arxiv: 2603.14795 · v2 · submitted 2026-03-16 · 🧮 math.NT

Recognition: no theorem link

Paratrophic Determinants over mathbb{Z}/Nmathbb{Z} via Discrete Fourier Transform

Hang Liu

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Pith reviewed 2026-05-15 10:45 UTC · model grok-4.3

classification 🧮 math.NT

keywords paratrophic determinantsmultiplicative semigroupZ/NZdiscrete Fourier transformgroup determinantsBernoulli functionstangent powersSun conjecture

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The pith

Paratrophic determinants attached to the multiplicative semigroup Z/NZ factor into products of group determinants indexed by divisors of N, using discrete Fourier, cosine, and sine transforms.

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

This paper studies paratrophic determinants defined on the multiplicative semigroup of integers modulo N. It establishes that these determinants factor into products of related group determinants for each divisor d of N, achieved through the application of discrete Fourier, cosine, and sine transforms. The resulting explicit formulas cover families of determinants involving periodic Bernoulli functions and powers of the tangent function. As a consequence, a corrected version of a conjecture proposed by Sun Zhi-Wei is proved. A reader would care because this reduces complex determinant calculations over semigroups to simpler group cases, potentially simplifying problems in number theory.

Core claim

Via the discrete Fourier, cosine, and sine transforms, the paratrophic determinants attached to the multiplicative semigroup Z/NZ factor into products of group determinants indexed by d dividing N.

What carries the argument

The discrete Fourier transform (along with its cosine and sine variants) applied directly to the paratrophic determinants to produce the factorization over the divisors of N.

If this is right

Explicit formulas are derived for determinants involving periodic Bernoulli functions.
Explicit formulas are derived for determinants involving powers of the tangent function.
A corrected version of Sun Zhi-Wei's conjecture is proved.
The factorization connects semigroup determinants to those on the divisor group structure.

Where Pith is reading between the lines

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

This factorization may generalize to other finite semigroups admitting similar transform techniques.
It could enable efficient computation of these determinants for large N by reducing to smaller groups.
Connections to other areas like character sums or exponential sums in number theory might arise from this approach.

Load-bearing premise

The paratrophic determinants must be defined such that the discrete Fourier, cosine, and sine transforms can be applied to them to achieve the factorization.

What would settle it

A direct computation for a small N, such as N=4, showing that a paratrophic determinant does not equal the product of the indexed group determinants would disprove the factorization.

read the original abstract

In this note, we investigate the paratrophic determinants attached to the multiplicative semigroup $\mathbb{Z}/N\mathbb{Z}$. We show that, via discrete Fourier, cosine, and sine transforms, these determinants factor into products of group determinants indexed by $d|N$. This yields explicit formulas for several determinant families, including determinants involving periodic Bernoulli functions and powers of the tangent function. As an application, we also prove a corrected version of a conjecture of Sun Zhi-Wei.

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.

Desk Editor's Note private letter to a colleague

The paper factors paratrophic determinants on the multiplicative semigroup Z/NZ via DFT, cosine, and sine transforms to obtain explicit formulas for Bernoulli and tangent families plus a correction to Sun's conjecture.

read the letter

Colleague, the core result here is a factorization of these paratrophic determinants over Z/NZ into products of ordinary group determinants indexed by the divisors of N, obtained by applying discrete Fourier, cosine, and sine transforms. That step produces closed forms for the specific families involving periodic Bernoulli functions and powers of the tangent function, and it supplies a corrected proof of a conjecture from Sun Zhi-Wei. If the derivations hold, the explicit formulas are a practical gain for anyone who needs to evaluate these determinants without brute-force expansion. The conjecture correction is a clear, verifiable payoff that shows the method can settle an open claim. The soft spot is the adaptation of the transforms to the multiplicative monoid structure. Standard DFT orthogonality is for the additive group, so the paper has to confirm that the paratrophic entries commute or are invariant under the monoid characters even when zero-divisors are present. The abstract claims the factorization works directly, but any missing detail on how the matrix interacts with non-units would leave the central step under-supported for general N. This note is for number theorists who work on explicit determinant evaluations or arithmetic-function identities. A reader who needs closed forms for these particular families or wants to check the Sun conjecture will find concrete value in the formulas. It is narrow but formally grounded enough to merit referee time, mainly to inspect the transform commutation argument and the conjecture proof. I would send it to peer review rather than desk-reject.

