论文标题
在功能字段中的ERDőS原始集合上
On the Erdős primitive set conjecture in function fields
论文作者
论文摘要
erdős证明了$ \ Mathcal {f}（a）：= \ sum_ {a \ in a} \ frac {1} {1} {a \ log a} $收敛于任何原始的整数$ a $ a $ a $ a $，后来猜测，当$ a $ a $ a $ a a $是普莱时的总和时，该总和被最大化。银行和马丁进一步指出，$ \ Mathcal {f}（\ Mathcal {p} _1）> \ ldots> \ ldots> \ Mathcal {f}（\ Mathcal {p} _k）> \ Mathcal> \ Mathcal {f} $ \ MATHCAL {P} _J $是一组具有$ j $ prime因子计算多重性的整数，尽管这最近被利奇曼（Lichtman）拒绝。我们考虑函数字段$ \ mathbb {f} _q [x] $上的相应问题，调查总和$ \ mathcal {f}（a）：= \ sum_ {f \ in} \ frac {1} {1} {\ frac {1} {\ text {deg} {deg} f \ cdot q^f \ cdot q^{\ ctext}我们在所有原始的多项式集合$ a \ subset \ mathbb {f} _q [x] $上建立了$ \ mathcal {f}（a）$的统一限制，并猜测它被一组Monic norric norric norric -norricble -dymials所最大化。我们发现，银行 - 马丁的模拟猜想是错误的，$ q = 2、3 $和$ 4 $，但我们发现其计算证据以$ q>> 4 $所示。
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 $q > 4$.
