Analytic Number Theory · Wessen Getachew
C(n) — Block-Coprime Density
$$C(n)\;=\;\zeta(2)\cdot\prod_{p\;\text{prime}}\!\left(1-\frac{\min(n+1,\,p)}{p^2}\right)$$
C(n)
=
ζ(2) · ∏p(1 − min(n+1, p) / p²)
=
—
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.
G=100
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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.
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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.
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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
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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).
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p=
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Per-ring:
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p=
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:
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Grid G:
20
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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.
C(n) vs n
— C(n)
--- 6/π² ≈ 0.6079
| Saturation events
● n = current
Hover chart to inspect values…
Euler Product Convergence
Partial product CP(n) = ζ(2)·∏p≤P(1 − min(n+1, p)/p²) converging to C(n) as P ranges over primes.
Teal = active primes (p > n+1),
coral = saturated (p ≤ n+1).
Primes shown:
30
—
Current:
—
Target C(n):
—
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) |
|---|
Goldbach Calculator
Even N =
S(N) singular series
—
2C₂ · ∏p|N,p>2(p−1)/(p−2)
HL prediction G(N)
—
S(N)·N / (ln N)²
Actual pairs
—
p+q=N, p≤q, both prime
HL ratio actual/pred
—
→ 1 on average (unproved)
■ both prime
■ p prime only
■ q=N−p prime only
■ neither
Anti-diagonal p + q = N visualized
Additive Number Theory
The multiplicative → additive bridge:
C(1) ≈ 0.5307 from this tool is the same constant C that appears as the
baseline in E(N) = R(N) − Ĝ(N). The Euler product computes the prediction;
the additive error E(N) measures the residual, oscillating around zero with
amplitudes driven by Riemann zeros. Prime counting and prime gaps are the
purely additive companions to the multiplicative structure here.
R(N) — Goldbach Pair Count & Error E(N) = R(N) − Ĝ(N)
— R(N) actual
— Ĝ(N) = S(N)·N/(ln N)² (H–L)
— E(N) = R(N) − Ĝ(N)
- - C(1) ≈ 0.5307 baseline
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 ·