2019/09/15 by Andrés Gómez-Colunga, Charlotte Kavaler, Gómez-Colunga, Andrés +5
Mathematics · Computer Science · #Analytic Number Theory Research #Coding theory and cryptography #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.1909.06740
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 \mathbbFq[x] with upper density arbitrarily close to (q - 1)/(q). Then, for a primitive set A, we consider the sum ∑a ∈ A \frac1q°a°a, the natural analogue in this setting of a sum considered by Erdős 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.