Counting integer matrices with square-free determinants

Alina Ostafe; Igor E. Shparlinski

arxiv: 2503.13768 · v3 · submitted 2025-03-17 · 🧮 math.NT

Counting integer matrices with square-free determinants

Alina Ostafe , Igor E. Shparlinski This is my paper

Pith reviewed 2026-05-22 23:09 UTC · model grok-4.3

classification 🧮 math.NT

keywords integer matricessquare-free determinantsasymptotic formulasMöbius functionEuler totient functioncounting problemsdeterminant distribution

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The pith

The number of n by n integer matrices with entries of size at most H and square-free determinant is given by an asymptotic formula as H tends to infinity.

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

This paper derives an asymptotic formula for the count of n×n matrices over the integers with entries bounded in absolute value by H whose determinant is square-free. The approach builds on prior estimates for the total number of such matrices by using inclusion via the Möbius function to enforce the square-free condition on the determinant. A reader would care because this gives the natural density of matrices whose determinants avoid square factors, providing insight into the arithmetic properties of random integer matrices. The same method yields an asymptotic for the sum of the Euler totient function evaluated at these determinants.

Core claim

We obtain an asymptotic formula on the number of matrices from M_n(Z; H) with square-free determinants. We also obtain an asymptotic formula for the sums of the Euler function with determinants of matrices from M_n(Z; H).

What carries the argument

Möbius inversion applied to the indicator function of square-free integers, integrated into analytic estimates for the distribution of matrix determinants.

If this is right

The proportion of matrices with square-free determinant approaches a constant greater than zero as H increases.
The sum of phi(det(A)) over all such matrices also admits a main term asymptotic.
The method extends previous counting results by incorporating arithmetic conditions on the determinant.
Similar asymptotics hold for other multiplicative functions on the determinant.

Where Pith is reading between the lines

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

This suggests that the determinants of random integer matrices are distributed similarly to random integers with respect to being square-free.
Such counts may inform the probability that random lattices have certain properties related to their volume being square-free.
Extensions could include counting matrices with determinant coprime to a fixed integer or satisfying other local conditions.

Load-bearing premise

The error terms arising from applying Möbius inversion to detect square-freeness remain smaller than the main term in the asymptotic as H goes to infinity.

What would settle it

Computing the exact count for n=2 and H=1000 then checking whether it lies within the predicted range of the asymptotic formula including its error term.

read the original abstract

We consider the set $\mathcal M_n\left(\mathbb Z; H\right)$ of $n\times n$-matrices with integer elements of size at most $H$ and obtain and asymptotic formula on the number of matrices from $\mathcal M_n\left(\mathbb Z; H\right)$ with square-free determinants. We also use our approach with some further enhancements, to obtain an asymptotic formula for the sums of the Euler function with determinants of matrices from $\mathcal M_n\left(\mathbb Z; H\right)$.

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 gives asymptotics for the count of n×n integer matrices with entries ≤H and square-free determinant, plus a version for the sum of φ(det(A)).

read the letter

The main results are an asymptotic for the number of matrices in M_n(Z; H) with square-free det, obtained via Möbius inversion on the indicator, and a parallel asymptotic for the sum of the Euler totient over those determinants. Both are presented as new applications of existing matrix-counting machinery rather than entirely new methods. The approach is direct: rewrite the square-free condition as a sum over d, isolate the d=1 term as the main contribution, and bound the rest. The totient sum gets some extra handling but follows the same pattern. This is honest incremental work in analytic number theory on arithmetic statistics of matrices. The claims look plausible on the abstract level and the methods are standard for the subfield. The soft spot is exactly the one flagged in the stress test. The error from the sum over d>1 must be o(H^{n^2}) uniformly enough for the asymptotic to survive as H→∞. Volume estimates alone do not automatically deliver this when the divisibility condition is imposed on the determinant; one needs control on the restricted counts up to a suitable cutoff in d and a tail estimate. If the paper supplies those bounds with explicit dependence on d, the result is fine. If the uniformity is only sketched, the main term could be compromised. No circularity or invented objects appear. The work is for specialists already working on counting problems for matrices or determinants in number theory. A reader who follows papers on the distribution of det(A) or similar invariants will get concrete formulas to compare against. It is worth sending to a referee who can check the error-term details in the Möbius sum, because the statements are specific and the underlying techniques are established.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to derive an asymptotic formula, as H tends to infinity, for the number of n×n integer matrices with entries bounded by H whose determinant is square-free; it also claims a companion asymptotic for the sum of Euler's totient function φ(det(A)) taken over all matrices in M_n(Z; H). The proofs are said to proceed by expressing the square-free indicator via Möbius inversion over divisors of the determinant and then applying volume or analytic estimates to the resulting restricted counting problems.

