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

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 Möbius 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.

28 pages, submitted for publication, comments are welcome