Ramanujan's function \(k(\tau)=r(\tau)r^2(2\tau)\) and its modularity
type: Item
instance of: scholarly article
title: Ramanujan's function \(k(\tau)=r(\tau)r^2(2\tau)\) and its modularity
zbMATH Open document ID: 1484.11109
author: Yoon Kyung Park
author: Yoonjin Lee
published in: Open Mathematics
publication date: 2021-11-29T00:00:00Z
review text: The content of this paper is related to the following topics: Ramanujan's function $k(\tau)$, modular function, class field theory, congruence subgroup, Rogers-Ramanujan continued fraction, Kronecker's congruence. These topics have been used in analytic number theory. The authors provide the modularity of Ramanujan's function $k(\tau)$ associated with the Rogers-Ramanujan continued fraction. They present not only the modular equation of the function $k(\tau)$ with some symmetry relations and some congruence relations which are satisfied by the modular equations, but also for some $\tau$ in an imaginary quadratic field, the value $k(\tau)$ generates the ray class field over an imaginary quadratic field modulo 10 since the function $k$ is a generator of the field of the modular function on $\Gamma_1(10)$. The authors give an optimal way of evaluating the singular values of the function $k(\tau)$ with the aid of the modular equations associated with the $j(\tau)$ which denotes the elliptic modular function. For the details, fallow the content of the paper. Moreover, the authors provide some calculations with the aid of the MAPLE program.
reviewed by: Yilmaz Simsek
Mathematics Subject Classification ID: 11F03
Mathematics Subject Classification ID: 11R37
Mathematics Subject Classification ID: 11R04
Mathematics Subject Classification ID: 14H55
zbMATH DE Number: 7435586
zbMATH Keywords: Ramanujan's function $k(\tau)$
zbMATH Keywords: modular function
zbMATH Keywords: class field theory
zbMATH Keywords: congruence subgroup
zbMATH Keywords: Rogers-Ramanujan continued fraction
zbMATH Keywords: Kronecker's congruence
MaRDI profile type: Publication
full work available at URL: math-2020-0105
OpenAlex ID: W3125961406
cites work: Traces of singular values of Hauptmoduln
cites work: Divisibility properties of coefficients of modular functions in genus zero levels
cites work: Q4000402
cites work: Explicit evaluations of a level 13 analogue of the Rogers-Ramanujan continued fraction
cites work: The level 13 analogue of the Rogers-Ramanujan continued fraction and its modularity
cites work: On Ramanujan's function \(k(q)=r(q)r^{2}(q^{2})\)
cites work: Q5513003
cites work: On some arithmetic properties of Siegel functions
cites work: Some modular relations for Ramanujan's function \(k(q)=r(q)r^2(q^2)\)
cites work: Ramanujan's Theta Functions
cites work: The equations for modular function fields of principal congruence subgroups of prime level
cites work: Arithmetic of the Ramanujan-Göllnitz-Gordon continued fraction
cites work: Q5625265
cites work: Modular curves and Ramanujan's continued fraction
cites work: CONSTRUCTION OF CLASS FIELDS OVER IMAGINARY QUADRATIC FIELDS AND APPLICATIONS
DOI: 10.1515/MATH-2020-0105
description: scientific article; zbMATH DE number 7435586