Jha, Prassanna NandPrassanna NandJhaSahoo, JagannathJagannathSahoo2026-04-162026-04-162026-04-012331-842210.48550/arXiv.2604.02469https://repository.iitgn.ac.in/handle/IITG2025/35013Alladi's duality identities (1977) provide a fundamental relation between the smallest and the -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 , 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 Möbius sums in the function field setting.en-USAlladi�s duality identityMobius functionPrime number theoremFunction fieldse-Print