On the Critical Exponent for k-Primitive Sets

Tsz Ho Chan, Kennesaw State University
Jared Duker Lichtman, University of Oxford
Carl Pomerance, Dartmouth College


A set of positive integers is primitive (or 1-primitive) if no member divides another. Erdős proved in 1935 that the weighted sum ∑1/(n log n) for n ranging over a primitive set A is universally bounded over all choices for A. In 1988 he asked if this universal bound is attained by the set of prime numbers. One source of difficulty in this conjecture is that ∑n−λ over a primitive set is maximized by the primes if and only if λ is at least the critical exponent τ1 ≈ 1.14. A set is k-primitive if no member divides any product of up to k other distinct members. One may similarly consider the critical exponent τk for which the primes are maximal among k-primitive sets. In recent work the authors showed that τ2 < 0.8, which directly implies the Erdős conjecture for 2-primitive sets. In this article we study the limiting behavior of the critical exponent, proving that τk tends to zero as k → ∞.