An asymptotic property on a reciprocity law for the Bettin--Conrey cotangent sum
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arXiv
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| Format: | Preprint |
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2024
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| author | Akatsuka, Hirotaka Murakami, Yuya |
| author_facet | Akatsuka, Hirotaka Murakami, Yuya |
| contents | In 2013 Bettin and Conrey have introduced a cotangent sum $c \colon \mathbb{Q}_{>0}\to \mathbb{R}$, which can be regarded as a variant of the Dedekind sum. They have discovered that the cotangent sum satisfies a kind of reciprocity laws. Roughly speaking, the reciprocity law for $c(x)$ means that there is a relation between $c(x)$ and $c(1/x)$ modulo holomorphic functions. Furthermore they have investigated Taylor coefficients $g_n$ of the implicit holomorphic function, which appears in the reciprocity law for $c(x)$, at $x=1$. As a result, they have obtained an asymptotic formula for $g_n$ as $n\to\infty$. In this paper we improve it to an asymptotic series expansion. This resolves a conjecture by Zagier. A new ingredient of this paper is to use the confluent hypergeometric function of the second kind. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_14216 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An asymptotic property on a reciprocity law for the Bettin--Conrey cotangent sum Akatsuka, Hirotaka Murakami, Yuya Number Theory 11F11, 11L03, 11F20, 33C15 In 2013 Bettin and Conrey have introduced a cotangent sum $c \colon \mathbb{Q}_{>0}\to \mathbb{R}$, which can be regarded as a variant of the Dedekind sum. They have discovered that the cotangent sum satisfies a kind of reciprocity laws. Roughly speaking, the reciprocity law for $c(x)$ means that there is a relation between $c(x)$ and $c(1/x)$ modulo holomorphic functions. Furthermore they have investigated Taylor coefficients $g_n$ of the implicit holomorphic function, which appears in the reciprocity law for $c(x)$, at $x=1$. As a result, they have obtained an asymptotic formula for $g_n$ as $n\to\infty$. In this paper we improve it to an asymptotic series expansion. This resolves a conjecture by Zagier. A new ingredient of this paper is to use the confluent hypergeometric function of the second kind. |
| title | An asymptotic property on a reciprocity law for the Bettin--Conrey cotangent sum |
| topic | Number Theory 11F11, 11L03, 11F20, 33C15 |
| url | https://arxiv.org/abs/2402.14216 |