Odd values of the Rogers-Ramanujan functions
type: Item
instance of: scholarly article
title: Odd values of the Rogers-Ramanujan functions
zbMATH Open document ID: 1411.11095
author: Shi-Chao Chen
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2018-12-17T00:00:00Z
review text: In this paper, the author considers \[ \gamma(N)= \# \{1\leq n \leq N: n\equiv 1\pmod 2\text{ and } g(n) \equiv 1 \pmod 2 \} \] and \[ \delta(N)= \# \{1\leq n \leq N: n\equiv 0\pmod 2\text{ and } h(n) \equiv 1 \pmod 2 \}, \] where \(g(n)\), respectively \(h(n)\) denote the the number of partitions of the positive integer \(n\) into parts \(\equiv \pm1 \bmod 5\), respectively \(\pm2 \bmod 5\). \textit{B. Gordon} [Dev. Math. 23, 83--93 (2012; Zbl 1264.11090)] proved the order of magnitude of \(\gamma(N)\) and \(\delta(N)\) is \(\frac{N}{\log N}\). For sufficiently large \(N\), the author proves that \[ \gamma(N)=\frac{\pi^2}{5}\cdot \frac{N}{\log N} + O\left(\frac{N \log \log N}{\log^2 N} \right) \] and \[ \delta(N)=\frac{\pi^2}{5}\cdot \frac{N}{\log N} + O\left(\frac{N \log \log N}{\log^2 N} \right), \] which improve Gordon's result.
reviewed by: Mircea Merca
Mathematics Subject Classification ID: 11P84
zbMATH DE Number: 6994049
zbMATH Keywords: Rogers-Ramanujan identities
zbMATH Keywords: odd values
zbMATH Keywords: lower bounds
MaRDI profile type: Publication
full work available at URL: j.crma.2018.10.002
OpenAlex ID: W2898116535
cites work: Nonzero coefficients of half-integral weight modular forms mod \(\ell \)
cites work: A Proof of some identities of Ramanujan using modular forms
cites work: On the Parity of the Rogers-Ramanujan Coefficients
cites work: On the 2- and 4-dissections of the Rogers-Ramanujan functions
cites work: Q4449110
Wikidata QID: Q129035193
DOI: 10.1016/J.CRMA.2018.10.002
description: scientific article; zbMATH DE number 6994049