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Article: Interesting identities involving weighted representations of integers as sums of arbitrarily many squares

TitleInteresting identities involving weighted representations of integers as sums of arbitrarily many squares
Authors
Keywordsrepresentations of integers
weighted representations
polygonal numbers
Jacobi triple product
identity
Issue Date2019
PublisherNational Academy of Sciences. The Journal's web site is located at http://www.pnas.org
Citation
Proceedings of the National Academy of Sciences, 2019, v. 116 n. 39, p. 19374-19379 How to Cite?
AbstractWe consider the number of ways to write an integer as a sum of squares, a problem with a long history going back at least to Fermat. The previous studies in this area generally fix the number of squares which may occur and then either use algebraic techniques or connect these to coefficients of certain complex analytic functions with many symmetries known as modular forms, from which one may use techniques in complex and real analysis to study these numbers. In this paper, we consider sums with arbitrarily many squares, but give a certain natural weighting to each representation. Although there are a very large number of such representations of each integer, we see that the weighting induces massive cancellation, and we furthermore prove that these weighted sums are again coefficients of modular forms, giving precise formulas for them in terms of sums of divisors of the integer being represented.
Persistent Identifierhttp://hdl.handle.net/10722/277448
ISSN
2023 Impact Factor: 9.4
2023 SCImago Journal Rankings: 3.737
PubMed Central ID
ISI Accession Number ID

 

DC FieldValueLanguage
dc.contributor.authorJang, MJ-
dc.contributor.authorKane, B-
dc.contributor.authorKohnen, W-
dc.contributor.authorMan, SH-
dc.date.accessioned2019-09-20T08:51:16Z-
dc.date.available2019-09-20T08:51:16Z-
dc.date.issued2019-
dc.identifier.citationProceedings of the National Academy of Sciences, 2019, v. 116 n. 39, p. 19374-19379-
dc.identifier.issn0027-8424-
dc.identifier.urihttp://hdl.handle.net/10722/277448-
dc.description.abstractWe consider the number of ways to write an integer as a sum of squares, a problem with a long history going back at least to Fermat. The previous studies in this area generally fix the number of squares which may occur and then either use algebraic techniques or connect these to coefficients of certain complex analytic functions with many symmetries known as modular forms, from which one may use techniques in complex and real analysis to study these numbers. In this paper, we consider sums with arbitrarily many squares, but give a certain natural weighting to each representation. Although there are a very large number of such representations of each integer, we see that the weighting induces massive cancellation, and we furthermore prove that these weighted sums are again coefficients of modular forms, giving precise formulas for them in terms of sums of divisors of the integer being represented.-
dc.languageeng-
dc.publisherNational Academy of Sciences. The Journal's web site is located at http://www.pnas.org-
dc.relation.ispartofProceedings of the National Academy of Sciences-
dc.rightsProceedings of the National Academy of Sciences. Copyright © National Academy of Sciences.-
dc.subjectrepresentations of integers-
dc.subjectweighted representations-
dc.subjectpolygonal numbers-
dc.subjectJacobi triple product-
dc.subjectidentity-
dc.titleInteresting identities involving weighted representations of integers as sums of arbitrarily many squares-
dc.typeArticle-
dc.identifier.emailJang, MJ: min-joo.jang@hku.hk-
dc.identifier.emailKane, B: bkane@hku.hk-
dc.identifier.authorityKane, B=rp01820-
dc.identifier.doi10.1073/pnas.1906632116-
dc.identifier.pmid31501318-
dc.identifier.pmcidPMC6765249-
dc.identifier.scopuseid_2-s2.0-85072637981-
dc.identifier.hkuros305463-
dc.identifier.volume116-
dc.identifier.issue39-
dc.identifier.spage19374-
dc.identifier.epage19379-
dc.identifier.isiWOS:000487532900031-
dc.publisher.placeUnited States-
dc.identifier.issnl0027-8424-

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