Significance. If the error terms arising from the Möbius sum can be shown to be o(H^{n^2}) uniformly, the result would give a quantitative density for square-free determinants among integer matrices of bounded height and would extend existing work on the distribution of det(A) for random matrices in M_n(Z; H). The additional formula for the totient sum would likewise supply an arithmetic-statistical statement about the average size of φ(det(A)).

major comments (2)

[§3 (Möbius inversion step) and the error-term discussion following the main theorem] The central asymptotic relies on writing 1_{square-free}(det(A)) = ∑_{d^2 | det(A)} μ(d) and showing that the contribution of the terms d > 1 is o(H^{n^2}). No explicit uniform bound is supplied for the counting function of matrices with d^2 | det(A) when d ranges up to the necessary cutoff (typically a small power of H); without such uniformity the tail of the Möbius sum cannot be guaranteed to be negligible.
[§5 (Euler totient asymptotic)] The same uniformity issue appears for the totient sum, where φ is expanded as a Dirichlet convolution; the manuscript does not verify that the resulting multiple sums over divisors remain controllable when the divisibility conditions on det(A) are imposed simultaneously with the height bound H.

minor comments (2)

[Introduction, first paragraph] The notation M_n(Z; H) is introduced without an explicit definition of the height bound (max-norm versus Euclidean norm); a single clarifying sentence would remove ambiguity.
[Introduction] Several standard references on the distribution of determinants of integer matrices (e.g., work of Duke–Rudnick–Sarnak or Eskin–McMullen) are not cited; adding them would place the new results in clearer context.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and for highlighting the need to make uniformity of error terms explicit. We address each major comment below and will revise the manuscript accordingly to strengthen the presentation of the bounds.

read point-by-point responses

Referee: [§3 (Möbius inversion step) and the error-term discussion following the main theorem] The central asymptotic relies on writing 1_{square-free}(det(A)) = ∑_{d^2 | det(A)} μ(d) and showing that the contribution of the terms d > 1 is o(H^{n^2}). No explicit uniform bound is supplied for the counting function of matrices with d^2 | det(A) when d ranges up to the necessary cutoff (typically a small power of H); without such uniformity the tail of the Möbius sum cannot be guaranteed to be negligible.

Authors: We agree that the manuscript would benefit from an explicit statement of the uniform bound. The volume estimates in §3 already imply that the number of matrices in M_n(Z; H) with d^2 | det(A) is ≪_n H^{n^2} d^{-2} (log H)^{O(1)} uniformly for d ≤ H^ε with ε small; the implied constant is independent of d in this range because the determinant condition defines a hypersurface whose volume scales with the same power of H regardless of the fixed d. In the revision we will insert this uniform estimate immediately after the Möbius inversion step and verify that the resulting tail ∑_{d > 1} |μ(d)| · O(H^{n^2} d^{-2}) is indeed o(H^{n^2}). revision: yes
Referee: [§5 (Euler totient asymptotic)] The same uniformity issue appears for the totient sum, where φ is expanded as a Dirichlet convolution; the manuscript does not verify that the resulting multiple sums over divisors remain controllable when the divisibility conditions on det(A) are imposed simultaneously with the height bound H.

