Relations between theta-functions, Hardy sums, Eisenstein and Lambert series in the transformation formula of log \(\eta_{g,h}(z)\).
type: Item
instance of: scholarly article
title: Relations between theta-functions, Hardy sums, Eisenstein and Lambert series in the transformation formula of log \(\eta_{g,h}(z)\).
zbMATH Open document ID: 1047.11042
DOI: 10.1016/S0022-314X(02)00072-0
author: Yilmaz Simsek
published in: Journal of Number Theory
publication date: 2003-05-25T00:00:00Z
review text: The author defines logarithms of generalized theta functions \(f_{n;g,h}(z, N)\) for \(n= 2,3,4\) from generalized Dedekind eta functions \(\eta_{g,h}(z, N)\), analogously to the expressions of classical theta functions \(\vartheta_n(z)\) for \(n= 2,3,4\) in terms of the Dedekind eta function \(\eta(z)\). When \((g, h)= (0, 0)\), \(f_{n;g,h}(z, N)\) coincides with \(2\log\vartheta_n(z)\). Then first, the transformation formula of \(f_{n;g,h}(z, N)\) under the appropriate congruence subgroup of \(\text{SL}_2(\mathbb{Z})\) is derived for each \(n\), where sums similar to Hardy sums appear. Secondly, the author considers the first derivative of \(f_{n;g,h}(z, N)\) with respect to \(z\) and finds the expression of it in terms of generalized Eisenstein series. Finally, the author considers Lambert series, which are connected with generalized Hardy sums. Here generalizations are given by the replacement of classical Dedekind sums with Apostol's generalized Dedekind sums.
reviewed by: Kaori Ota
Mathematics Subject Classification ID: 11F20
Mathematics Subject Classification ID: 14K25
Mathematics Subject Classification ID: 11M36
zbMATH DE Number: 1915510
zbMATH Keywords: generalized Dedekind eta function
zbMATH Keywords: Dedekind sum
zbMATH Keywords: theta function
zbMATH Keywords: Hardy sum
zbMATH Keywords: Eisenstein series
zbMATH Keywords: Lambert series
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1915510