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arXiv · 2604.02469

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

Prassanna Nand Jha·Jagannath Sahoo
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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 Alladi-type duality identities over global function fields, extending a result of Duan, Wang, and Yi. We also prove the asymptotic vanishing of a function field analogue of the weighted M\"{o}bius sum $\sum \mu(n)\omega(n)/n$, where $\omega(n)$ denotes the number of distinct prime divisors of $n$. By estimating $\Psi_2(n,m)$, the number of monics of degree $n$ whose second largest prime divisor degree is at most $m$, we further show that this vanishing persists when the sum is restricted to monics with a unique prime divisor of smallest degree belonging to a set of primes with natural density. As a corollary, we show that the unrestricted sum decomposes into infinitely many sub-series, each vanishing asymptotically.

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Prassanna Nand Jha, Jagannath Sahoo. 2026-04-02. Higher Order Dualities over Global Function Fields and Weighted M\"{o}bius Sums over $\mathbb{F}_q{[T]}$. https://arxiv.org/abs/2604.02469

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