On the critical exponent for $k$-primitive sets

Abstract

A set of positive integers is primitive (or 1-primitive) if no member divides another. Erdős proved in 1935 that the weighted sum $\sum 1/(n\log n)$ 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 $\sum n^{-\lambda }$ over a primitive set is maximized by the primes if and only if $\lambda$ is at least the critical exponent ${\tau }_1\approx 1.14$. 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 ${\tau }_k$ for which the primes are maximal among $k$-primitive sets. In recent work the authors showed that ${\tau }_2<0.8$, which directly implies the Erdős conjecture for 2-primitive sets. In this article we study the limiting behavior of the critical exponent, proving that ${\tau }_k$ tends to zero as $k\to \infty$.

14 pages

References in corpus (1)

• Almost primes and the Banks-Martin conjecture

Cited by in corpus (1)

• A proof of the Erdős primitive set conjecture