论文标题
在关键指数上的$ k $ promitive套装
On the critical exponent for $k$-primitive sets
论文作者
论文摘要
如果没有成员划分另一个成员，则一组正整数是原始的（或1个主要）。 Erdő在1935年证明了加权总和$ \ sum1/（n \ log n）$ n $在原始集合上$ a $ n $ n $ n $ a $在$ a $的所有选择上普遍限制。 1988年，他询问该通用界限是否通过质数集获得。在此猜想中的难度之一是，当原始集合上的$ \ sum n^{ - λ} $被素数最大化时，并且仅当$λ$至少是关键指数$τ_1\大约1.14 $。 如果没有成员将最多$ k $其他不同成员的任何产品划分，则一套为$ k $ timimitive。人们可以同样考虑关键指数$τ_k$，在$ k $ promimimive的集合中，素数是最大的。在最近的工作中，作者表明，$τ_2<0.8 $，这直接暗示了2个主要集合的Erdő猜想。在本文中，我们研究了关键指数的限制行为，证明$τ_k$倾向于零为$ k \ to \ infty $。
A set of positive integers is primitive (or 1-primitive) if no member divides another. Erdős 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 $\sum n^{-λ}$ over a primitive set is maximized by the primes if and only if $λ$ is at least the critical exponent $τ_1 \approx 1.14$. 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 $τ_k$ for which the primes are maximal among $k$-primitive sets. In recent work the authors showed that $τ_2 < 0.8$, which directly implies the Erdős conjecture for 2-primitive sets. In this article we study the limiting behavior of the critical exponent, proving that $τ_k$ tends to zero as $k\to\infty$.
