The prime k-tuplets in arithmetic progressions

Abstract

In this paper we discuss a problem on the distribution of prime multiplets in arithmetic progressions. Before mentioning our problem we need to introduce the following notation. ( In connection with our problem, see also the introduc- tion of Balog's tract [1].) For an integer $k\geqq 2$ , we let $a_{j}(0\leqq j\leqq k-1)$ be non-zero integers, and let $b_{j}(0\leqq]\leqq k-1)$ be integers, and put $a=(a_{0}, a_{1}, \cdots, a_{k-1}, b_{0}),$ $b=(b_{1}, \cdots, b_{k-1})$ , (Later, we will fix all the coordinates of $a$ , and treat an average over $b$ . This is why the unsymmetry of the definitions of $a$ and $b$ occurs.), $R(b)=R(a, b)=\prod_{j=0}^{k-1}|a_{j}|\prod_{0\leq i<J\leq k-1}|a_{\iota}b_{j}-a_{j}b_{t}|$ , $N(x;b)=N(x;a, b)=\{n;1\leqq a_{j}n+b_{j}\leqq xforall0\leqq j\leqq k-1\}$ ,