Formal group laws for multiple sine functions and applications
type: Item
instance of: scholarly article
title: Formal group laws for multiple sine functions and applications
zbMATH Open document ID: 1316.11080
author: Shin-ya Koyama
author: Nobushige Kurokawa
published in: Kodai Mathematical Journal
publication date: 2013-05-24T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.kmj/1364562723
review text: Let \[ E_1(\tau)=-\frac{1}{4} +\sum_{n=1}^{\infty} d(n)q^n=-\frac{1}{4}+\sum_{m,n \geq 1} q^{mn}, \] where \(q=\exp(2\pi i \tau)\) and \(\tau\) belongs to the upper half-plane. Suppose \(M,N \geq 1\) are two coprime integers. The authors prove that the limit \(\lim_{\tau \to M/N} (E_{1}(-1/\tau)-\tau E_1(\tau))\) is equal to \[ \frac{i}{2N} \sum_{m,n} \Big|\frac{m}{M}-\frac{n}{N} \Big| \cot \Big(\pi \Big|\frac{m}{M}-\frac{n}{N} \Big| \Big) +\frac{i}{2\pi N}, \] where the sum is taken over every pair of integers \((m,n)\) satisfying \(0 \leq m<M\) and \(0 \leq n <N \) except for the pair \((0,0)\).
Mathematics Subject Classification ID: 11M32
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 33E20
zbMATH DE Number: 6165616
zbMATH Keywords: multiple sine function
zbMATH Keywords: Eisenstein series
zbMATH Keywords: modular function
reviewed by: A. Dubickas
MaRDI profile type: Publication
OpenAlex ID: W1986156504
DOI: 10.2996/KMJ/1364562723
description: scientific article; zbMATH DE number 6165616