On the Asymptotic Density of a GCD-based Map

Malcolm Tan Jun Xi; Thang Pang Ern

arxiv: 2506.19812 · v2 · submitted 2025-06-24 · 🧮 math.NT

On the Asymptotic Density of a GCD-based Map

Thang Pang Ern , Malcolm Tan Jun Xi This is my paper

Pith reviewed 2026-05-19 07:56 UTC · model grok-4.3

classification 🧮 math.NT

keywords gcdasymptotic densitySL(2,Z)natural densityprimitive pairsarithmetic progressionsnumber theory

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The pith

The function f(a,b) = gcd(ab, a+b)/gcd(a,b) has all solutions for each fixed n classified by a uniform three-parameter family coming from an SL_2(Z) action on primitive pairs.

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

This paper shows that a certain ratio of greatest common divisors, called f(a,b), has a hidden symmetry explained by the action of the modular group SL_2(Z) on pairs of coprime integers. It supplies a single three-parameter recipe that generates every integer pair satisfying f(a,b) equals any given positive integer n. When n is square-free the recipe reduces to families of arithmetic progressions linked by the Chinese remainder theorem. Using the parametrization the authors compute that the proportion of pairs with f(a,b) equal to 1 approaches a specific infinite product over all primes, numerically about 0.88151. A higher-order version of the same map settles at the familiar density 6 over pi squared once the order is two or more. Readers may find the result interesting because it turns an apparently ad-hoc GCD expression into an object whose frequency can be read off from a group action and a simple counting argument.

Core claim

The symmetry of f(a,b) = gcd(ab, a+b)/gcd(a,b) stems from an SL_2(Z) action on primitive pairs and that all solutions to f(a,b)=n admit a uniform three-parameter description. This recovers arithmetic-progression families via the Chinese remainder theorem when n is squarefree. The density of pairs with f(a,b)=1 tends to the product over primes p of (1 - p^{-2}(p+1)^{-1}) approximately 0.88151, and the higher-order analogue f_r has a limiting density of 6/pi^2 for r greater than or equal to 2.

What carries the argument

SL_2(Z) action on primitive pairs together with the three-parameter description that classifies solutions to f(a,b)=n

If this is right

When n is squarefree the solutions appear as arithmetic-progression families recovered by the Chinese remainder theorem.
The natural density of pairs satisfying f(a,b)=1 equals the infinite product over primes of (1 minus p to the minus 2 times (p plus 1) to the minus 1).
The higher-order map f_r possesses limiting density exactly 6 over pi squared whenever r is at least 2.

Where Pith is reading between the lines

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

The same three-parameter counting technique could be applied to other GCD ratios built from linear forms in a and b.
Direct enumeration of pairs up to moderate bounds offers a practical check on the predicted density value 0.88151.
Analogous densities may exist for similar maps on tuples of integers rather than pairs.

Load-bearing premise

The SL(2,Z) action on primitive pairs together with the three-parameter description exhaustively classifies all integer solutions without omissions or overcounts.

What would settle it

Compute the proportion of pairs (a,b) with absolute values up to a large bound N for which f(a,b) equals 1 and check whether the proportion converges to approximately 0.88151 as N grows.

read the original abstract

We show that the symmetry of \[f\left(a,b\right)=\frac{\operatorname{gcd}\left(ab,a+b\right)}{\operatorname{gcd}\left(a,b\right)}\] stems from an $\operatorname{SL}_2\left(\mathbb{Z}\right)$ action on primitive pairs and that all solutions to $f\left(a,b\right)=n$ admit a uniform three-parameter description -- recovering arithmetic-progression families via the Chinese remainder theorem when $n$ is squarefree. It shows that the density of pairs with $f\left(a,b\right)=1$ tends to $\prod_p\left(1-p^{-2}(p+1)^{-1}\right)\approx0.88151$, and that its higher-order analogue $f_r$ has a limiting density $6/\pi^2$ for $r\ge2$.

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 parametrizes solutions to f(a,b)=n via SL(2,Z) action on primitive pairs and extracts an explicit Euler product density for the case f=1.

read the letter

The main point is that this work connects the symmetry in f(a,b) = gcd(ab, a+b)/gcd(a,b) to an SL(2,Z) action and supplies a uniform three-parameter description of all solutions. That description lets them pull out the natural density of pairs with f(a,b)=1 as the product over primes of (1 - p^{-2}(p+1)^{-1}), which evaluates to roughly 0.88151. They also note that the higher-order versions f_r settle at the coprime density 6/π² once r is at least 2. The parametrization recovers arithmetic-progression families through the Chinese remainder theorem when n is squarefree, and the derivation rests on standard facts about the group action and CRT rather than any internal fitting. This is a concrete step forward for counting solutions to this particular gcd ratio. The approach looks reproducible once the bijection or covering property between parameters and pairs is verified in detail. The central risk is whether the three-parameter map really accounts for every positive integer pair without systematic omissions or extra multiplicities; if it does, the local densities feed straight into the product and the limit follows. If the full argument closes that step cleanly, the result stands. This is aimed at analytic number theorists who track gcd statistics or lattice actions on pairs. A reader already working on similar arithmetic functions would pick up the explicit formula and the group-action viewpoint. It is worth sending to peer review because the claim is explicit, the method is standard, and the computation is falsifiable in principle.

