Number Theory Seminar: Greg Martin

Abstract

A primitive set is a set of positive integers with the property that no element of the set divides another. We review a 1934 result of Besikovitch, showing that certain primitive sets can have large upper density even though their counting function is usually small, and a 1935 result of Erdős, showing that the sum of 1/(n log n) over all elements n of a primitive set is bounded by an absolute constant. We go on to describe two new theorems on primitive sets. First, in joint work with Carl Pomerance, we construct primitive sets with consistently large counting functions (as opposed to occasionally large as in Besikovitch's example), essentially providing a converse to Erdős's theorem. Second, the optimal absolute constant in Erdős's theorem is conjectured to be the sum of 1/(p log p) over all primes p, but this conjecture is still open; we describe current joint work with Bill Banks that makes progress towards this conjecture.