Certain eta-quotients and arithmetic density of Andrews' singular overpartitions
type: Item
instance of: scholarly article
title: Certain eta-quotients and arithmetic density of Andrews' singular overpartitions
zbMATH Open document ID: 1479.05026
author: Yanyan Li
published in: Journal of Number Theory
publication date: 2021-09-06T00:00:00Z
review text: Singular overpartitions were introduced by \textit{G. E. Andrews} [Int. J. Number Theory 11, No. 5, 1523--1533 (2015; Zbl 1325.11107)] in his study of analogues of the Rogers-Ramanujan identities. In particular, Andrews' singular overpartition function \(\overline{C}_{k,i}(n)\) counts the number of overpartitions of \(n\) in which no part is divisible by \(k\) and only parts congruent to \(\pm i\) modulo \(k\) may be overlined. The object of this paper under review is to investigate the arithmetic densities of \(\overline{C}_{3\ell,\ell}(n)\) modulo arbitrary powers of \(2\) and \(3\) when \(\ell=p\cdot 2^\alpha\) with \(p\) prime, by applying the modularity of certain eta-quotients. For example, the authors show that, for all \(\alpha\ge 1\) with \(2^\alpha\ge p\), the set \(\{n\in\mathbb{N}:\overline{C}_{3p\cdot 2^\alpha,p\cdot 2^\alpha}(n) \equiv 0 \pmod{2^k}\}\) has arithmetic density \(1\).
reviewed by: Shane Chern
Mathematics Subject Classification ID: 05A17
Mathematics Subject Classification ID: 11P83
zbMATH DE Number: 7391387
zbMATH Keywords: singular overpartitions
zbMATH Keywords: eta-quotients
zbMATH Keywords: modular forms
zbMATH Keywords: arithmetic density
Wikidata QID: Q114157123
MaRDI profile type: Publication
full work available at URL: j.jnt.2020.11.018
OpenAlex ID: W3116570261
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DOI: 10.1016/J.JNT.2020.11.018
description: scientific article; zbMATH DE number 7391387