On the size of primitive sets in function fields

摘要

A set is primitive if no element of the set divides another. We consider primitive sets of monk polynomials over a finite field and find natural generalizations of many of the results known for primitive sets of integers. In particular, we show that primitive sets in the function field have lower density zero by showing that the sum Sigma(alpha is an element of A) 1/q (deg a) deg a, an analogue of a sum considered by Erdos, is uniformly bounded over all primitive sets A. We then adapt a method of Besicovitch to construct primitive sets in F-q[x] with upper density arbitrarily close to q-1/q and generalize a result 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. (C) 2020 Elsevier Inc. All rights reserved.