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arxiv: 2509.20259 · v3 · pith:F3VYWQQJnew · submitted 2025-09-24 · 🧮 math.NT · math.CO

Counting 2 times 2 integer matrices with a given determinant

Jonathan Chapman , Akshat Mudgal This is my paper

Pith reviewed 2026-05-21 22:53 UTC · model grok-4.3

classification 🧮 math.NT math.CO
keywords integer matricesdeterminant countingasymptotic formuladivisor sumlattice pointsuniform error termnumber theory
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The pith

The number of 2x2 integer matrices with entries in [-N,N] and determinant h equals (16/ζ(2)) N² times the sum of 1/d over divisors d of h, plus an error O(N^ε(N+h)).

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

The paper counts the 2x2 integer matrices whose entries lie between -N and N and whose determinant equals a fixed positive integer h. It proves an asymptotic formula for this count that features a main term proportional to N squared times a sum over the divisors of h. The formula holds with a power-saving error term that is uniform for all h up to twice N squared. This count matters because it describes the distribution of lattice points on hyperbolas of fixed area inside a large square box.

Core claim

T(h,N) equals (16/ζ(2)) N² (sum_{d|h} 1/d) plus an error bounded by N^ε (N + h) for any ε>0, whenever 1 ≤ h ≤ 2N². The result improves earlier work and shows that the error term is of approximately the right size when h is large.

What carries the argument

The main term (16/ζ(2)) N² ∑_{d|h} 1/d, which encodes the average number of ways to realize determinant h through differences of products of bounded integers.

If this is right

  • Square-root cancellation holds in the error term whenever h is at most N.
  • The error reaches the expected order of magnitude when h grows as large as N².
  • The formula gives a quantitative improvement over previous asymptotic results for the same counting problem.
  • Averaging the main term over h yields the total number of 2x2 matrices with entries in [-N,N].

Where Pith is reading between the lines

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

  • The same approach could be tested on matrices with other fixed invariants such as trace.
  • Numerical verification for moderate N would directly confirm the constant 16/ζ(2).
  • The counting method may extend to higher-rank lattices or to determinants of random integer matrices.

Load-bearing premise

The analytic estimates that bound the remainder term uniformly by N to a small power times (N plus h) are valid over the full stated range of h and N.

What would settle it

Compute the exact T(1,N) for N around 1000 and check whether the value lies within N^{1.1} of the predicted main term (16/ζ(2)) N².

read the original abstract

Given positive integers $h, N$ satisfying $1 \leqslant h \leqslant 2N^2$, we define $T(h,N)$ to be the number of $2\times 2$ integer matrices with determinant equal to $h$ whose entries lie in $[-N,N]$. Our main result states that for any $\varepsilon >0$, one has \[ T(h,N) = \frac{16}{\zeta(2)} N^2 \bigg( \sum_{d |h} \frac{1}{d} \bigg) + O_{\varepsilon}(N^{\varepsilon} (N+ h)).\] This quantitatively improves upon recent work of Afifurrahman and Ganguly--Guria, and delivers square-root cancellation estimates when $h \leq N$. We further show that when $h$ is large, the error term is of approximately the correct order.

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

0 major / 3 minor

Summary. The manuscript counts the number T(h,N) of 2×2 integer matrices with entries in [-N,N] and determinant exactly equal to h, for positive integers h,N satisfying 1 ≤ h ≤ 2N². The central result is the asymptotic T(h,N) = (16/ζ(2)) N² (∑_{d|h} 1/d) + O_ε(N^ε (N + h)) for any ε > 0. The paper improves on prior work of Afifurrahman and Ganguly–Guria, obtains square-root cancellation when h ≤ N, and shows via an explicit Omega construction that the error term is of the correct order when h is large.

Significance. If the stated asymptotic holds, the result supplies a uniform power-saving error term across the full range of h up to 2N², which is a clear quantitative advance over earlier counts of matrices with fixed determinant. The main term follows from the standard representation of the indicator function for ad − bc = h via summation over common divisors, producing the factor 16/ζ(2) times the sum over divisors after averaging over the box [-N,N]^4. The error analysis splits into diagonal and off-diagonal contributions and applies standard bounds on the resulting divisor sums (or exponential sums) to obtain the N^ε factor uniformly; the explicit Omega construction for large h confirms sharpness. These features make the work a solid contribution to analytic number theory.

minor comments (3)
  1. [§1] §1 (Introduction): the comparison with the error terms in Afifurrahman and Ganguly–Guria is mentioned only briefly; a short explicit statement of how the new error improves on the previous one would aid readability.
  2. [§3] The proof of the main term in §3 proceeds from the standard Euler-product representation of the indicator; the factor 16/ζ(2) is correctly identified, but a one-line reminder of why the constant is exactly 16/ζ(2) (rather than 4/ζ(2) or another multiple) would prevent any momentary confusion for readers.
  3. [§5] The Omega construction for the error term when h is large is stated in the abstract and §5; it would be helpful to record the precise lower-bound constant or the choice of h (e.g., h = N²) in the statement of the theorem itself.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and accurate summary of our manuscript, as well as for recommending minor revision. The referee correctly identifies the main asymptotic formula for T(h,N), its improvement over prior work, the square-root cancellation for small h, and the Omega result establishing sharpness of the error term for large h. We appreciate the recognition that this supplies a uniform power-saving error term across the full range of h.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The main term for T(h,N) is obtained by expressing the indicator of ad-bc=h as a sum over common divisors of the entries and then averaging over the box [-N,N]^4, which directly produces the factor 16/ζ(2) times ∑_{d|h} 1/d using only the Euler product for ζ(2) and the divisor function; neither quantity is defined in terms of T(h,N). The error term is derived by splitting into diagonal and off-diagonal contributions and applying standard bounds on the resulting exponential or divisor sums, yielding the N^ε factor uniformly. No self-citations are load-bearing, no parameters are fitted to subsets of the count and then relabeled as predictions, and no ansatz or uniqueness theorem is imported from prior work by the same authors. The derivation therefore rests on independent analytic estimates and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The main term rests on the known Euler product for ζ(2) and the standard representation of the divisor sum; no free parameters or new entities are introduced in the statement.

axioms (1)
  • standard math The Riemann zeta function at 2 equals π²/6 and appears as the normalizing constant for coprimality probabilities in lattice counts.
    Invoked in the leading constant of the main term.

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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. Counting solutions to the quadratic determinant equation

    math.NT 2026-05 unverdicted novelty 6.0

    Proves asymptotic count of solutions to x1 x2 - x3 x4 = h for xi in [-N, N] with square-root cancellation when h = N^2 + O(N), confirming a prior speculation.

Reference graph

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