A Partition Identity Related to Stanley's Theorem

In this paper, we use the Lambert series generating function for Euler's totient function to introduce a new identity for the number of \(1\)'s in the partitions of \(n\). A new expansion for Euler's partition function \(p(n)\) is derived in this context. These surprising new results...

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Bibliographic Details
Published in:	arXiv.org
Main Authors:	Merca, Mircea, Schmidt, Maxie D
Format:	Paper
Language:	English
Published:	Ithaca Cornell University Library, arXiv.org 20.10.2023
Subjects:	Number theory Partitions (mathematics)
ISSN:	2331-8422
Online Access:	Get full text
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Summary:	In this paper, we use the Lambert series generating function for Euler's totient function to introduce a new identity for the number of \(1\)'s in the partitions of \(n\). A new expansion for Euler's partition function \(p(n)\) is derived in this context. These surprising new results connect the famous classical totient function from multiplicative number theory to the additive theory of partitions.
Bibliography:	SourceType-Working Papers-1 ObjectType-Working Paper/Pre-Print-1 content type line 50
ISSN:	2331-8422
DOI:	10.48550/arxiv.2310.13672