Asymptotics of $D(q)$-pairs and triples via $L$-functions of Dirichlet charaters

Let $q$ be an integer. A $D(q)$-$m$-tuple is a set of $m$ distinct positive integers $a_1,a_2,\mathrm{.}\mathrm{.}\mathrm{.},a_m$ such that $a_i{a}_j+q$ is a perfect square for all $1\le i<j\le m$. By counting integer solutions $x\in [1,b]$ of congruences $x^2\equiv q(\mathrm{m}\mathrm{o}\mathrm{d}b)$ with $b\le N$, we count $D(q)$-pairs with both elements up to $N$, and give estimates on asymptotic behaviour. We show that for prime $q$, the number of such $D(q)$-pairs and $D(q)$-triples grows linearly with $N$. Up to a factor of $2$, the slope of this linear function is the quotient of the value of the $L$-function of an appropriate Dirichlet character (usually a Kronecker symbol) and of $\zeta (2)$.