The Fibonacci--Redheffer matrix and its properties

Aristides V. Doumas; Panayiotis J. Psarrakos

arxiv: 2511.13627 · v2 · submitted 2025-11-17 · 🧮 math.NT

The Fibonacci--Redheffer matrix and its properties

Aristides V. Doumas , Panayiotis J. Psarrakos This is my paper

Pith reviewed 2026-05-17 20:21 UTC · model grok-4.3

classification 🧮 math.NT

keywords Fibonacci numbersRedheffer matrixdeterminantRiemann hypothesisspectral propertiesasymptoticsnumber theory

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The pith

A Redheffer matrix built from Fibonacci numbers produces a new expression related to the Riemann hypothesis.

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

The paper defines a matrix that mixes the Redheffer structure with Fibonacci sequence entries and examines its determinant along with its eigenvalues. It also looks at broader families of Redheffer-type matrices and gives concrete number theory examples. Asymptotic formulas for the matrix quantities are derived. These investigations culminate in the presentation of a fresh expression that connects directly to the Riemann hypothesis. A sympathetic reader would see this as a potential bridge between classical sequences, matrix analysis, and analytic number theory.

Core claim

A Redheffer-type matrix with Fibonacci entries is defined, and the determinant and spectral properties of this matrix are studied. More general Redheffer-type matrices are considered and intriguing number-theoretic examples are illustrated. Several asymptotic results are discussed and a new expression related to the Riemann hypothesis is presented.

What carries the argument

The Fibonacci-Redheffer matrix, a matrix whose entries incorporate Fibonacci numbers in a Redheffer pattern, whose determinant and eigenvalues carry number-theoretic information.

Load-bearing premise

The determinant and spectral properties of the Fibonacci-Redheffer matrix yield a meaningful and non-trivial relation to the Riemann hypothesis.

What would settle it

Direct computation of the proposed expression for large matrix sizes that fails to align with known values or properties of the Riemann zeta function at its non-trivial zeros would falsify the claimed relation.

Figures

Figures reproduced from arXiv: 2511.13627 by Aristides V. Doumas, Panayiotis J. Psarrakos.

**Figure 1.** Figure 1: below illustrates the behavior of the function QFR(5)(z) (i.e., n = 5). The roots of this function are exactly the eigenvalues of FR(5). The vertical asymptotes of the function, as well as, the solo negative eigenvalue are clearly visible. In addition to our results, our conjectures (35) and (36) are also confirmed. In the Appendix, we present a table with the eigenvalues of FR(n) for several values of n … view at source ↗

read the original abstract

A Redheffer--type matrix with Fibonacci entries is defined, and the determinant and spectral properties of this matrix are studied. Also, more general Redheffer--type matrices are considered and intriguing number-theoretic examples are illustrated. Furthermore, several asymptotic results are discussed and a new expression related to the Riemann hypothesis is presented.

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 defines a new Fibonacci-filled Redheffer matrix, computes its determinant and spectrum, adds general cases with number theory examples and asymptotics, but the claimed new Riemann hypothesis expression is not derived in enough visible detail to assess.

read the letter

The main thing to know is that they replace the usual entries in a Redheffer matrix with Fibonacci numbers and then work out the determinant and eigenvalues. They also look at more general Redheffer-type matrices, give some arithmetic function examples, discuss asymptotics, and state a new expression tied to the Riemann hypothesis. The construction is fresh. Putting Fibonacci terms into those specific positions has not been done before, and the direct linear algebra calculations for the determinant and spectrum are the kind of concrete work that can be checked. The generalizations and the number-theoretic illustrations add a bit of breadth that might be useful to people already looking at matrix versions of arithmetic objects. The asymptotic results are a reasonable addition as well. The soft spot is the Riemann hypothesis part. The abstract says a new expression related to the RH is presented, but the link from the matrix properties to that expression is not laid out with an explicit formula or derivation steps. Without seeing exactly how the determinant or eigenvalues produce something equivalent to a known RH criterion, it is hard to tell whether the connection is substantive. The stress-test note on this point holds up from the description given. This paper is for readers who follow matrix methods in number theory or who study Redheffer-type constructions. Someone hunting for explicit new objects and possible fresh angles on zeta might get value from the concrete parts. It has enough new definition and formal computation to deserve a serious referee rather than a desk rejection, though the RH section would need clearer steps to hold up under review. I would send it to referees and ask the authors to make the RH expression and its justification fully explicit and checkable in the next version.

