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Analytic Number Theory · Wessen Getachew
C(n) — Block-Coprime Density
$$C(n)\;=\;\zeta(2)\cdot\prod_{p\;\text{prime}}\!\left(1-\frac{\min(n+1,\,p)}{p^2}\right)$$
Natural density of pairs (r, M) with gcd(r, M+j) = 1 for all j = 0,…,n, rescaled by ζ(2) so that C(0) = 1.
At n = 1: C(1) = ζ(2)·DFT ≈ 0.5307, where DFT = ∏p(1−2/p²) is the Feller–Tornier Euler product.
The classical Feller–Tornier constant CFT = ½(1 + DFT) ≈ 0.6613 is a distinct object.
C(n)
=
ζ(2) · ∏p(1 − min(n+1, p) / p²)
=
—
C(n)
n
→sat —
→sat —
Speed
nth prime:
p≤2 (1)
loop to:
<0.001%
<0.1%
<1%
far
▸ Advanced — custom rings & χ(r) filter applies to every view
Dirichlet Character Controls
Advanced Modulus Selection — restrict which rings (M) draw, across all views
Multi-select:
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Range:
–
All rings 1–G active (default)
▸ Sector Path b·r − a·M = s · s=0 is the ray r/M=a/b, s=±1 are Farey neighbours
a
/ b
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Point Detail
▸ Show Working prime-by-prime Euler product · fraction or decimal · export PNG / CSV
Display:
Primes:
Calculator
Decimals
12
C(n) and D(n) are accurate to ~12 dp
digit precision key
■ 1–14
fully reliable (IEEE-754 double)
■ 15–16
last float digits — may carry rounding noise
■ 17+
beyond float precision — toFixed padding
C(n)
—
ζ(2) × D(n)
D(n) = raw density
—
ζ(2) = 1.644934…
D(n) = C(n)/ζ(2)
D(n) = C(n)/ζ(2)
Block length
2
n+1 consecutive integers
checked for coprimality
checked for coprimality
Saturated primes
1
p ≤ n+1 → factor = 1−1/p
next at n = —
next at n = —
Saturation: prime p is active (p > n+1) contributing 1−(n+1)/p², or
saturated (p ≤ n+1) contributing 1−1/p.
As n→∞, D(n)=C(n)/ζ(2) ∼ e−γ/ln(n+1) (Mertens), hence C(n) ∼ ζ(2)·e−γ/ln(n+1).
Feller–Tornier note: at n=1, D(1)=∏p(1−2/p²)≈0.3226 is the FT Euler product DFT;
the classical constant CFT=½(1+DFT)≈0.6613 is distinct — our C(1)=ζ(2)·DFT≈0.5307.
Export
▸ Block-Coprimality Canvas Grid · Ring · Farey · Lift · 3D Cube
Block-Coprimality Canvas
GLOBAL
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Grid view — cell (r, M): block-coprime · coprime-only · neither. Scroll to zoom, drag to pan, hover to inspect.
▸ Grid structure, barriers & view modes
Grid structure: Column r = residue, row M = modulus. Cell (r,M) is
gold if gcd(r,M+j)=1 for all j=0…n (block-coprime at current n),
teal if coprime to M only, dark otherwise.
The fraction of gold cells converges to D(n) = C(n)/ζ(2).
Barriers overlay Farey fraction paths r/M = k/j; these are the "lift barriers" where block-coprimality fails.
View modes (Euclidean, Fermat, Farey, Ulam spiral) rearrange the cells to reveal different structural patterns.
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empirical D(n): — | theory: —
Ring view — concentric rings M=2…mMax. Dot r at angle 2πr/M. Hover to inspect.
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▸ Ring structure & connection modes
Ring structure: Each ring M shows M residue classes r ∈ {1,…,M} at angles 2πr/M.
Gold dots = block-coprime (gcd(r,M+j)=1 for all j=0…n).
Teal dots = coprime to M only (gcd(r,M)=1 but fails later).
The density of gold dots across all rings converges to C(n).
Connection mode lets you trace structural patterns: Same GCD shows shared divisors across rings;
Gap-2 connects r to r+2 within each ring (twin prime analog);
Lift arcs connect same fraction r/M across adjacent rings — the "lift survival" paths central to C(n).
Max M:
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1
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p=
a/b rot:
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Per-ring preset:
Per-ring:
= 0.00000
per ring
Ring color:
M=2…128 · n=1
ρ: — | theory: —
3D Cube view — block-coprime lattice on all 6 faces of a cube.
Drag to rotate · scroll to zoom · right-click/two-finger to pan. Enter Rx/Ry/Rz or choose a preset.
