GCD

The Boundary Cancellation Principle

A Discovery Engine for Arithmetic Lattice Residues and Riemann Hypothesis Connections

Wessen Getachew

Twitter: @7dview

Abstract

In the limit as \(R \to \infty\), the density of coprime \(k\)-tuples in an integer lattice \(\mathbb{Z}^k\) converges to \(1/\zeta(k)\). However, for any finite search radius \(R\), a residual error term \(\Delta(R)\) exists. This discovery engine identifies this error not as stochastic noise, but as a deterministic geometric residue. We demonstrate that \(\Delta(R)\) is a function of the \((k-1)\) boundary shell of the \(k\)-cube, where the Möbius inversion engine is truncated by the finite search radius \(R\).

Key Discoveries

Geometric Origin

The error term \(\Delta(R)\) is structurally concentrated at the truncation limits of the lattice, not randomly distributed.

Boundary Cancellation

The Möbius function's cancellation cycle is incomplete at boundary divisors, creating a deterministic residue.

Dimensional Stability

Relative error decreases as \(k\) increases: \(\Delta(R)/R^k \to 0\) faster for higher dimensions.

RH Connection

The structure of \(\Delta(R)\) is governed by oscillations of \(\mu(n)\), linking directly to the Riemann Hypothesis.

Mathematical Foundation

Density Identity

\[ P(k) = \frac{1}{\zeta(k)} \]

Probability that \(k\) randomly chosen integers are coprime.

Counting Function

\[ N(R) = \sum_{d=1}^{R} \mu(d) \left\lfloor \frac{R}{d} \right\rfloor^k \]

Exact count of coprime \(k\)-tuples in \([1, R]^k\).

Error Term

\[ \Delta(R) = N(R) - \frac{R^k}{\zeta(k)} \]

Boundary residue from incomplete Möbius cancellation.

2D Lattice Visualization

Click on points to inspect GCD values

Coprime Points (GCD = 1)

Non-coprime Points

Boundary Points

Origin (1,1)

2D Coprime Lattice (k=2)

Selected Point

Lattice Parameters

Radius \(R\): 50

Point Size: 10px

Visual Effects

Grid Opacity: 0.3

Show Grid Highlight Boundary Animate

Statistics

2500

Total Points

1519

Coprime Points

-0.70

Error Δ(R)

196

Boundary Points

3D Lattice Visualization

Drag to rotate, scroll to zoom. Colors indicate GCD.

3D Coprime Lattice (k=3)

3D Parameters

Radius \(R\): 15

Cube Size: 0.7

Auto Rotate Show Axes

Point Inspector

Point (1, 1, 1): GCD = 1, Coprime = Yes

3D Statistics

3375

Total Points

2805

Coprime Points

-0.42

Error Δ(R)

1256

Boundary Points

Error Analysis Chart

Visualization of Δ(R) across dimensions and radii

Error Term Δ(R) Analysis

Chart Parameters

Max Radius: 200

Step Size: 10

Absolute Error Relative Error Running Average

Dimensions

k = 2 (Δ(R) ∝ R log R) k = 3 (Δ(R) ∝ R²) k = 4 (Δ(R) ∝ R³) k = 5 (Δ(R) ∝ R⁴)

Point Analysis

k=2 k=3 k=4 k=5

For R=100, k=2: N(R)=6087, Expected=6079.30, Δ(R)=7.70

Real-time Diagnostics

Live analysis of boundary cancellation patterns

Möbius Wave Analysis

M(R) = Σ μ(d) from d=1 to R

Wave Intensity: 1.0

Show Möbius Wave Show Cumulative Sum

Growth Exponent

α in Δ(R) ∝ R^α

-

R² = -

Boundary Self-Correction

Test: lim(R→∞) (1/R) Σ Δ(i) → 0?

-

Running Avg

-

Autocorr

GCD Calculator

Compute greatest common divisors with prime factorization

Input Numbers

Result

GCD(12, 18, 24) = 6

Prime factorization:
12 = 2² × 3
18 = 2 × 3²
24 = 2³ × 3

Properties

6

GCD

72

LCM

No

Coprime?

4

# of Factors

Number Theory Tools

Additional utilities for number analysis

Möbius Function

μ(12) = 0

Prime Test

17 is prime

Euler's Totient

φ(12) = 4

Export Data & Visualizations

Save high-resolution images and datasets for publication

Ready to export. Select an option above.

Export Settings

Image Settings

Image Quality: 100%

Include Watermark Include Timestamp Include Metadata

Data Settings

Precision: 6 decimals

Include Formulas Include Statistics

Batch Export

Research Mode: Diagnostic Overlays

Möbius Wave Overlay

Visualizes the cumulative sum of μ(d) (Mertens function) to reveal cancellation patterns.

M(R) = Σ μ(d) = 0

Oscillation Range: [0, 0]

Log-Log Scaling & Exponent Hunter

Calculates growth exponent α in Δ(R) ∝ R^α using linear regression on log-log plot.

Growth Exponent α

-

CI: ±-, R² = -

Critical Strip Projection

Maps Δ(R) onto complex plane to test Riemann Hypothesis predictions.

Point: Re = 0, Im = 0

Distance from ½-line: -

High-Precision Arithmetic

Compares 64-bit floating point with arbitrary precision Decimal.js calculations.

Standard: -

High-Precision: -

Relative Error: -

Research Console

[System] Research Mode Initialized [System] Diagnostic overlays ready [System] High-precision arithmetic enabled

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