A Computational Framework for Boundary Cancellation Analysis
By Wessen Getachew | @7dview
Explore mathematically significant radii, dimensions, and configurations. Each preset demonstrates important number-theoretic properties.
Density 6/π² ≈ 0.6079
Euler's solution to ζ(2) = π²/6. Shows fundamental coprime probability.
ζ(3) ≈ 1.202, Density ≈ 0.832
3D primitive point density. Apéry proved ζ(3) is irrational.
Error Δ(R) = O(R^(2/3))
Classic lattice point problem. Connected to Riemann Hypothesis.
89th Fibonacci number (prime)
Intersection of Fibonacci sequence and primes.
2⁵ - 1 = 31 (prime)
Mersenne prime. Related to perfect numbers.
28 = 1+2+4+7+14
Second perfect number. Sum of divisors equals 2×28.
R ≈ 100φ ≈ 161.8
Golden ratio approximation. Appears in Fibonacci spirals.
Density ≈ 0.924
Four-dimensional sphere. Most points are primitive.
Density ≈ 0.996
Related to E₈ exceptional Lie group. Nearly all points primitive.
Density ≈ 0.9998
As k→∞, density→1. Almost all points are primitive.
φ(12) = 4, modular patterns
Rich divisor structure (2²×3). Patterns mod 12.
Forms field ℤ/17ℤ
All non-zero elements invertible. Field structure.
64 = 2⁶, binary structure
Pure power of 2. Dyadic patterns visible.
We develop a comprehensive study of primitive lattice points in Z^n, generalizing the classical 2-dimensional coprime lattice problem. Using geometric decomposition into primitive rays and the Möbius inversion formula, we rigorously derive the asymptotic density of coprime points in any dimension. The framework connects number theory, geometry of numbers, and classical constants like the Riemann zeta function, providing both intuition and formal proofs.
Counting lattice points inside a ball is a classical problem in number theory. Restricting to primitive points, i.e., points with coordinates coprime, leads to deep insights in analytic number theory and the geometry of numbers. In two dimensions, this relates to the Basel problem (ζ(2) = π²/6) and visible points from the origin. We extend the problem to arbitrary dimensions, n ≥ 2, developing both geometric intuition and rigorous algebraic derivation.
A vector v = (a₁, ..., aₙ) ∈ Z^n \ {0} is called primitive if
$$\gcd(a_1, \dots, a_n) = 1$$Every nonzero lattice point v ∈ Z^n lies on a unique primitive ray:
$$\mathbf{v} = k \mathbf{u}, \quad k \in \mathbb{N}, \ \mathbf{u} \text{ primitive}$$Let d = gcd(a₁, ..., aₙ). Then v = du, where u is primitive. Uniqueness follows from the uniqueness of the gcd.
Each primitive vector acts as the "gatekeeper" for the entire ray of lattice points extending from the origin.
Let C_n(R) denote the number of primitive points inside the n-dimensional ball of radius R:
For a primitive vector u, the lattice points along its ray are
Counting only the first point on each ray yields exactly C_n(R).
The density of primitive points in Z^n equals the probability that n integers are coprime:
Thus, the leading term for C_n(R) is
where
is the n-dimensional ball volume. The error term arises from the boundary: O(R^(n-1)).
The key insight is that non-primitive points (those sharing a common factor d > 1) can be expressed as d·(primitive point). Through Möbius inversion, we show that boundary contributions from non-primitive points cancel systematically, leaving only the volume term divided by ζ(k).
The Möbius function μ(d) = (-1)^ω(d) for square-free d (where ω counts distinct prime factors) ensures alternating cancellation of boundary terms.
This combines geometric intuition, rigorous number theory, and multi-dimensional generalization. It provides a unified framework for analyzing primitive lattice points in any dimension.
Case n=1: Only primitive points are ±1. ζ(1) diverges; density is zero.
2D Case (Classical): Density: 1/ζ(2) = 6/π² ≈ 0.6079. Corresponds to "visible" points in the plane.
3D Case: Density: 1/ζ(3) ≈ 0.832. Visualization: "particles in space," approximately 83% visible from the origin.
High Dimensions: As n → ∞, ζ(n) → 1, so almost all points are primitive.
Precision: 6 decimal places (k > 6 shows up to 17 decimals)
| k | ζ(k) | 1/ζ(k) | V_k (unit sphere) |
|---|
The 2D case directly connects to Euler's solution of the Basel problem:
$$\zeta(2) = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \cdots = \frac{\pi^2}{6}$$Thus the density of primitive points in Z² is exactly 6/π².
A lattice point (a, b) is visible from the origin if and only if gcd(a,b) = 1. The number of visible points in a circle of radius R is asymptotically:
$$V(R) \sim \frac{6}{\pi^2} \cdot \pi R^2 = \frac{6R^2}{\pi}$$Primitive lattice points in 2D correspond to reduced fractions. The Farey sequence F_n contains all fractions p/q in lowest terms with 0 ≤ p ≤ q ≤ n. The number of terms in F_n is:
$$|F_n| = 1 + \sum_{k=1}^{n} \phi(k)$$where φ(k) is Euler's totient function, which counts integers coprime to k.
