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An asymptotic property on a reciprocity law for the Bettin--Conrey cotangent sum

Hirotaka Akatsuka, Yuya Murakami

Abstract

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.

An asymptotic property on a reciprocity law for the Bettin--Conrey cotangent sum

Abstract

In 2013 Bettin and Conrey have introduced a cotangent sum , 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 means that there is a relation between and modulo holomorphic functions. Furthermore they have investigated Taylor coefficients of the implicit holomorphic function, which appears in the reciprocity law for , at . As a result, they have obtained an asymptotic formula for as . 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.
Paper Structure (6 sections, 14 theorems, 105 equations, 1 figure, 1 table)

This paper contains 6 sections, 14 theorems, 105 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. The $K$-Bessel functions
  3. The discussion on $g(x)$ and $g_n$ by Bettin--Conrey
  4. Confluent hypergeometric functions $U(\alpha;\beta;z)$ and their relations with $\mathcal{I}_n(x)$
  5. Proof of Theorem \ref{['Thm:main']}
  6. A remark on numerical computation of $g_n$

Key Result

Theorem 1.1

There are $\widetilde{C}_l\in\langle\pi^{2m}:m=0,1,2,\ldots,l\rangle_{\mathbb {Q}}\setminus\{0\}$ such that as $n\to\infty$ for each $L\in\mathbb {Z}_{\geq 1}$, where the implied constant depends only on $L$.

Figures (1)

  • Figure 1: The plot of (\ref{['eq:mainRmk']}) for $8000\leq n\leq 10001$. The gray curve is the first term of (\ref{['eq:sin']}). Here, $(2\pi)^{-5/2}\widetilde{C}_5=1.49962\cdots$.

Theorems & Definitions (27)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • proof : Proof of Lemma \ref{['Lem:Inest']}
  • ...and 17 more