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Highly composite polynomials and the maximum order of the divisor function in Fq[t]

Afshar, A; (2020) Highly composite polynomials and the maximum order of the divisor function in Fq[t]. The Ramanujan Journal 10.1007/s11139-020-00299-2. (In press). Green open access

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Abstract

We investigate the analogues, in Fq[t], 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.

Type: Article
Title: Highly composite polynomials and the maximum order of the divisor function in Fq[t]
Open access status: An open access version is available from UCL Discovery
DOI: 10.1007/s11139-020-00299-2
Publisher version: https://doi.org/10.1007/s11139-020-00299-2
Language: English
Additional information: This 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/.
Keywords: Highly composite numbers, Divisor function, Arithmetic of polynomials over finite fields
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/10119951
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