Extremal eigenvalues and singular values of sparse matrices related to the prime number theorem

Status: Version 0.1.0, admitted August 2, 2026. The immutable release is tagged v0.1.0 and permanently archived at doi:10.5281/zenodo.21764114. The release passed the project's public research standard and is listed on Experimental Mathematics.

This repository studies two sparse matrices introduced by Kline in 2020. Their entries come from divisibility and the harmonic weight 1/k; changing the first row makes their determinant

m(n) = sum_{k <= n} mu(k)/k,

where mu is the Möbius function. The 2020 paper asked whether the eigenvalue nearest zero is asymptotic to this determinant, and whether its eigenvector is asymptotic to the Möbius vector.

The paper here settles both questions:

Main result. For each of the two matrices, the eigenvalue nearest zero is real, simple, and equal to m(n)(1 + O(exp(-c sqrt(log n)))). The proposed eigenvector estimate is false, already in its second coordinate; a two-term expansion gives the correct replacement.

The work then studies the larger family obtained by replacing 1/k with k^(-s). It identifies a phase change at Re(s) = 1, determines the extreme singular values and their vectors for 1/2 < Re(s) <= 1, and shows exactly where the method records classical prime-number-theorem information without providing new leverage on that theorem.

The authoritative source is the paper (55 pages); paper/extremal-eigenvalues.tex is its source. The files in proofs/ give structured derivations, and the scripts in code/ check exact identities and representative numerical cases.

The matrices

Let A(f) be the lower-triangular divisor matrix associated with an arithmetic function f. For h(k) = 1/k, write H = A(h). Thus

H(i,j) = j/i    if j divides i,
         0      otherwise.

The 2020 paper replaces the first row of H_n in two ways:

• Hbar_n has first row (1,1,...,1);
• Htilde_n has first row (|mu(1)|,...,|mu(n)|).

Both determinants equal m(n). The two source conjectures were:

1. C-EIG: the eigenvalue of smallest modulus is asymptotic to the determinant;
2. C-VEC: after normalizing the corresponding eigenvector by v_n(1)=1, one has v_n(i) = mu(i)/i + o(1/n) uniformly for 1 <= i <= n.

Results

The two conjectures

The paper proves C-EIG for both matrices, with an explicit asymptotic rate. For all sufficiently large n, each matrix has one eigenvalue in a shrinking disk about zero. That eigenvalue is real and simple, and

lambda_n = m(n) (1 + O(exp(-c sqrt(log n)))).

C-VEC is false. At i = 2, its error has order |m(n)|, and no estimate O(n^(-theta)) can hold for any theta > 1/2. The corrected expansion is

j v_n(j) = mu(j)
           + lambda_n ((mu * mu)(j) - mu(j))
           + O(lambda_n^2 (log_2(2j))^2 A(j)),

where A(j) counts ordered factorizations of j into factors at least two.

The interpolating family

The matrices sit in the family Hbar_n^(sigma) = Abar(k^(-sigma))_n, which joins the classical Redheffer matrix at sigma = 0 to Hbar_n at sigma = 1. For each fixed 0 <= sigma < 1, two eigenvalues dominate and the product of the remaining nontrivial eigenvalues satisfies

Pi_small^(sigma)
  = -(1-sigma) m_sigma(n) / n^(1-sigma) (1 + o(1)).

The statement Pi_small^(sigma) -> 0 is equivalent to the prime number theorem. This is a new matrix formulation of a classical criterion, not a new proof of the prime number theorem. At sigma = 1, the behavior changes: a single eigenvalue, rather than a product, is asymptotic to the determinant.

Singular values and vectors

For complex s with 1/2 < Re(s) <= 1, the paper determines both extremes:

sigma_max(Hbar_n^(s)) ~ sqrt(n),
sigma_min(Hbar_n^(s)) ~ |m_s(n)| / sqrt(B_Re(s) K_s n).

It also identifies the associated singular vectors. The right vector at the small end approaches a weighted Möbius vector; the left vector approaches a weighted Mertens-sum profile. At s = 1, the smallest singular value is smaller than the smallest eigenvalue by a factor of order sqrt(n), which quantifies the matrices' non-normality. The condition number gives another equivalent form of the prime number theorem.

For every fixed nonprincipal Dirichlet character chi, the same argument applies to the weight chi(k)/k and gives

kappa(Hbar_n^(chi)) / n -> |L(1,chi)| sqrt(B_chi K_chi).

