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arxiv: 2512.22494 · v2 · submitted 2025-12-27 · 🧮 math.NT · math.RA

On the Limiting Density of a gcd Map

Thang Pang Ern , Malcolm Tan Jun Xi , Loh Wei Xuan Ryan This is my paper

Pith reviewed 2026-05-16 19:42 UTC · model grok-4.3

classification 🧮 math.NT math.RA
keywords gcd functionnatural densityEuler productquadratic class numberarithmetic statisticscoprime pairs
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The pith

The gcd ratio function f(a,b) equals 1 with natural density approximately 0.88151 for integer pairs.

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

The paper defines the function f(a,b) as the gcd of a plus b and a times b divided by the gcd of a and b. It determines the natural density of pairs where this equals one by computing an infinite product over all primes. The resulting value is approximately 0.88151 and coincides with the quadratic class number constant from real quadratic fields. A higher-order analogue of the function reduces the problem to ordinary coprimality, giving the density six over pi squared.

Core claim

The limiting density of pairs (a,b) such that f(a,b)=1 is given by the Euler product over primes of one minus one over p squared times p plus one, which evaluates to approximately 0.88151 and matches the quadratic class number constant.

What carries the argument

The gcd ratio map f(a,b) = gcd(a+b, ab)/gcd(a,b), whose frequency of equaling one is found by multiplying local densities at each prime.

If this is right

  • The density equals the explicit Euler product over all primes.
  • This value is identical to the constant appearing in the class number formula for real quadratic fields.
  • The higher-order version f_r yields the density of coprime pairs, equal to one over zeta of two.
  • The conditions at distinct primes become independent in the limit.

Where Pith is reading between the lines

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

  • The match to the quadratic class number constant may point to an underlying arithmetic connection between this gcd expression and quadratic fields.
  • Similar product formulas could be derived for other functions built from gcds of linear and quadratic combinations.
  • Numerical verification for large bounds on a and b would test the predicted convergence rate of the density.

Load-bearing premise

The natural density of pairs where f(a,b) equals 1 exists and equals the product of the local densities at each prime.

What would settle it

A direct enumeration of the fraction of pairs with maximum absolute value at most N where f(a,b) equals one, and checking whether the fraction converges to 0.88151 as N tends to infinity.

Figures

Figures reproduced from arXiv: 2512.22494 by Loh Wei Xuan Ryan, Malcolm Tan Jun Xi, Thang Pang Ern.

Figure 1
Figure 1. Figure 1: shows the heat map of the function f (a, b) for 1 ≤ a, b ≤ 50. 5 10 15 20 25 30 35 40 45 50 5 10 15 20 25 30 35 40 45 50 Legend: f = 1 f = 2 f = 3 f = 4 f = 5 f = 6 f = 7 f = 8 f = 9 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Geometric interpretation of the Dirichlet hyperbola method Also, the geometric interpretation of the Dirichlet hyperbola method generally in￾volves the principle of inclusion and inclusion. Let a ∈ R such that 1 < a < n, and let b = n/a. Then the lattice points (x, y) can be split into three overlapping regions: one region is bounded by 1 ≤ x ≤ a and 1 ≤ y ≤ n/x, another region is bounded by [PITH_FULL_IM… view at source ↗
read the original abstract

The function \[f(a,b)=\frac{\gcd(a+b,ab)}{\gcd(a,b)}\] is of interest in this paper. We then ask a natural question regarding how often $f(a,b)=1$ is. We yield the limiting density $\rho=\prod_{p}\left(1-\frac{1}{p^2(p+1)}\right)\approx 0.88151$ which is an Euler product that unexpectedly matches the quadratic class number constant from the theory of real quadratic fields. We also consider its higher-order analogue $f_r$, where the problem collapses to coprimality and the density becomes $1/\zeta(2)=6/\pi^2$.

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 defines the function f(a,b) = gcd(a+b, ab)/gcd(a,b) for positive integers a,b and determines the natural density of pairs where f(a,b)=1. It proves that this density exists and equals the Euler product ∏_p (1 - 1/(p²(p+1))) ≈ 0.88151. The authors note that this product coincides with a constant from the theory of class numbers of real quadratic fields. The paper also treats a higher-order analogue f_r, showing that its density equals 1/ζ(2) = 6/π².

Significance. The result supplies an explicit Euler-product formula for the density of a naturally occurring gcd-derived function, obtained through standard local-density calculations at each prime. The derivation confirms that the bad local event at p (equal positive valuations together with a residue condition) has measure exactly 1/(p²(p+1)), so the complementary density is 1 - 1/(p²(p+1)). The global density then follows from the Chinese Remainder Theorem. The observed numerical match with the quadratic class-number constant is interesting and may suggest further links between gcd statistics and real quadratic fields, although the manuscript presents the equality as an observation rather than a derived identity.

minor comments (3)
  1. [Abstract] Abstract: the verb 'yield' in 'We yield the limiting density' is slightly nonstandard in mathematical English; 'We obtain' or 'The limiting density equals' would read more naturally.
  2. [Abstract] Abstract: a brief parenthetical reference to the precise class-number constant being matched (e.g., the constant in the asymptotic for the sum of class numbers of real quadratic fields) would help readers locate the connection.
  3. The definition and precise statement of the higher-order function f_r should be stated explicitly in the main text before the density claim is made, to avoid any ambiguity about the reduction to coprimality.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and insightful report, which accurately summarizes the main results on the density of pairs with f(a,b)=1 and the higher-order analogue f_r. We appreciate the recognition of the Euler-product derivation and the interesting observation regarding the quadratic class-number constant.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives the density ρ = ∏_p (1 − 1/(p²(p+1))) directly by computing the local density at each prime p that f(a,b) ≠ 1. This is done by summing the exact probabilities over equal valuations k ≥ 1 of the event v_p(a) = v_p(b) = k together with the conditional probability 1/(p−1) that the reduced a′ ≡ −b′ (mod p). The resulting geometric series yields the local factor exactly, and the global density is the product by independence via the Chinese Remainder Theorem on the finite set of p-adic conditions. No parameters are fitted to data, no self-citations are invoked for the central existence or value of the density, and the numerical match to the quadratic class-number constant is presented only as an observation after the derivation. The result is therefore obtained from first-principles counting and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the existence of the natural density as the product of independent local densities at each prime; no free parameters are introduced and no new entities are postulated.

axioms (1)
  • domain assumption The natural density of the set {(a,b) : f(a,b)=1} exists and equals the Euler product over primes of the local probabilities.
    Standard assumption when computing densities of gcd conditions via Chinese Remainder Theorem and independence across primes.

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Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

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