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- Publisher Website: 10.1073/pnas.1906632116
- Scopus: eid_2-s2.0-85072637981
- PMID: 31501318
- WOS: WOS:000487532900031
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Article: Interesting identities involving weighted representations of integers as sums of arbitrarily many squares
Title | Interesting identities involving weighted representations of integers as sums of arbitrarily many squares |
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Authors | |
Keywords | representations of integers weighted representations polygonal numbers Jacobi triple product identity |
Issue Date | 2019 |
Publisher | National 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? |
Abstract | We 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 Identifier | http://hdl.handle.net/10722/277448 |
ISSN | 2021 Impact Factor: 12.779 2020 SCImago Journal Rankings: 5.011 |
PubMed Central ID | |
ISI Accession Number ID |
DC Field | Value | Language |
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dc.contributor.author | Jang, MJ | - |
dc.contributor.author | Kane, B | - |
dc.contributor.author | Kohnen, W | - |
dc.contributor.author | Man, SH | - |
dc.date.accessioned | 2019-09-20T08:51:16Z | - |
dc.date.available | 2019-09-20T08:51:16Z | - |
dc.date.issued | 2019 | - |
dc.identifier.citation | Proceedings of the National Academy of Sciences, 2019, v. 116 n. 39, p. 19374-19379 | - |
dc.identifier.issn | 0027-8424 | - |
dc.identifier.uri | http://hdl.handle.net/10722/277448 | - |
dc.description.abstract | We 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.language | eng | - |
dc.publisher | National Academy of Sciences. The Journal's web site is located at http://www.pnas.org | - |
dc.relation.ispartof | Proceedings of the National Academy of Sciences | - |
dc.rights | Proceedings of the National Academy of Sciences. Copyright © National Academy of Sciences. | - |
dc.subject | representations of integers | - |
dc.subject | weighted representations | - |
dc.subject | polygonal numbers | - |
dc.subject | Jacobi triple product | - |
dc.subject | identity | - |
dc.title | Interesting identities involving weighted representations of integers as sums of arbitrarily many squares | - |
dc.type | Article | - |
dc.identifier.email | Jang, MJ: min-joo.jang@hku.hk | - |
dc.identifier.email | Kane, B: bkane@hku.hk | - |
dc.identifier.authority | Kane, B=rp01820 | - |
dc.identifier.doi | 10.1073/pnas.1906632116 | - |
dc.identifier.pmid | 31501318 | - |
dc.identifier.pmcid | PMC6765249 | - |
dc.identifier.scopus | eid_2-s2.0-85072637981 | - |
dc.identifier.hkuros | 305463 | - |
dc.identifier.volume | 116 | - |
dc.identifier.issue | 39 | - |
dc.identifier.spage | 19374 | - |
dc.identifier.epage | 19379 | - |
dc.identifier.isi | WOS:000487532900031 | - |
dc.publisher.place | United States | - |
dc.identifier.issnl | 0027-8424 | - |