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

Prassanna Nand Jha, Jagannath Sahoo

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 $\sum μ(n)ω(n)/n$, 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 Möbius sums in the function field setting.

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 -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.

Paper Structure

This paper contains 9 sections, 18 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Statement of results
  3. Higher order Dualities over Global Function Fields
  4. Prerequisites
  5. Proof of Theorem \ref{['thm:duality in ff']}
  6. Proof of Theorem \ref{['thm:landau_two_FF']}
  7. Proof of Theorem \ref{['thm:mainthm AJ']}
  8. Proof of Corollary \ref{['cor:to main thm in ff']}
  9. Proof of Theorem \ref{['thm:global_duality']}

Key Result

Theorem 2.1

Let $A\in \mathbb{F}_q[T]$ be monic. If $f:\mathbb{N}\to\mathbb{C}$ is an arithmetic function with $f(0)=0$, then for any positive integer $k$, where $\Omega(A)=\#\{\deg P:\text{ prime }P\mid A\}$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (28)

  • Theorem 2.1
  • remark 1
  • Theorem 2.2
  • Theorem 2.3
  • remark 2
  • Corollary 2.4
  • Theorem 3.1
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • ...and 18 more