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

TL;DR

This work analyzes the asymptotic growth of D(q)-pairs and D(q)-triples by reducing to quadratic congruences $x^2 \equiv q \pmod{b}$ and encoding solution counts into weighted divisor sums. The authors develop Dirichlet-series frameworks $\beta(s)$ built from $2^{\omega(n)}$ and good-prime indicators, relate them to zeta and Kronecker-character $L$-functions, and apply Wiener–Ikehara tauberian theorems to derive linear-in-$N$ asymptotics with explicit constants depending on $q$ (modulo 8). For prime $q$, they give precise slopes in terms of $L(1,\cdot)$, such as $D_{2,2}(N) \sim \frac{L(1,\chi_{8,5})}{\zeta(2)}N$ and $D_{2,-2}(N) \sim \frac{L(1,\chi_{8,3})}{\zeta(2)}N$, and establish that the number of D(n)-triples is asymptotically half the number of D(n)-pairs. They also show that, in general, D(n)-triples are dominated by regular triples with irregular contributions being negligible, leveraging Pell-type parametrizations. Together, these results illuminate a deep connection between Diophantine $D(q)$-tuples and analytic objects like L-functions and Dirichlet series.

Abstract

Let $q$ be an integer. A $D(q)$-$m$-tuple is a set of $m$ distinct positive integers ${a_1, a_2, . . . , a_m}$ such that $a_ia_j + q$ is a perfect square for all $1 \leq i < j \leq m$. By counting integer solutions $x \in [1, b]$ of congruences $x^2 \equiv q (\mod b)$ with $b \leq N$, 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 $ζ(2)$.

Theorems & Definitions (50)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof
• Corollary 7