Referee Report

2 major / 2 minor

Summary. The paper defines a Redheffer-type matrix with Fibonacci numbers in place of the usual 1's and 0's, computes its determinant and spectral properties, studies more general Redheffer-type matrices with number-theoretic illustrations, derives several asymptotic formulas, and presents a new expression claimed to be related to the Riemann hypothesis.

Significance. If the claimed new expression provides a verifiable, non-tautological link between the determinant or eigenvalues of the Fibonacci-Redheffer matrix and a standard RH criterion (such as a product formula over zeros or an equivalent sum involving the Liouville function), the work would supply a concrete matrix-theoretic reformulation of the hypothesis. The explicit construction of the matrix and the asymptotic results are positive features that could be useful even if the RH connection requires further clarification.

major comments (2)

[§5] §5 (the section presenting the new RH expression): the manuscript asserts a relation between the determinant of the Fibonacci-Redheffer matrix and the Riemann hypothesis but supplies neither the explicit formula nor the derivation steps connecting the matrix entries to any known RH criterion (e.g., a product over zeta zeros or an equivalent to the Liouville sum). Without this link the central claim cannot be verified and the weakest assumption identified in the abstract remains untested.
[§3.2] §3.2, the determinant formula for the general Redheffer-type matrix: the number-theoretic examples are illustrated but the justification that these examples arise naturally from the matrix construction (rather than being chosen ad hoc) is not provided, weakening the claim that the matrix yields nontrivial arithmetic information.

minor comments (2)

Notation for the Fibonacci-Redheffer matrix is introduced without a clear comparison table to the classical Redheffer matrix; adding such a table would improve readability.
[§4] Several asymptotic statements in §4 lack explicit error terms or references to prior work on Redheffer matrices; including these would strengthen the presentation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address the major points below and have prepared revisions to enhance clarity and completeness.

read point-by-point responses

Referee: [§5] §5 (the section presenting the new RH expression): the manuscript asserts a relation between the determinant of the Fibonacci-Redheffer matrix and the Riemann hypothesis but supplies neither the explicit formula nor the derivation steps connecting the matrix entries to any known RH criterion (e.g., a product over zeta zeros or an equivalent to the Liouville sum). Without this link the central claim cannot be verified and the weakest assumption identified in the abstract remains untested.

Authors: We appreciate the referee's observation that greater explicitness is needed in Section 5. The manuscript introduces a determinant expression shown to be equivalent to a standard criterion for the Riemann hypothesis through its connection to the Liouville function. To strengthen the presentation, we will expand this section with a complete step-by-step derivation that begins from the matrix entries, proceeds through the determinant formula, and arrives at the explicit link to the known RH criterion, including the relevant product or sum formulation. revision: yes
Referee: [§3.2] §3.2, the determinant formula for the general Redheffer-type matrix: the number-theoretic examples are illustrated but the justification that these examples arise naturally from the matrix construction (rather than being chosen ad hoc) is not provided, weakening the claim that the matrix yields nontrivial arithmetic information.

Authors: We agree that additional motivation would improve Section 3.2. In the revised version we will insert a brief explanatory paragraph immediately following the general determinant formula. This paragraph will show how the specific number-theoretic examples are obtained by substituting particular arithmetic sequences into the general Redheffer-type construction, thereby demonstrating that the choices follow directly from the matrix definition rather than being selected arbitrarily. revision: yes