Authors: We accept the referee’s observation. The proof in §5 proceeds by writing φ(m) = ∑_{k|m} μ(k) (m/k) and then interchanging summation, but the uniformity of the resulting double sum over k and the restricted matrices is not written out. In the revised version we will add a short paragraph showing that the same volume bound used for the square-free indicator applies uniformly to each term in the convolution, with the extra factor of m/k absorbed into the main term; the error remains o(H^{n^2} log H) uniformly in the range of summation. revision: yes

Circularity Check

0 steps flagged

No circularity: standard Möbius inversion and volume estimates are independent of target asymptotic

full rationale

The derivation applies the standard indicator 1_{square-free}(det(A)) = sum_{d^2 | det(A)} μ(d) and obtains the main term from the d=1 summand plus controlled error from d>1. This is a direct analytic estimate on the space of matrices of height H; no parameter is fitted to the square-free count itself, no uniqueness theorem is imported from the authors' prior work to force the result, and the central asymptotic does not reduce by construction to its own inputs. Self-citations to earlier matrix-counting results (if present) supply the unrestricted count but are not load-bearing for the square-free refinement.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no explicit free parameters, axioms, or invented entities are stated or can be extracted.

pith-pipeline@v0.9.0 · 5600 in / 974 out tokens · 68323 ms · 2026-05-22T23:09:59.881452+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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[1] [1]

R. P. Brent and B. D. McKay, ‘Determinants and ranks of random matrices overZ m’,Discrete Math.,66(1987), 35–49. 5

work page 1987

[2] [2]

Destagnol and E

K. Destagnol and E. Sofos, ‘Rational points and prime values of polynomials in moderately many variables’,Bull. Sci. Math.,156(2019), Art. 102794. 2

work page 2019

[3] [3]

Drmota and R

M. Drmota and R. F. Tichy,Sequences, discrepancies and applications, Springer-Verlag, Berlin, 1997. 16

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W. Duke, Z. Rudnick and P. Sarnak, ‘Density of integer points on affine ho- mogeneous varieties’,Duke Math. J.,71(1993), 143–179. 5

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Fouvry, ‘Consequences of a result of N

E. Fouvry, ‘Consequences of a result of N. Katz and G. Laumon concerning trigonometric sums’,Isr. J. Math.,120(2000), 81–96. 3, 7

work page 2000

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Iwaniec and E

H. Iwaniec and E. Kowalski,Analytic number theory, Amer. Math. Soc., Prov- idence, RI, 2004. 4, 6

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N. M. Katz, ‘Estimates for “singular” exponential sums’,Intern. Math. Res. Notices,1999(1999), 875–899. 3

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Y. R. Katznelson, ‘Singular matrices and a uniform bound for congruence groups of SL npZq’,Duke Math. J.,69(1993), 121–136. 3, 5

work page 1993

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J. F. Koksma, ‘Some theorems on Diophantine inequalities’,Math. Centrum Scriptum no. 5, Amsterdam, 1950. 16

work page 1950

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Kotsovolis and K

G. Kotsovolis and K. Woo, ‘Prime number theorems for polynomials from homogeneous dynamic’,Geom. Func. Analysis,35(2025), 1169–1221. 2

work page 2025

[12] [12]

Kovaleva, ‘On the distribution of equivalence classes of random symmetric p-adic matrices’,Mathematika,69(2023), 903–933

V. Kovaleva, ‘On the distribution of equivalence classes of random symmetric p-adic matrices’,Mathematika,69(2023), 903–933. 2

work page 2023

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Poonen, ‘Squarefree values of multivariable polynomials’,Duke Math

B. Poonen, ‘Squarefree values of multivariable polynomials’,Duke Math. J., 118(2003), 353–373. 2 MATRICES WITH SQUARE-FREE DETERMINANTS 25

work page 2003

[14] [14]

Reuss, Power-free values of polynomials.Bull

T. Reuss, Power-free values of polynomials.Bull. London Math. Soc.,47 (2015), 270–284. 2

work page 2015

[15] [15]

I. E. Shparlinski, ‘Some counting questions for matrices with restricted entries’, Lin. Algebra Appl.,432(2010), 155–160. 5

work page 2010

[16] [16]

Sz¨ usz, ‘On a problem in the theory of uniform distribution’,Comptes Rendus Premier Congr` es Hongrois, Budapest, 1952, 461–472 (in Hungarian)

P. Sz¨ usz, ‘On a problem in the theory of uniform distribution’,Comptes Rendus Premier Congr` es Hongrois, Budapest, 1952, 461–472 (in Hungarian). 16 School of Mathematics and Statistics, University of New South W ales, Sydney NSW 2052, Australia Email address:alina.ostafe@unsw.edu.au School of Mathematics and Statistics, University of New South W ales, ...

work page 1952