Primitive sets in function fields

Abstract: A set of integers is \emph{primitive} if no element divides another. Erdős showed that $f(A) = \sum_{a \in A}\frac{1}{a\log a}$ converges for any primitive set $A$ of integers greater than one, and later conjectured this sum is maximized when $A$ is the set $P_1$ of primes. Banks and Martin further conjectured that $f(\mathcal{P}_1) > \ldots > f(\mathcal{P}_k) > f(\mathcal{P}_{k+1}) > \ldots$, where $\mathcal{P}_j$ denotes the integers with exactly $j$ prime factors. However, this was recently disproven by Lichtman. We consider the analogous questions for polynomials over a finite field $\mathbb{F}_q[x]$, obtaining bounds on the analogous sum, and find that while the analogue of the Banks and Martin conjecture similarly fails for small values of $q$, it seems likely to hold for larger values.

Joint work with Andrés Gómez-Colunga, Charlotte Kavaler and Mirilla Zhu.

number theory

Audience: researchers in the topic

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Combinatorial and additive number theory (CANT 2021)

Series comments: This is the nineteenth in a series of annual workshops sponsored by the New York Number Theory Seminar on problems in combinatorial and additive number theory and related parts of mathematics.

Registration for the conference is free. Register at cant2021.eventbrite.com.

The conference website is www.theoryofnumbers.com/cant/ Lectures will be broadcast on Zoom. The Zoom login will be emailed daily to everyone who has registered on eventbrite. To join the meeting, you may need to download the free software from www.zoom.us.

The conference program, list of speakers, and abstracts are posted on the external website.

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