On the critical exponent for $k$-primitive sets

TL;DR

This paper investigates the critical exponent for k-primitive sets, showing that it approaches zero as k increases, which advances understanding of Erdős's conjecture related to primitive sets.

Contribution

The authors prove that the critical exponent for k-primitive sets tends to zero as k approaches infinity, providing new insights into the structure of primitive sets.

Findings

The critical exponent τ_k approaches zero as k increases.

For 2-primitive sets, τ_2 is less than 0.8.

The result supports the conjecture that primes maximize certain sums over primitive sets.

Abstract

A set of positive integers is primitive (or 1-primitive) if no member divides another. Erd\H{o}s proved in 1935 that the weighted sum $\sum1/(n \log n)$ 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 Erd\H{o}s 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 $k\to\infty$.