Higher Order Dualities over Global Function Fields and Weighted M\"{o}bius Sums over $\mathbb{F}_q{[T]}$

TL;DR

This paper extends Alladi's duality identities to global function fields and analyzes weighted Möbius sums over finite fields, revealing their asymptotic behavior influenced by second-order dualities.

Contribution

It generalizes duality identities to higher orders in the setting of global function fields and applies them to study asymptotics of weighted Möbius sums.

Findings

Established higher order dualities in global function fields.

Derived asymptotic formulas for weighted Möbius sums.

Connected duality identities with sum behavior over prime subsets.

Abstract

Alladi's duality identities (1977) provide a fundamental relation between the smallest and the $k$-th largest prime factors of integers. In this paper, we establish these dualities in the setting of global function fields, extending a result of Duan, Wang, and Yi (2021) to higher orders. We apply this to study a function field analogue of the sum $\sum \mu(n)\omega(n)/n$, when restricted to integers whose smallest prime factor lies in an arbitrary subset of primes possessing a natural density. These results demonstrate how the second-order duality identity governs the asymptotic behaviour of these weighted M\"{o}bius sums in the function field setting.