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.),$R(b)=R(a, b)=\prod_{j=0}^{k-1}|a_{j}|\prod_{0\leq i<J\leq k-1}|a_{\iota}b_{j}-a_{j}b_{t}|$ , $N(x;b)=N(x;a, b)=\{n;1\leqq a_{j}n+b_{j}\leqq xforall0\leqq j\leqq k-1\}$ ,and define $\Psi(x;b, a, q)=\Psi(x;a, b;a, q)=\sum_{Jn\in Ntxb_{q^{)})}=0 ,n\cong a(m\dot{o}d}$ $k1\Pi^{-}\Lambda(a_{j}n+b_{j})$ , where $\Lambda$ denotes the von Mangoldt function.And, we let, for any prime $p$ , $\rho(p)=\rho(p;a, b)$ be the number of solutions of the congruence $\prod_{j=0}^{k-1}(a_{j}n+b_{j})\equiv 0$ $(mod p)$ , and set, if $R(b)\neq 0,$ $\rho(p)<p$ for all prime $p$ , and $(a_{j}a+b_{j}, q)=1$ for all $ 0\leqq j\leqq$ $k-1$ , $\sigma(b;a, q)=\sigma(a, b;a, q)=\frac{1}{q}\prod_{p1q}(1-\frac{\rho(p)}{p})^{-1}\prod_{p}(1-\frac{\rho(p)}{p})(1-\frac{1}{p})^{-k}$ and $\sigma(b;a, q)=0$ otherwise.Further, we put $Z(x)=Z(x;a)=\{b;|N(x;b)|\neq 0\}$ , where $|N(x;b)|$ denote the length of the interval $N(x;b)$ .

Locations

• Tsukuba Journal of Mathematics - View - PDF
• Terrestrial Environment Research Center (University of Tsukuba) - View - PDF