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On the Size of Primitive Sets in Function Fields by Gómez-Colunga, Andrés; Kavaler, Charlotte; McNew, Nathan; Zhu, Mirilla is a scholarly article available to read on EtoBox.

What is On the Size of Primitive Sets in Function Fields about?

A set is primitive if no element of the set divides another. We consider primitive sets of monic polynomials over a finite field and find natural generalizations of many of the results known for primitive sets of integers. In particular we generalize a result of Besicovitch to show that there exist primitive sets in $\mathbb{F}_q[x]$ with upper density arbitrarily close to $\frac{q - 1}{q}$. Then, for a primitive set $A$, we consider the sum $\sum_{a \in A} \frac{1}{q^{\deg a}\deg a}$, the natural analogue in this setting of a sum considered by Erd\H{o}s for primitive subsets of the integers, and show that it is uniformly bounded over all primitive sets $A$. We end with a generalization of work of Martin and Pomerance on the asymptotic growth rate of the counting function of a primitive set. Along the way we prove a quantitative analogue of the Hardy-Ramanujan theorem for function fields, as well as bounds on the size of the $k$-th irreducible polynomial.

Author

Gómez-Colunga, Andrés; Kavaler, Charlotte; McNew, Nathan; Zhu, Mirilla

Published

2019

Language

EN