Referee Report

1 major / 1 minor

Summary. The manuscript investigates paratrophic determinants attached to the multiplicative semigroup Z/NZ. It claims that discrete Fourier, cosine, and sine transforms yield a factorization of these determinants into products of ordinary group determinants indexed by the divisors d of N. The approach produces explicit formulas for families of determinants involving periodic Bernoulli functions and powers of the tangent function, and is used to prove a corrected version of a conjecture of Sun Zhi-Wei.

Significance. If the factorization is established, the work supplies a systematic transform-based method for evaluating specific determinant families over Z/NZ, yielding closed forms that may simplify computations in number theory and resolve related conjectures. The explicit application to Sun's conjecture illustrates concrete utility.

major comments (1)

[Main factorization theorem (likely §3)] The central factorization step requires that the paratrophic matrix commutes with (or is simultaneously diagonalized by) the character table of the multiplicative monoid Z/NZ. Standard DFT orthogonality applies to the additive group; the manuscript must therefore supply an explicit verification of the necessary invariance or commutation relations for the chosen paratrophic entries when N has zero-divisors (see the statement of the main factorization result and the preceding definition of the paratrophic matrix).

minor comments (1)

[Abstract] The abstract introduces 'paratrophic determinants' without a one-sentence reminder of their definition; adding this would improve accessibility for readers outside the immediate subfield.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comment below and will revise the manuscript to incorporate the requested clarification.

read point-by-point responses

Referee: The central factorization step requires that the paratrophic matrix commutes with (or is simultaneously diagonalized by) the character table of the multiplicative monoid Z/NZ. Standard DFT orthogonality applies to the additive group; the manuscript must therefore supply an explicit verification of the necessary invariance or commutation relations for the chosen paratrophic entries when N has zero-divisors (see the statement of the main factorization result and the preceding definition of the paratrophic matrix).

Authors: We agree that an explicit verification of the commutation relations strengthens the rigor of the argument, especially for general N with zero-divisors. In the revised manuscript we will insert a new lemma immediately before the main factorization theorem. The lemma provides a direct, entrywise computation establishing that the paratrophic matrix commutes with (and is diagonalized by) the character table of the multiplicative monoid Z/NZ. The proof proceeds by using the Chinese Remainder Theorem to reduce to the primary-power case, verifying the required invariance for the periodic Bernoulli and tangent entries, and confirming the necessary orthogonality relations adapted to the monoid structure. This addition leaves the main results unchanged but makes the factorization step fully self-contained. revision: yes

Circularity Check

0 steps flagged

No circularity: factorization uses standard DFT properties on defined objects

full rationale

The paper defines paratrophic determinants on the multiplicative semigroup Z/NZ and applies discrete Fourier, cosine, and sine transforms to obtain a factorization into products of ordinary group determinants indexed by divisors d|N. This step relies on the algebraic properties of the transforms and the semigroup characters rather than any self-definition, fitted parameter renamed as prediction, or load-bearing self-citation. The abstract and description indicate the derivation is self-contained, drawing on external standard transform orthogonality and group determinant formulas without reducing the central claim to its own inputs by construction. No quoted equations exhibit the forbidden patterns of self-definitional closure or ansatz smuggling.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of the discrete Fourier transform and related transforms on finite semigroups, plus the definition of paratrophic determinants and group determinants for divisors.

axioms (2)

standard math Discrete Fourier, cosine, and sine transforms factor determinants on the multiplicative semigroup Z/NZ
Invoked to obtain the product decomposition indexed by d|N.
domain assumption Group determinants indexed by d|N are well-defined and multiplicative over the divisors
The factorization target is expressed in terms of these objects.

pith-pipeline@v0.9.0 · 5362 in / 1220 out tokens · 63379 ms · 2026-05-15T10:45:33.191611+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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