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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) — live value & survival ratios
C(n)
=
ζ(2) · ∏p(1 − min(n+1, p) / p²)
=
—
∞ asymptotic: 114,155 primes (≤1.5M) + tail correction, ~11 dp reliable · n=1 shown exact (OEIS A065474/A065469)
gcd(r,M)=1
—
→ 1/ζ(2) ≈ 0.60793 as G→∞
lift: gcd(r,M)=gcd(r,M+1)=1
—
→ D(1)=C(1)/ζ(2) ≈ 0.32263 as G→∞
lift / gcd=1
—
→ C(1) ≈ 0.53071 as G→∞
▸ Lattice Controls
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G=31
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▸ 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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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
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/ b
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Point size:
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▸ Active Gaps twin / cousin / sexy-prime highlighting — applies to every view
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SEMANTICS:
Dynamic Gap Generation
Max Gap:
Add Consecutive Range — From:
To:
▸ Show Working prime-by-prime Euler product · fraction or decimal · export PNG / CSV
Display:
Primes:
▸ Goldbach Overlay every even N = p + q with p, q prime, highlighted as (r,M) pairs on the lattice
N =
N=—
N = —
▸ Intro — what is C(n)?
Pick a random whole number. Now ask: how likely is it — and the next n numbers right after it — to all share no common factor with a fixed modulus M? That probability has a name, C(n), and it drops in clean, predictable steps — one jump at every prime.
It starts at certainty: C(0) = 1, exactly. The first step lands on C(1) ≈ 0.530711…, and nothing about it is mysterious — it's ζ(2) times the Feller–Tornier product ∏p(1−2/p²), and the result has its own classical entry. Two known constants multiplied, one known answer — everything after that first step is where this site begins.
OEIS: A013661 (ζ(2)) × A065474 (∏(1−2/p²)) = A065469 (C(1)).
How the ζ(2) cancels — baseline C(0) & D(0)
C(n) — normalized (× ζ(2))
C(0) = ζ(2)·∏p(1−1/p²)
= ∏p(1−1/p²)−1·∏p(1−1/p²) [Euler]
= 1 exactly — every factor cancels its own inverse.
C(0) = ζ(2)·∏p(1−1/p²)
= ∏p(1−1/p²)−1·∏p(1−1/p²) [Euler]
= 1 exactly — every factor cancels its own inverse.
D(n) = C(n)/ζ(2) — raw density
D(0) = C(0)/ζ(2) = 1/ζ(2) = ∏p(1−1/p²)
= (3/4)·(8/9)·(24/25)·(48/49)·…
= 0.607927101854 — no cancellation left, it's just 1/ζ(2).
D(0) = C(0)/ζ(2) = 1/ζ(2) = ∏p(1−1/p²)
= (3/4)·(8/9)·(24/25)·(48/49)·…
= 0.607927101854 — no cancellation left, it's just 1/ζ(2).
First step — C(1) & D(1)
C(1) = ζ(2)·∏p(1−2/p²)
= ∏p (1−2/p²)/(1−1/p²) = ∏p (p²−2)/(p²−1)
= (2/3)·(7/8)·(23/24)·(47/48)·…
≈ 0.530711820472
= ∏p (1−2/p²)/(1−1/p²) = ∏p (p²−2)/(p²−1)
= (2/3)·(7/8)·(23/24)·(47/48)·…
≈ 0.530711820472
D(1) = C(1)/ζ(2) = ∏p(1−2/p²)
same product, no ζ(2) factor — the Feller–Tornier product DFT (the classical F–T constant is ½(1+DFT))
= (1/2)·(7/9)·(23/25)·(47/49)·…
≈ 0.322634098939
same product, no ζ(2) factor — the Feller–Tornier product DFT (the classical F–T constant is ½(1+DFT))
= (1/2)·(7/9)·(23/25)·(47/49)·…
≈ 0.322634098939
check: C(1)/D(1) = ζ(2) = 1.644934… ✓ · C(0)/D(0) = ζ(2) ✓
Try any n — step-by-step (C(n) & D(n))
n =
Calculator
Decimals
12
C(n)/D(n): ~11 dp reliable (114,155 primes ≤ 1.5M, asymptotic) · n=1 exact via OEIS A065474/A065469
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
*for n≠1, actual reliability is ~11 dp (float summation error), not the full 14 shown — n=1 is the exception: exact, OEIS-sourced.
mode
— exact analytic value
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
Color mode:
Labels:
Label size:
22
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.
Grid G:
Grid n:
1
Zoom:
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View:
a/b rot:
/
Per-mod:
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Preset:
Gap g:
off
max 64
Label:
zoom 1× · G=128 · n=0
empirical D(n): — | theory: —
Ring view — concentric rings M=2…mMax. Dot r at angle 2πr/M. Hover to inspect.
Speed:
180
Gap g:
off
max 64
Label:
▸ 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:
Dot size:
1
Connect:
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p=
a/b rot:
/
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 actual 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.
R(N) — Goldbach Partition Count vs Hardy–Littlewood Prediction
— R(N), actual prime-pair count
- - Ĝ(N), Hardy–Littlewood prediction
- - C(1) baseline
— E(N) error, when shown
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:
$$A(G) = \frac{2}{G^2}\sum_{g=1}^{G-1}(G-g)\cdot\frac{\varphi(g)}{g} \;\to\; \frac{6}{\pi^2}$$
Each offset g corresponds to two mirrored strips M−r=±g (hence the factor 2), each of length (G−g) and coprime density φ(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|