Quadratic characters of odd prime conductor provide an infinite family. The claim is for each fixed character; it is not uniform in the conductor.

What the method cannot see

The paper records two boundaries.

• The eigenvalue argument uses the resolvent at lambda = 0. For fixed lambda outside {0,1}, the relevant partial sums diverge unconditionally, so the method does not produce fixed-lambda limit laws.
• Twisting the unmodified Möbius divisor matrix by the Liouville function is a diagonal unitary similarity. Its singular values therefore depend on the squarefree indicator mu^2 and Re(s), not on the signs of mu. The all-ones row is the part of the modified matrix that retains Möbius cancellation.

Sign-blindness does not imply that every function of mu^2 is weaker than the prime number theorem or the Riemann hypothesis. The paper makes no such claim.

Evidence and limits

The repository separates four kinds of support:

• Proof. The paper proves the eigenvalue, eigenvector, interpolating-family, singular-value, character-transfer, and obstruction results. Several analytic estimates are quoted from the cited literature.
• Exact computation. Scripts check determinant, inverse, characteristic- polynomial, kernel, and finite matrix identities in exact arithmetic where possible.
• Numerical computation. The probes test representative finite cases and compare their output with tracked references. These checks support the formulas but cannot verify asymptotic behavior at unreachable values of n.
• Literature record. audit/ledger.md records prior-work comparisons and the limits of the search. Failure to locate an earlier result is not proof of global novelty.

Important qualifications:

• This is not an attack on the prime number theorem or the Riemann hypothesis. The equivalences encode classical criteria.
• The unconditional range 1/2 < Re(s) <= 1 for the smallest singular value uses a section-norm theorem of Hilberdink.
• The character result gives new spectral information for a supplied family but no new matrix dimensions, and its priority remains bounded by the literature search recorded in the audit.
• The Liouville-twist invariance is classical. The claimed contribution is the finite-dimensional obstruction and its consequences in this setting.
• Process-separated AI checks are useful for finding errors, but they are not peer review or independent expert validation.

Reproduce the numerical checks

The main probes require Python 3 and NumPy:

python3 -m venv .venv
.venv/bin/pip install -r requirements.txt
.venv/bin/python code/run_all.py

The harness runs eleven probes, compares their output with the tracked files in code/out/, restores those files, and exits with status 0 only when every probe passes. On macOS and Linux it also prevents two harness runs from writing the reference files at once. See code/REPRODUCE.md for the scope of each probe and audit/reports/reproduction.md for the release-candidate reproduction record.

To build the paper:

cd paper
pdflatex extremal-eigenvalues.tex
pdflatex extremal-eigenvalues.tex

The bibliography is inline, so no BibTeX step is needed.

Repository map

• paper/extremal-eigenvalues.tex and paper/extremal-eigenvalues.pdf — the paper and reading copy.
• proofs/ — structured proof sources for E1, E2, E3, E5, E7, E8, and E12.
• code/ — eleven numerical and exact-identity probes, the harness, and tracked reference outputs.
• audit/ledger.md — statement-level audit history, including repaired and withdrawn claims.
• audit/reports/ and audit/code/ — earlier read-only reports and separately written check scripts.
• AUDIT.md — the consolidated verification summary.
• ADMISSION.md, VERIFICATION.md, and CORRECTIONS.md — release gate, reproduction, and stewardship records.

AI assistance and responsibility

Large language models substantially assisted with the mathematics, code, literature search, exposition, and adversarial checks. Their agreement is evidence about the checking process, not a certificate of correctness. Jeffery Kline directs the work, is responsible for the claims released under his name, and will record material corrections or withdrawals in the public history.

Citation

For reproducible citation, cite the archived version:

@software{kline2026extremal,
  author  = {Kline, Jeffery},
  title   = {Extremal eigenvalues and singular values of sparse matrices
             related to the prime number theorem},
  version = {0.1.0},
  year    = {2026},
  doi     = {10.5281/zenodo.21764114},
  url     = {https://doi.org/10.5281/zenodo.21764114}
}

The concept DOI 10.5281/zenodo.21764113 resolves to the latest archived version. Use the version DOI above when exact reproducibility matters.

The source conjectures appear in:

J. Kline, “On the eigenstructure of sparse matrices related to the prime number theorem,” Linear Algebra and its Applications 584 (2020), 409–430.

License

Copyright (C) 2026 Jeffery Kline. Released under the GNU General Public License, version 3 (GPL-3.0); see LICENSE. The software is provided without warranty; see the license for details.