Referee Report

1 major / 2 minor

Summary. The manuscript establishes that the symmetry in the function f(a, b) = gcd(ab, a + b)/gcd(a, b) arises from an SL(2, Z) action on primitive pairs of integers. It further provides a uniform three-parameter description for all solutions to the equation f(a, b) = n for positive integers a, b, n. When n is squarefree, this description recovers families in arithmetic progressions using the Chinese Remainder Theorem. From this parametrization, the paper derives the natural density of the set of pairs (a, b) with f(a, b) = 1 as the infinite product over primes ∏_p (1 - p^{-2} (p + 1)^{-1}) which approximates 0.88151, and shows that a higher-order version f_r has asymptotic density 6/π² for all r ≥ 2.

Significance. If the three-parameter classification is complete and bijective (or accounts properly for multiplicities), the results offer a precise computation of asymptotic densities for this gcd-related map, which may have implications for the distribution of gcd values in integer lattices. The connection to SL(2, Z) and the recovery of arithmetic progressions via CRT highlights an elegant group-theoretic perspective on what appears to be a purely arithmetic problem. The explicit Euler product and the appearance of the coprimality density 6/π² for higher orders are notable features that align with classical results in analytic number theory.

major comments (1)

[Abstract and paragraph on solutions and density] The density claims, including the Euler product for pairs with f(a,b)=1 and the constant 6/π² for f_r (r≥2), are derived directly from the uniform three-parameter description of solutions. The manuscript asserts that this description, stemming from the SL(2,Z) action on primitive pairs and extended via CRT for squarefree n, exhaustively classifies all positive integer solutions without omissions or uncontrolled multiplicities. However, to confirm that the local densities used in the product are accurate, an explicit argument or verification showing that the parametrization induces a measure-preserving cover (or bijection) on the set of all pairs would be necessary, as any systematic omission would invalidate the asymptotic limit.

minor comments (2)

[Introduction] The definition of the higher-order analogue f_r should be stated explicitly early in the paper to avoid ambiguity for readers.
Consider providing more decimal places for the numerical approximation 0.88151 or indicating how it was computed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the constructive comment regarding the completeness and measure-preserving properties of the three-parameter parametrization. We address this point directly below.

read point-by-point responses

Referee: [Abstract and paragraph on solutions and density] The density claims, including the Euler product for pairs with f(a,b)=1 and the constant 6/π² for f_r (r≥2), are derived directly from the uniform three-parameter description of solutions. The manuscript asserts that this description, stemming from the SL(2,Z) action on primitive pairs and extended via CRT for squarefree n, exhaustively classifies all positive integer solutions without omissions or uncontrolled multiplicities. However, to confirm that the local densities used in the product are accurate, an explicit argument or verification showing that the parametrization induces a measure-preserving cover (or bijection) on the set of all pairs would be necessary, as any systematic omission would invalidate the asymptotic limit.

Authors: The parametrization is constructed explicitly from the SL(2, Z) action on primitive pairs: every pair (a, b) is mapped to a unique primitive pair (a', b') via an invertible group element such that f(a, b) equals f(a', b'), after which the equation f(a', b') = n is solved by three integer parameters subject to explicit coprimality conditions derived from the gcd definitions. This correspondence is bijective by construction, as the group action is invertible on the set of pairs and the three-parameter solution recovers every primitive solution without omission or multiplicity beyond what is controlled by the parameters. For squarefree n the CRT extension then yields disjoint arithmetic-progression families. Consequently the natural density factors as an Euler product of local densities, each computed as the proportion of admissible parameter triples modulo powers of p; the factor (1 - p^{-2}(p + 1)^{-1}) is the exact local density at p, and the same local analysis yields the constant 6/π² for the higher-order maps f_r when r ≥ 2. We will add a short subsection that explicitly records the inverse map from (a, b) back to the three parameters and verifies that the induced map on the unit square [0, 1]^2 preserves Lebesgue measure, thereby confirming that the asymptotic densities are unaffected by omissions or overcounting. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses external SL(2,Z) and CRT facts

full rationale

The paper derives the symmetry of f from the standard SL(2,Z) action on primitive pairs and asserts a uniform three-parameter description of all solutions to f(a,b)=n, recovering AP families via CRT for squarefree n. These inputs are independent external facts from group theory and elementary number theory, not defined in terms of the target densities. The Euler-product density for f=1 and the 6/π² limit for f_r then follow by direct counting over the parametrization without any fitted parameters, self-definitional loops, or load-bearing self-citations. The classification step is presented as exhaustive but does not reduce to a tautology or prior result by the same authors; the chain remains self-contained against standard benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of the gcd function, the modular group SL(2,Z), and the Chinese remainder theorem; no free parameters are introduced and no new entities are postulated.

axioms (2)

standard math Standard properties of the gcd function and the natural action of SL(2,Z) on primitive pairs
Invoked to establish symmetry of f (abstract opening sentence).
standard math Chinese remainder theorem applies to arithmetic-progression families when n is squarefree
Used to recover known families from the three-parameter description.

pith-pipeline@v0.9.0 · 5664 in / 1487 out tokens · 50803 ms · 2026-05-19T07:56:25.198016+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 unclear

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

the symmetry of f(a,b)=gcd(ab,a+b)/gcd(a,b) stems from an SL_2(Z) action on primitive pairs and that all solutions to f(a,b)=n admit a uniform three-parameter description
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

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

ρ = ∏_p (1 - 1/(p²(p+1))) ≈ 0.88151

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