Higher-order dualities yield ∑ μ(n) ω(n)^k / n = 0 for k ≥ 2 and conditional sums over smallest prime factor p1(n) ≡ j mod ℓ equal to zero or 1/φ(ℓ) for coprime j, ℓ and k ≥ 3.
Higher Order Dualities over Global Function Fields and Weighted M\"{o}bius Sums over $\mathbb{F}_q{[T]}$
1 Pith paper cite this work. Polarity classification is still indexing.
1
Pith paper citing it
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.
fields
math.NT 1years
2026 1verdicts
UNVERDICTED 1representative citing papers
citing papers explorer
-
Duality Between Prime Factors and The Prime Number Theorem For Arithmetic Progressions -- Higher Order Dualities
Higher-order dualities yield ∑ μ(n) ω(n)^k / n = 0 for k ≥ 2 and conditional sums over smallest prime factor p1(n) ≡ j mod ℓ equal to zero or 1/φ(ℓ) for coprime j, ℓ and k ≥ 3.