Highly Composite Polynomials and the maximum order of the Divisor Function in $\mathbb{F}_q[t]$
Number Theory
2020-08-05 v3
Abstract
We investigate the analogues, in , 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.
Keywords
Cite
@article{arxiv.2001.05635,
title = {Highly Composite Polynomials and the maximum order of the Divisor Function in $\mathbb{F}_q[t]$},
author = {Ardavan Afshar},
journal= {arXiv preprint arXiv:2001.05635},
year = {2020}
}
Comments
10 pages. Major revisions incorporated to the presentation of the main argument, and an appendix with a table of highly composite polynomials added. Accepted for publication in The Ramanujan Journal