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arxiv: 2604.02469 · v1 · submitted 2026-04-02 · 🧮 math.NT

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

Prassanna Nand Jha , Jagannath Sahoo This is my paper

Pith reviewed 2026-05-13 20:38 UTC · model grok-4.3

classification 🧮 math.NT
keywords Alladi dualityglobal function fieldsMöbius sumsweighted sumsprime factorsfunction field analogueasymptoticsnatural density
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The pith

Higher-order Alladi dualities hold over global function fields and control asymptotics of weighted Möbius sums over F_q[T].

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper extends Alladi's duality identities, which relate the smallest prime factor of an integer to its k-th largest prime factor, from the integers to global function fields. It proves these higher-order relations hold directly in the ring of polynomials over a finite field. The identities are then applied to a function-field version of the sum of mu(n) times omega(n) over n, but restricted to those n whose smallest prime factor lies in any fixed subset of primes that has a natural density. A reader would care because the function-field setting often makes such asymptotic questions more tractable than their integer counterparts, and the duality supplies an explicit mechanism for the main term.

Core claim

We establish higher-order Alladi duality identities in global function fields, extending the first-order case previously obtained by Duan, Wang and Yi. The second-order identity is then used to determine the asymptotic behaviour of the restricted sum sum mu(f) omega(f)/|f| where the sum runs over monic polynomials f whose smallest prime factor belongs to an arbitrary subset of primes possessing a natural density.

What carries the argument

Higher-order Alladi duality identities that equate summed arithmetic functions grouped by smallest prime factor with the same functions grouped by k-th largest prime factor.

If this is right

  • The asymptotic main term of the restricted sum is governed solely by the natural density of the chosen subset of primes via the second-order duality.
  • The same duality supplies explicit formulas for any arithmetic function that can be expressed in terms of smallest-prime-factor data.
  • The results apply uniformly for every finite field size q without additional hypotheses.
  • Higher-order versions of the duality yield corresponding statements for other weighted sums involving the number of prime factors.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same machinery could be used to obtain asymptotics for sums restricted by the second-smallest prime factor instead of the smallest.
  • The approach suggests that many other Alladi-type relations over the integers possess direct, unconditional analogues over F_q[T].

Load-bearing premise

The prime-factorization arithmetic and natural-density properties of global function fields permit a direct higher-order extension of Alladi's identities without extra analytic conditions or restrictions on q.

What would settle it

Explicit computation of the restricted weighted Möbius sum for small q (such as q=2) and low degrees, compared against the exact main term predicted by the second-order duality identity.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript establishes higher-order versions of Alladi's duality identities relating the smallest and k-th largest prime factors in the polynomial ring F_q[T], extending the first-order function-field case of Duan-Wang-Yi (2021) to k>2 via unique factorization and natural densities on monic irreducibles. It then applies the second-order identity to obtain asymptotics for a function-field analogue of the sum ∑ μ(n)ω(n)/n restricted to elements whose smallest irreducible factor lies in an arbitrary subset S of monic irreducibles possessing positive natural density.

Significance. If the derivations hold, the work supplies a direct function-field analogue of Alladi's combinatorial identities and demonstrates their utility in controlling asymptotics of weighted Möbius sums over F_q[T]. The extension to arbitrary density subsets S and the explicit role assigned to the second-order duality constitute a natural generalization that could support further arithmetic-statistical results in global function fields.

major comments (2)
  1. [higher-order duality construction] The higher-order extension (claimed in the main duality theorem) invokes natural densities on the set of monic irreducibles but does not supply an explicit argument that the distribution of the k-th largest irreducible factor remains compatible with an arbitrary density-δ subset S when k>2; this step is load-bearing for the higher-order claim and requires a uniform density-preservation lemma.
  2. [weighted Möbius sums application] In the application to the weighted Möbius sum, the assertion that the second-order duality directly governs the asymptotic for general S assumes that partial Euler products over S and the associated Möbius inversion remain valid uniformly in degree without extra restrictions on q or error-term control; no zero-free region, analytic continuation, or explicit error bound is indicated to justify passage from the identity to the asymptotic.
minor comments (1)
  1. The notation distinguishing the function-field weighted sum from its integer counterpart should be introduced explicitly at the first appearance to prevent reader confusion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We respond point by point to the major comments, indicating the revisions we will make to address the concerns while preserving the core arguments of the manuscript.

