Asymptotics of $D(q)$-pairs and triples via $L$-functions of Dirichlet charaters
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866912204046991360 |
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| author | Adžaga, Nikola Dražić, Goran Dujella, Andrej Pethő, Attila |
| author_facet | Adžaga, Nikola Dražić, Goran Dujella, Andrej Pethő, Attila |
| contents | 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 $ζ(2)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_01775 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Asymptotics of $D(q)$-pairs and triples via $L$-functions of Dirichlet charaters Adžaga, Nikola Dražić, Goran Dujella, Andrej Pethő, Attila Number Theory 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 $ζ(2)$. |
| title | Asymptotics of $D(q)$-pairs and triples via $L$-functions of Dirichlet charaters |
| topic | Number Theory |
| url | https://arxiv.org/abs/2304.01775 |