▸ What the cube shows & color modes
What this shows: The block-coprime lattice is projected onto all 6 faces of a cube.
Each face is a G×G grid. Front/back faces show (r, M) at fixed n.
Left/right faces show (n, M) at fixed r. Top/bottom faces show (r, n) at fixed M.
Gold cells pass the block-coprime test at the current n-depth on that face.
GCD mode colors by gcd value — non-coprime pairs form lines at r=kp.
Saturation mode shows which prime first blocks: p=2 (coral), p=3 (orange), p=5 (yellow), p=7 (lime).
n-depth:
1
Grid G:
20
Color:
off
Rx:
Ry:
Rz:
Zoom:
Presets:
G=20 · n=1 · 0 pts
C(n) = — | survivors: — | density: —
Speed:
n = —
Prime Factor Table
| p | Status | min(n+1,p) | p² | Local factor | log factor | Partial D(n) | Partial C(n) | Block corr. |
|---|
First 60 primes. Partial C(n) = ζ(2) × running product. Click any row for local factor matrix and step-by-step detail. Block corr. = (1−min(n+1,p)/p²)/(1−1/p)^(n+1): how much the 2D block-coprime density exceeds the independence baseline at each prime. Always > 1 and diverges as a product — not the HL singular series. The C₂ local factor p(p−2)/(p−1)² is a separate 1D quantity visible in the HL Connection panel.
Data Table — n = 0 to current
| n | Block | C(n) | D(n) | C(n)/C(n−1) | ΔC | Sat. primes | C(n)/C₂ | Event |
|---|
Click any row to see step-by-step derivation of C(n) and D(n) for that value of n.
Step-by-Step Derivation
Hardy–Littlewood Connection
The Euler product structure of C(n) is the combinatorial bedrock of the Hardy–Littlewood prime k-tuple
conjectures. The following constants all live in the same family.
| Symbol | Value | Formula | Meaning |
|---|
Structural parallel — same exclusion count, different space:
The C(n) local factor at prime $p$ is
$$f_p(n) = 1 - \frac{\min(n+1,p)}{p^2} \quad (\text{2D: counts pairs }(r,M)\in(\mathbb{Z}/p\mathbb{Z})^2)$$
The HL singular series local factor for an admissible tuple $\mathcal{H}$ with $\nu_p=|\mathcal{H}\bmod p|$ is
$$\frac{1-\nu_p/p}{(1-1/p)^k} \quad (\text{1D: counts residues in }\mathbb{Z}/p\mathbb{Z})$$
Both share the exclusion count $\nu_p = \min(n+1,p)$, but operate in different ambient dimensions. The
Block corr. column shows the ratio $f_p(n)/(1-1/p)^{n+1}$ — always > 1, diverging —
while the C₂ local (1D) column shows the genuine twin prime factor $p(p-2)/(p-1)^2$, which is distinct.
Per-Prime HL Factors at current n
hlFactor(p,n) = (1−min(n+1,p)/p²)/(1−1/p)^(n+1)
| p | Status | Raw f_p(n) | (1−1/p)^(n+1) | Block corr. | C₂ local (1D) |
|---|
Additive Number Theory
The multiplicative → additive bridge:
C(1) ≈ 0.5307 from this tool's Euler product is provably the same constant as the
lift-survival constant C = ∏p(p²−2)/(p²−1): since ζ(2) = ∏p p²/(p²−1),
substituting gives C(1) = ζ(2)·∏p(1−2/p²) = ∏p(p²−2)/(p²−1) exactly — an
algebraic identity, not a numerical coincidence. Prime counting and prime gaps below are the
purely additive companions to the multiplicative structure here.
π(x) — Prime Counting Function vs Li(x)
gₙ = pₙ₊₁ − pₙ — Prime Gaps
● Maximal gaps (record gaps)
— ln(pₙ) predicted average gap (PNT)
All gaps ·
Strip Average — Diagonal Decomposition of C(0)
The coprime density 6/π² = D(0) is a weighted average of φ(g)/g over all diagonal strips of the G×G lattice:
$$A(G) = \frac{1}{G^2}\sum_{g=1}^{G-1}(G-g)\cdot\frac{\varphi(g)}{g} \;\to\; \frac{6}{\pi^2}$$
Each strip |M−r|=g contributes φ(g)/g coprime pairs, weighted by strip length (G−g).
The block-coprime extension replaces φ(g)/g with B(g,n) to recover D(n) = C(n)/ζ(2) for any n.
This is the explicit bridge to the Gap Diagonal Identity (page2.html).
G max:
— A(G) strip average
- - 6/π² = 0.60793…
- - D(n) block-extended
· φ(g)/g per strip
— error |A−6/π²|