On the exact location of the non-trivial zeros of Riemann’s zeta function
In this paper we introduce the real valued real analytic function κ(t) implicitly defined by e 2πiκ(t) = −e −2iϑ(t) ζ 0 ( 1 2 − it) ζ 0( 1 2 + it) , (κ(0) = − 1 2 ). By studying the equation κ(t) = n (without making any unproved hypotheses), we will show that (and how) this function is closely relat...
| Autores: | , |
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| Tipo de documento: | artigo |
| Estado: | Versão publicada |
| Data de publicação: | 2014 |
| País: | España |
| Recursos: | Universidad de Sevilla (US) |
| Repositório: | idUS. Depósito de Investigación de la Universidad de Sevilla |
| OAI Identifier: | oai:idus.us.es:11441/43000 |
| Acesso em linha: | http://hdl.handle.net/11441/43000 https://doi.org/10.4064/aa163-3-3 |
| Access Level: | Acceso aberto |
| Palavra-chave: | Zeta function Non-trivial zeros Distribution of zeros |
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On the exact location of the non-trivial zeros of Riemann’s zeta functionArias de Reyna Martínez, JuanLune, Jan van deZeta functionNon-trivial zerosDistribution of zerosIn this paper we introduce the real valued real analytic function κ(t) implicitly defined by e 2πiκ(t) = −e −2iϑ(t) ζ 0 ( 1 2 − it) ζ 0( 1 2 + it) , (κ(0) = − 1 2 ). By studying the equation κ(t) = n (without making any unproved hypotheses), we will show that (and how) this function is closely related to the (exact) position of the zeros of Riemann’s ζ(s) and ζ 0 (s). Assuming the Riemann hypothesis and the simplicity of the zeros of ζ(s), it will follow that the ordinate of the zero 1/2 + iγn of ζ(s) will be the unique solution to the equation κ(t) = n.Ministerio de Economía y CompetitividadPolish Academy of Sciences, Institute of MathematicsAnálisis MatemáticoFQM104: Analisis MatematicoMinisterio de Economía y Competitividad (MINECO). España2014info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionapplication/pdfapplication/pdfhttp://hdl.handle.net/11441/43000https://doi.org/10.4064/aa163-3-3reponame:idUS. Depósito de Investigación de la Universidad de Sevillainstname:Universidad de Sevilla (US)InglésActa Arithmetica, 163 (3), 215-245.info:eu-repo/grantAgreement/MINECO/MTM2012-30748/http://dx.doi.org/10.4064/aa163-3-3info:eu-repo/semantics/openAccessoai:idus.us.es:11441/430002026-06-17T12:51:07Z |
| dc.title.none.fl_str_mv |
On the exact location of the non-trivial zeros of Riemann’s zeta function |
| title |
On the exact location of the non-trivial zeros of Riemann’s zeta function |
| spellingShingle |
On the exact location of the non-trivial zeros of Riemann’s zeta function Arias de Reyna Martínez, Juan Zeta function Non-trivial zeros Distribution of zeros |
| title_short |
On the exact location of the non-trivial zeros of Riemann’s zeta function |
| title_full |
On the exact location of the non-trivial zeros of Riemann’s zeta function |
| title_fullStr |
On the exact location of the non-trivial zeros of Riemann’s zeta function |
| title_full_unstemmed |
On the exact location of the non-trivial zeros of Riemann’s zeta function |
| title_sort |
On the exact location of the non-trivial zeros of Riemann’s zeta function |
| dc.creator.none.fl_str_mv |
Arias de Reyna Martínez, Juan Lune, Jan van de |
| author |
Arias de Reyna Martínez, Juan |
| author_facet |
Arias de Reyna Martínez, Juan Lune, Jan van de |
| author_role |
author |
| author2 |
Lune, Jan van de |
| author2_role |
author |
| dc.contributor.none.fl_str_mv |
Análisis Matemático FQM104: Analisis Matematico Ministerio de Economía y Competitividad (MINECO). España |
| dc.subject.none.fl_str_mv |
Zeta function Non-trivial zeros Distribution of zeros |
| topic |
Zeta function Non-trivial zeros Distribution of zeros |
| description |
In this paper we introduce the real valued real analytic function κ(t) implicitly defined by e 2πiκ(t) = −e −2iϑ(t) ζ 0 ( 1 2 − it) ζ 0( 1 2 + it) , (κ(0) = − 1 2 ). By studying the equation κ(t) = n (without making any unproved hypotheses), we will show that (and how) this function is closely related to the (exact) position of the zeros of Riemann’s ζ(s) and ζ 0 (s). Assuming the Riemann hypothesis and the simplicity of the zeros of ζ(s), it will follow that the ordinate of the zero 1/2 + iγn of ζ(s) will be the unique solution to the equation κ(t) = n. |
| publishDate |
2014 |
| dc.date.none.fl_str_mv |
2014 |
| dc.type.none.fl_str_mv |
info:eu-repo/semantics/article info:eu-repo/semantics/publishedVersion |
| format |
article |
| status_str |
publishedVersion |
| dc.identifier.none.fl_str_mv |
http://hdl.handle.net/11441/43000 https://doi.org/10.4064/aa163-3-3 |
| url |
http://hdl.handle.net/11441/43000 https://doi.org/10.4064/aa163-3-3 |
| dc.language.none.fl_str_mv |
Inglés |
| language_invalid_str_mv |
Inglés |
| dc.relation.none.fl_str_mv |
Acta Arithmetica, 163 (3), 215-245. info:eu-repo/grantAgreement/MINECO/MTM2012-30748/ http://dx.doi.org/10.4064/aa163-3-3 |
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info:eu-repo/semantics/openAccess |
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openAccess |
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application/pdf application/pdf |
| dc.publisher.none.fl_str_mv |
Polish Academy of Sciences, Institute of Mathematics |
| publisher.none.fl_str_mv |
Polish Academy of Sciences, Institute of Mathematics |
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reponame:idUS. Depósito de Investigación de la Universidad de Sevilla instname:Universidad de Sevilla (US) |
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Universidad de Sevilla (US) |
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idUS. Depósito de Investigación de la Universidad de Sevilla |
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idUS. Depósito de Investigación de la Universidad de Sevilla |
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