Analogues of Alladi's formula over global function fields

TL;DR

This paper establishes an analogue of Alladi's formula for global function fields, linking prime divisor densities with M"obius function sums, and extends classical results like Chebotarev's theorem and the Prime Polynomial Theorem to this setting.

Contribution

It introduces a new Alladi-type formula for function fields and applies it to classical theorems, connecting divisor densities, M"obius functions, and modular forms.

Findings

Established an Alladi analogue for global function fields.

Extended Chebotarev and Prime Polynomial Theorem results to function fields.

Connected M"obius function with Fourier coefficients of modular forms.

Abstract

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 $\delta(S)$ within prime divisors, then $$-\lim_{n\to\infty} \sum_{\substack{1\le \deg D\le n\\ D\in \mathfrak{D}(K,S)}}\frac{\mu(D)}{|D|}=\delta(S),$$ where $\mu(D)$ is the M\"{o}bius 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 M\"obius 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.