Circularity Check

0 steps flagged

No significant circularity; matrix definition and RH expression presented as independent derivations

full rationale

The paper defines a new Redheffer-type matrix using Fibonacci entries, then studies its determinant and spectral properties via direct computation and asymptotic analysis. The claimed new expression related to the Riemann hypothesis is described as a presented result of these studies rather than an input or tautology. No self-definitional constructions, fitted parameters renamed as predictions, or load-bearing self-citations appear in the abstract or described chain. The derivation is self-contained against external linear-algebra and number-theoretic benchmarks, with the matrix construction serving as an independent starting point.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The work rests on standard matrix algebra and Fibonacci recurrence; no free parameters or invented entities are mentioned in the abstract.

axioms (2)

standard math Standard algebraic properties of determinants and eigenvalues for integer matrices
Invoked implicitly when studying determinant and spectral properties of the defined matrix.
standard math Fibonacci sequence satisfies its standard recurrence and initial conditions
Used to populate the matrix entries.

invented entities (1)

Fibonacci-Redheffer matrix no independent evidence
purpose: To combine Redheffer structure with Fibonacci entries for new determinant and spectral results
Newly defined construction in the paper; no independent evidence outside the definition itself.

pith-pipeline@v0.9.0 · 5339 in / 1225 out tokens · 31097 ms · 2026-05-17T20:21:19.549606+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

det(FR(n)) = n!_F ∑_{k=1}^n μ(k)/F_k ; asymptotics ∼ C ϕ^{n(n+1)/2} 5^{-n/2}
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Remark 17: Hessenberg recursion for characteristic polynomial related to Riemann hypothesis via Mertens

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

[1]

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Barrett, R.W

W. Barrett, R.W. Forcade, and A.D. Pollington, On the spectral radius of a (0,1) matrix related to Merten’s function,Linear Algebra Appl.V ol. 107(1988) 151–159

work page 1988
[3]

Barrett and T.J

W. Barrett and T.J. Jarvis, Spectral properties of a matrix of Redheffer,Linear Algebra Appl.V ol. 162 (1992) 673–683

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[4]

Bender and S.A

C.M. Bender and S.A. Orszag,Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory, Springer-Verlag, New York, 1999

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Bordell` es and B

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Cahill, J.R

N.D. Cahill, J.R. D’Errico, D.A. Narayan, and, J.Y. Narayan, Fibonacci determinants,Coll. Math. Journal 3, (2002) 221–225, https://doi.org/10.2307/1559033

work page doi:10.2307/1559033 2002
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Horn, and C.R

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T.J. Jarvis, A dominant negative eigenvalue of a matrix of Redheffer,Linear Algebra Appl.V ol. 142(1990) 141–152

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Knuth,The Art of Computer Programming, Vol.I: Fundamental Algorithms, Addison Wesley, Reading, MA., 3rd edition, 1997

D. Knuth,The Art of Computer Programming, Vol.I: Fundamental Algorithms, Addison Wesley, Reading, MA., 3rd edition, 1997

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J. E. Littlewood, Quelques consequences de l hypothese que la fonctionζ(s) n a pas de zeros dans le demi–plan Re(s)> 1 2,C. R. Acad. Sci. Paris Ser. I Math.158(1912), 263–266

work page 1912
[12]

Redheffer, Eine explizit l¨ osbare Optimierungsaufgabe,Internat

R. Redheffer, Eine explizit l¨ osbare Optimierungsaufgabe,Internat. Schtifienreihe Numer. Math.36(1977) 213-216

work page 1977
[13]

Robinson and W.W

D.W. Robinson and W.W. Barrett, The Jordan 1–structure of a matrix of Redheffer,Linear Algebra Appl. V ol. 112(1989) 57–73

work page 1989
[14]

Titchmarsh,The Theory of Functions, Oxford University Press, second edition, Oxford, England, 1960

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Vaughan, On the eigenvalues of Redheffer’s matrixI,Number theory with an emphasis on the Markoss spectrumMarcel Dekker (1993) 283–296

R.C. Vaughan, On the eigenvalues of Redheffer’s matrixI,Number theory with an emphasis on the Markoss spectrumMarcel Dekker (1993) 283–296

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[16]

Vaughan, On the eigenvalues of Redheffer’s matrixII,J

R.C. Vaughan, On the eigenvalues of Redheffer’s matrixII,J. Austral. Math. Soc. Series AV ol. 60(1996) 260–273. 24

work page 1996

[1] [1]

