arXiv 2007.023012020-07-05math.NT

关于函数域上的Erdés原集猜想

On the Erdős primitive set conjecture in function fields

Andrés Gómez-Colunga, Charlotte Kavaler, Nathan McNew, Mirilla Zhu

备注：: 20 pages, 1 table

摘要：

Erdens证明了$\mathcal{F}(A) := \sum_{a \in A}\frac{1}{a\log a}$收敛于任何整数本原集$A$，后来证明了当$A$是素数集时，这个和是最大化的。Banks和Martin进一步证明了$\mathcal{F}(\mathcal{P}_1) > \ldots > \mathcal{F}(\mathcal{P}_k) > \mathcal{F}(\mathcal{P}_{k+1}) > \ldots$，其中$\mathcal{P}_j$是整数的集合，$j$素因子计数多重性，尽管这最近被Lichtman推翻了。我们考虑了函数域$\mathbb{F}_q[x]$上的相应问题，研究了和$\mathcal{F}(A) := \sum_{f \in A} \frac{1}{\text{deg} f \cdot q^{\text{deg} f}}$ .我们建立了$\mathcal{F}(A)$在所有本原多项式集$A \subset \mathbb{F}_q[x]$上的一致界，并猜想它是由一元不可约多项式集最大化的。我们发现，类似的班克斯-马丁猜想是假的$q = 2, 3$，和$4$，但我们发现的计算证据，它持有$q > 4$。

英文摘要：

Erdős proved that $\mathcal{F}(A) := \sum_{a \in A}\frac{1}{a\log a}$ converges for any primitive set of integers $A$ and later conjectured this sum is maximized when $A$ is the set of primes. Banks and Martin further conjectured that $\mathcal{F}(\mathcal{P}_1) > \ldots > \mathcal{F}(\mathcal{P}_k) > \mathcal{F}(\mathcal{P}_{k+1}) > \ldots$, where $\mathcal{P}_j$ is the set of integers with $j$ prime factors counting multiplicity, though this was recently disproven by Lichtman. We consider the corresponding problems over the function field $\mathbb{F}_q[x]$, investigating the sum $\mathcal{F}(A) := \sum_{f \in A} \frac{1}{\text{deg} f \cdot q^{\text{deg} f}}$. We establish a uniform bound for $\mathcal{F}(A)$ over all primitive sets of polynomials $A \subset \mathbb{F}_q[x]$ and conjecture that it is maximized by the set of monic irreducible polynomials. We find that the analogue of the Banks-Martin conjecture is false for $q = 2, 3$, and $4$, but we find computational evidence that it holds for $q > 4$.

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