Certain modular functions similar to the Dedekind eta function.
type: Item
instance of: scholarly article
title: Certain modular functions similar to the Dedekind eta function.
zbMATH Open document ID: 1044.11026
DOI: 10.1007/BF02941667
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 2003-02-13T00:00:00Z
review text: For any even and primitive Dirichlet character \(\psi\) the authors study properties of the function \[ \eta_\psi(z)=q^{-\tfrac 12 L(-1,\psi)} \prod^\infty_{n=1} (1-q^n)^{\psi(n)}, \] where \(q=\exp(2\pi iz)\) and \(L(s,\psi)\) is the Dirichlet \(L\)-function attached to \(\psi\). When \(\psi= \mathbf{1}\) (the trivial character) \(\eta_\psi(z)\) becomes the classical Dedekind eta function \(\eta(z)\), which has weight \(1/2\). But for \(\psi= \mathbf{1}\) the function \(\eta_\psi(z)\) has weight 0 and is not a product or quotient of eta functions. The authors show that for quadratic characters the functions \(\eta_\psi(z)\) enjoy properties similar to those of \(\eta(z)\).
Mathematics Subject Classification ID: 11F20
Mathematics Subject Classification ID: 11F37
zbMATH DE Number: 1868568
author: Taro Horie
author: Naruo Kanou
reviewed by: Tom M. Apostol
MaRDI profile type: Publication
cites work: Q5682393
cites work: On Eisenstein series with characters and the values of Dirichlet L-functions
cites work: Q4000402
cites work: Automorphic forms on \(O_{s+2,2}(\mathbb R)\) and infinite products
cites work: Monstrous Moonshine
cites work: Q3828097
cites work: The theory of Hecke integrals
cites work: Q3953865
cites work: Q4693061
cites work: Character analogues of Dedekind sums and transformations of analytic Eisenstein series
cites work: On generalized Dedekind sums attached to Dirichlet characters
cites work: Q5666618
cites work: Ramanujan's continued fraction
cites work: Q5513003
cites work: On modular forms of half integral weight
cites work: Q3934463
description: scientific article; zbMATH DE number 1868568