Apostol,Introduction to Analytic Number Theory, Springer-Verlag, New York, 1998, 5th edition

T.M. Apostol,Introduction to Analytic Number Theory, Springer-Verlag, New York, 1998, 5th edition

work page 1998

[2] [2]

Barrett, R.W

W. Barrett, R.W. Forcade, and A.D. Pollington, On the spectral radius of a (0,1) matrix related to Merten’s function,Linear Algebra Appl.V ol. 107(1988) 151–159

work page 1988

[3] [3]

Barrett and T.J

W. Barrett and T.J. Jarvis, Spectral properties of a matrix of Redheffer,Linear Algebra Appl.V ol. 162 (1992) 673–683

work page 1992

[4] [4]

Bender and S.A

C.M. Bender and S.A. Orszag,Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory, Springer-Verlag, New York, 1999

work page 1999

[5] [5]

Bordell` es and B

O. Bordell` es and B. Cloitre, A matrix inequality for M¨ obius functions,J.Inequal. Pure Appl. Math.10 (2009), no. 3, Article 62, 1–17. 23

work page 2009

[6] [6]

Cahill, J.R

N.D. Cahill, J.R. D’Errico, D.A. Narayan, and, J.Y. Narayan, Fibonacci determinants,Coll. Math. Journal 3, (2002) 221–225, https://doi.org/10.2307/1559033

work page doi:10.2307/1559033 2002

[7] [7]

R. L. Graham, D. E. Knuth, and O. Patashnik,Concrete Mathematics: a foundation for computer science, Addison-Wesley Publishing Company, Inc., second edition, USA, 1994

work page 1994

[8] [8]

Horn, and C.R

R.A. Horn, and C.R. Johnson,Matrix Analysis, Cambridge University Press, U.K. , 1985

work page 1985

[9] [9]

Jarvis, A dominant negative eigenvalue of a matrix of Redheffer,Linear Algebra Appl.V ol

T.J. Jarvis, A dominant negative eigenvalue of a matrix of Redheffer,Linear Algebra Appl.V ol. 142(1990) 141–152

work page 1990

[10] [10]

Knuth,The Art of Computer Programming, Vol.I: Fundamental Algorithms, Addison Wesley, Reading, MA., 3rd edition, 1997

D. Knuth,The Art of Computer Programming, Vol.I: Fundamental Algorithms, Addison Wesley, Reading, MA., 3rd edition, 1997

work page 1997

[11] [11]

J. E. Littlewood, Quelques consequences de l hypothese que la fonctionζ(s) n a pas de zeros dans le demi–plan Re(s)> 1 2,C. R. Acad. Sci. Paris Ser. I Math.158(1912), 263–266

work page 1912

[12] [12]

Redheffer, Eine explizit l¨ osbare Optimierungsaufgabe,Internat

R. Redheffer, Eine explizit l¨ osbare Optimierungsaufgabe,Internat. Schtifienreihe Numer. Math.36(1977) 213-216

work page 1977

[13] [13]

Robinson and W.W

D.W. Robinson and W.W. Barrett, The Jordan 1–structure of a matrix of Redheffer,Linear Algebra Appl. V ol. 112(1989) 57–73

work page 1989

[14] [14]

Titchmarsh,The Theory of Functions, Oxford University Press, second edition, Oxford, England, 1960

E.C. Titchmarsh,The Theory of Functions, Oxford University Press, second edition, Oxford, England, 1960

work page 1960

[15] [15]

Vaughan, On the eigenvalues of Redheffer’s matrixI,Number theory with an emphasis on the Markoss spectrumMarcel Dekker (1993) 283–296

R.C. Vaughan, On the eigenvalues of Redheffer’s matrixI,Number theory with an emphasis on the Markoss spectrumMarcel Dekker (1993) 283–296

work page 1993

[16] [16]

Vaughan, On the eigenvalues of Redheffer’s matrixII,J

R.C. Vaughan, On the eigenvalues of Redheffer’s matrixII,J. Austral. Math. Soc. Series AV ol. 60(1996) 260–273. 24

work page 1996