Highly composite polynomials and the maximum order of the divisor function in [Formula: see text]

We investigate the analogues, in [Formula: see text] , of highly composite numbers and the maximum order of the divisor function, as studied by Ramanujan. In particular, we determine a family of highly composite polynomials which is not too sparse, and we use it to compute the logarithm of the maxim...

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Detalles Bibliográficos
Autor principal: Afshar, Ardavan
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Springer US 2020
Materias:
Article
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8550348/
https://www.ncbi.nlm.nih.gov/pubmed/34720673
http://dx.doi.org/10.1007/s11139-020-00299-2
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author Afshar, Ardavan
author_facet Afshar, Ardavan
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description We investigate the analogues, in [Formula: see text] , of highly composite numbers and the maximum order of the divisor function, as studied by Ramanujan. In particular, we determine a family of highly composite polynomials which is not too sparse, and we use it to compute the logarithm of the maximum of the divisor function at every degree up to an error of a constant, which is significantly smaller than in the case of the integers, even assuming the Riemann Hypothesis.
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spelling pubmed-85503482021-10-29 Highly composite polynomials and the maximum order of the divisor function in [Formula: see text] Afshar, Ardavan Ramanujan J Article We investigate the analogues, in [Formula: see text] , of highly composite numbers and the maximum order of the divisor function, as studied by Ramanujan. In particular, we determine a family of highly composite polynomials which is not too sparse, and we use it to compute the logarithm of the maximum of the divisor function at every degree up to an error of a constant, which is significantly smaller than in the case of the integers, even assuming the Riemann Hypothesis. Springer US 2020-08-03 2021 /pmc/articles/PMC8550348/ /pubmed/34720673 http://dx.doi.org/10.1007/s11139-020-00299-2 Text en © The Author(s) 2020 https://creativecommons.org/licenses/by/4.0/Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ (https://creativecommons.org/licenses/by/4.0/) .
spellingShingle Article
Afshar, Ardavan
Highly composite polynomials and the maximum order of the divisor function in [Formula: see text]
title Highly composite polynomials and the maximum order of the divisor function in [Formula: see text]
title_full Highly composite polynomials and the maximum order of the divisor function in [Formula: see text]
title_fullStr Highly composite polynomials and the maximum order of the divisor function in [Formula: see text]
title_full_unstemmed Highly composite polynomials and the maximum order of the divisor function in [Formula: see text]
title_short Highly composite polynomials and the maximum order of the divisor function in [Formula: see text]
title_sort highly composite polynomials and the maximum order of the divisor function in [formula: see text]
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8550348/
https://www.ncbi.nlm.nih.gov/pubmed/34720673
http://dx.doi.org/10.1007/s11139-020-00299-2
work_keys_str_mv AT afsharardavan highlycompositepolynomialsandthemaximumorderofthedivisorfunctioninformulaseetext

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