA 1. Introduction.Let $\pi(x;q, a)$ denote the number of primes not exceeding $x$ and being congruent to $a$ modulo $q$ .In 1936 P. Tur\'an [6] showed that, under the ex- tended Riemann hypothesis, $\pi(x;q, a)\sim\frac{x}{\varphi(q)\log x}$ as $ x\rightarrow\infty$ for all $q\leqq x(\log x)^{-2-\epsilon}(\epsilon>0)$ and almost-all reduced residue classes $a$ modulo $q$ .The terminology "almost-all" means that the number of exceptional reduced classes is $o(\varphi(q))$ as $ q\rightarrow\infty$ .In 1972 C. Hooley [1] demonstrated that there holds the inequality $\pi(x;q, a)\leqq\frac{(4+\epsilon)x}{\varphi(q)\log(x^{2}/q)}$ $(\epsilon>0, x>x_{0}(\epsilon))$ for all $q\leqq x^{2/3}$ and almost-all $a$ .Later Y. Motohashi [4] proved that the same is valid for $x^{2/3} x_{0}(\epsilon)$ .If $q$ be given and $x^{6/7}\leqq q\leqq x(\log x)^{A}$ with $A>5$ , then we have $\pi(x;q, a)\leqq\frac{(18+\epsilon)x}{\varphi(q)\log(x^{6}/q)}$for almost-all reduced classes a modulo $q$ .REMARK.It is of some interest to note that, using the argument cf H. Iwaniec [3, section 2], one may easily show that $\pi(x;q, a)\leqq\left\{\begin{array}{l}\frac{(2+\epsilon)x}{\varphi(q)1og(xq^{-3/8})} if q\leqq x^{5/6-\delta}\\\frac{(1/2+\epsilon)x}{\varphi(q)1og(x/q)} if x^{5/6-\delta}\leqq q\leqq x^{6/7-\delta}\end{array}\right.