论文标题
Alladi公式在全球功能字段上的类似物
Analogues of Alladi's formula over global function fields
论文作者
论文摘要
在本文中，我们展示了Kural，McDonald和SAH对Alladi的全球功能领域公式的类似物。明确地，我们表明，对于全球函数字段$ k $，如果prime除数的集合$ s $在prime divisors内具有自然密度$δ（s）$，则$$ - \ lim_ {n \ to \ infty} \ sum _ { \ Mathfrak {d}（k，s）}}} \ frac {μ（d）} {| d |} =δ（s），$$，其中$μ（d）$是möbius在分隔师上的功能，$ \ mathfrak {d}（d}（k，k，k，s）$是所有有效的分区的组合。作为应用，我们将Dawsey's和Sweeting和Woo的结果的类似物带到了Chebotarev密度定理的功能领域，以及Alladi结果对算术进行的主要多项式定理的类似物。我们还显示了Möbius函数与与椭圆曲线相关的模块化形式的傅立叶系数之间的连接。我们主要定理的证明与Kural等人的文章中的方法相似。
In this paper, we show an analogue of Kural, McDonald and Sah's result on Alladi's formula for global function fields. Explicitly, we show that for a global function field $K$, if a set $S$ of prime divisors has a natural density $δ(S)$ within prime divisors, then $$-\lim_{n\to\infty} \sum_{\substack{1\le °D\le n\\ D\in \mathfrak{D}(K,S)}}\frac{μ(D)}{|D|}=δ(S),$$ where $μ(D)$ is the Möbius function on divisors and $\mathfrak{D}(K,S)$ is the set of all effective distinguishable divisors whose smallest prime factors are in $S$. As applications, we get the analogue of Dawsey's and Sweeting and Woo's results to the Chebotarev Density Theorem for function fields, and the analogue of Alladi's result to the Prime Polynomial Theorem for arithmetic progressions. We also display a connection between the Möbius function and the Fourier coefficients of modular form associated to elliptic curves. The proof of our main theorem is similar to the approach in Kural et al.'s article.
