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Analytic Number Theory · Wessen Getachew · 2026

Gap Diagonal Identity

$$\text{coprime density on } |M - r| = g \;=\; \frac{\varphi(g)}{g}$$
diagonal  g = —   density  — = drag g slider ↓
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Grid G g depth n Color PHASE — Zoom Cell
Grid G8
Depth n0
C(n) field density0.60793
Diagonal goff
φ(g)/g theory—
Strip observed—
Block strip C(n,g)—
GRID — selected point
Theory — Gap Diagonal Identity ▶
The Gap Diagonal Identity

For the lattice $\{(r,M) : 1 \le r \le M \le G\}$, the fraction of pairs on the diagonal $|M-r|=g$ that are coprime equals $\varphi(g)/g$ — a direct consequence of Euler's totient function. The identity is classical; reading the lattice as a family of diagonal strips, each carrying its own density, gives a geometric way to decompose and visualize C(n).

$$\text{coprime density on } \{|M - r| = g\} = \frac{\varphi(g)}{g}$$
g=11.000
g=20.500
g=60.333
g=120.333
Proof Sketch
$$\gcd(r, r+g) = \gcd(r, g)$$
Step 01 · Parametrize the diagonal▶
Points on diagonal $|M-r|=g$ are exactly the pairs $(r, r+g)$ for $r=1,\ldots,G-g$. The coprimality condition is $\gcd(r, r+g)=1$.
Step 02 · Reduce the coprimality condition▶
By the key identity $\gcd(r, r+g) = \gcd(r, g)$, coprimality with $r+g$ reduces to coprimality with $g$ alone.
Step 03 · Count via periodicity▶
The condition $\gcd(r,g)=1$ is periodic in $r$ with period $g$. Among any $g$ consecutive integers, exactly $\varphi(g)$ are coprime to $g$, giving asymptotic density $\varphi(g)/g$.
Step 04 · Exactness and error bound▶
The density $\varphi(g)/g$ is exact when $G-g$ is a multiple of $g$; otherwise the error is bounded by $\varphi(g)/(G-g)$, vanishing as $G\to\infty$. The formula is thus asymptotically exact for all fixed $g$ as $G$ grows.
Extension to Block-Coprimality — The C(n) Setting

At $n=0$, diagonal strip $g$ has coprime density exactly $\varphi(g)/g$ — a classical fact. As $n$ grows, the block-coprime condition imposes additional constraints: a pair $(r, r+g)$ must now satisfy $\gcd(r, r+g+j) = 1$ for $j = 0, \ldots, n$. This modifies the local density on each strip by eliminating extra residue classes at each prime dividing $g, g+1, \ldots, g+n$.

The formula below gives the block-coprime density restricted to diagonal strip $g$ at depth $n$:

$$\prod_{p \mid g(g+1)\cdots(g+n)} \!\!\left(1 - \frac{1}{p}\right)$$
n=0φ(g)/g
n=1product
n→∞C(∞)=0

This per-strip formula describes how each diagonal behaves individually. The global C(n) is not exactly a weighted average of these per-strip densities, since that would require the strips to be independent — they aren't, as they share prime factors. The diagonal decomposition is a visualization and a close approximation rather than an exact identity once n>0.

Why This Matters
  • Geometric decomposition: reading the lattice as diagonal strips, each with its own density φ(g)/g, gives a visual explanation for why C(n)'s Euler product looks the way it does — each prime p contributes most strongly through the diagonals it divides.
  • Classical foundation: the density φ(g)/g on diagonal g is a direct consequence of Euler's totient formula and periodicity.
  • Hardy–Littlewood parallel: the per-strip block-coprime density is structurally similar to the local factors in Hardy–Littlewood's singular series for prime constellations. The Hardy–Littlewood conjecture itself remains unproven, and the parallel here is suggestive rather than a formal connection.
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