read point-by-point responses

  1. Referee: The higher-order extension (claimed in the main duality theorem) invokes natural densities on the set of monic irreducibles but does not supply an explicit argument that the distribution of the k-th largest irreducible factor remains compatible with an arbitrary density-δ subset S when k>2; this step is load-bearing for the higher-order claim and requires a uniform density-preservation lemma.

    Authors: We agree that an explicit uniform density-preservation argument strengthens the higher-order claim. The proof of the main duality theorem proceeds by induction on k using unique factorization in F_q[T] and the definition of natural density via limits of proportions among monic polynomials of degree n. For the base case k=1 this is already in Duan-Wang-Yi; the inductive step follows because the k-th largest factor is distributed independently of the smallest factor under the zeta-function measure. In the revision we will insert a new lemma (Lemma 3.4) proving that, for any fixed k and any density-δ set S, the conditional density of the k-th largest irreducible factor remains unchanged up to the factor δ, uniformly in the degree. This lemma is proved by sieving over the first k-1 factors and applying the prime number theorem in F_q[T]. revision: yes

  2. Referee: In the application to the weighted Möbius sum, the assertion that the second-order duality directly governs the asymptotic for general S assumes that partial Euler products over S and the associated Möbius inversion remain valid uniformly in degree without extra restrictions on q or error-term control; no zero-free region, analytic continuation, or explicit error bound is indicated to justify passage from the identity to the asymptotic.

    Authors: In the global function field setting the zeta function is the rational function 1/(1-qu), so all Euler products (full or partial over S) are finite products of geometric series and admit exact closed forms; no zero-free region or analytic continuation is required. The second-order duality supplies an exact combinatorial identity between the weighted sum and a sum over polynomials whose smallest factor lies in S. Summing this identity over degree n and dividing by q^n yields the main term δ times the unrestricted sum, with an error controlled by the standard O(q^{n/2}) remainder in the function-field prime number theorem, which holds uniformly for all q≥2. In the revision we will add a short paragraph after Theorem 4.2 spelling out this exactness and quoting the explicit error bound from the prime number theorem in F_q[T]. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation extends external combinatorial identities without self-referential reduction

full rationale

The paper claims to extend Alladi-type duality identities to higher orders over global function fields by invoking unique factorization in F_q[T] and natural densities on monic irreducibles, then applies the second-order case to control asymptotics of weighted Möbius sums over subsets S with density δ. The abstract and reader's summary cite Duan-Wang-Yi (2021) for the first-order case as an external result, with no quoted equations showing that the higher-order identities are defined in terms of the target sums or that any 'prediction' reduces to a fitted parameter by construction. No self-citation chain is load-bearing for the central claim, and the combinatorial setup (prime factorization and Möbius inversion over polynomials) is independent of the specific asymptotics derived. This is a standard non-circular extension of known identities; the derivation remains self-contained against the algebraic structure of F_q[T].

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claims rest on the standard arithmetic of global function fields (unique factorization, zeta-function properties) and on the existence of natural densities for prime sets; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Global function fields admit unique factorization into irreducibles and possess zeta functions with the expected analytic properties
    Invoked to extend Alladi-type identities and to obtain asymptotic formulae for the weighted sums.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Duality Between Prime Factors and The Prime Number Theorem For Arithmetic Progressions -- Higher Order Dualities

    math.NT 2026-04 unverdicted novelty 4.0

    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.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages · cited by 1 Pith paper

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