The Möbius function μ(n) is defined as:
$$\mu(n) = \begin{cases} 1 & \text{if } n = 1 \\ (-1)^k & \text{if } n \text{ is a product of } k \text{ distinct primes} \\ 0 & \text{if } n \text{ has a squared prime factor} \end{cases}$$The key property for our application:
$$\sum_{d|n} \mu(d) = \begin{cases} 1 & \text{if } n=1 \\ 0 & \text{if } n>1 \end{cases}$$The error term O(R^(n-1)) can be understood as arising from the boundary of the ball. More precisely:
$$C_n(R) = \frac{\operatorname{Vol}(B_n(R))}{\zeta(n)} + E_n(R)$$where |E_n(R)| ≤ C · R^(n-1) for some constant C depending on n.
This error reflects the discrete nature of the lattice versus the continuous ball boundary.
Explore mathematical patterns and structures within primitive lattice point distributions. Interactive tools reveal modular patterns, density variations, and symmetry properties.
Analyze how primitive points distribute across residue classes modulo m. This reveals deep connections between coprimality and modular arithmetic.
Theoretical Background:
For modulus m and primitive points with gcd(x,y) = 1, the distribution across residue classes (x+y) mod m reveals patterns related to Euler's totient function φ(m). Residue classes coprime to m tend to have higher concentrations of primitive points.
Explore GCD patterns in the lattice. Click on any point to see detailed information about its coordinates and GCD value.
Click on a point to see details
GCD Patterns:
- Points with GCD = d form a scaled copy of the primitive lattice, scaled by factor d
- The number of points with GCD = d in radius R is approximately N₁(R/d) where N₁ is the primitive count
- GCD values reveal the arithmetic structure of the lattice - patterns repeat at each GCD level
Explore various algebraic structures and patterns in modular arithmetic. Choose different table types to visualize GCD, addition, multiplication, and structural properties.
Rotate to view table from different quadrants
Click on a cell to see GCD(row, col) details
Patterns to Observe:
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About Grid Rotation:
Rotating the table by 90° increments reveals different symmetries and patterns:
• 0°: Standard view - rows increase downward, columns increase rightward
• 90°: Rotated clockwise - observe vertical symmetry patterns
• 180°: Upside down - reveals inverse patterns and reflections
• 270°: Rotated counter-clockwise - horizontal symmetry emphasis
Perfect for analyzing table structure from all four quadrants!
Analyzes the error term Δ(R) from the Gauss Circle Problem and its connection to the Riemann Hypothesis. The growth rate of |Δ(R)| is deeply connected to the distribution of zeros of the Riemann zeta function.
Computing error analysis...
Click on a point to see detailed error information
Gauss Circle Problem:
Let N(R) = |{(x,y) ∈ ℤ² : x² + y² ≤ R²}| be the number of lattice points in a circle of radius R.
We have N(R) = πR² + Δ(R), where Δ(R) is the error term.
Connection to Riemann Hypothesis:
- Known: |Δ(R)| = O(R^(2/3)) (classical result by van der Corput, 1923)
- Conjectured (Hardy, 1915): |Δ(R)| = O(R^(1/2 + ε)) for any ε > 0
- If RH is true: This bound is expected to hold, though not yet proven
- Lower bound: |Δ(R)| = Ω(R^(1/4)) (Sierpiński, 1906)
For Primitive Points:
N_prim(R) = πR²/ζ(2) + Δ_prim(R), with similar error behavior scaled by 1/ζ(2) ≈ 0.608
Key Observations:
- The error oscillates around zero with increasing amplitude
- Normalized error Δ(R)/R^(1/2) should be bounded if RH is true
- The oscillation frequency relates to the imaginary parts of zeta zeros
Computing lattice points...
Generating 3D lattice...
Compare how primitive lattice point counts change across dimensions for a fixed radius, or analyze how different radii affect each dimension.
Dimensions to compare:
Display options:
Key Observations:
- As dimension k increases, ζ(k) approaches 1, meaning the density 1/ζ(k) approaches 100%
- Higher dimensions have exponentially more lattice points (grows as R^k)
- The ratio of primitive to total points stabilizes at 1/ζ(k) for each dimension
- Sphere volume grows dramatically with dimension, following the gamma function pattern
Explore how the boundary term R^(k-1) becomes negligible compared to the main term R^k as dimension increases.
| k | Boundary Geometry | Density 1/ζ(k) | Stability |
|---|---|---|---|
| 2 | Perimeter (R¹) | 0.6079 | Baseline |
| 3 | Surface Area (R²) | 0.8319 | High |
| 4 | 3D Hypersurface (R³) | 0.9239 | Very High |
| 5 | 4D Hypersurface (R⁴) | 0.9644 | Extreme |
| 6 | 5D Hypersurface (R⁵) | 0.9829 | Extreme |
| 8 | 7D Hypersurface (R⁷) | 0.9959 | Extreme+ |
| 10 | 9D Hypersurface (R⁹) | 0.9990 | Approaching 1 |
| 12 | 11D Hypersurface (R¹¹) | 0.9998 | Nearly 1 |
Key Insight: As k increases, the relative error (Boundary/Main) = R^(k-1)/R^k = 1/R decreases.
For fixed R, higher dimensions have proportionally smaller boundary effects, leading to more stable density convergence to 1/ζ(k).
Visualization of the error term Δ(R) = N(R) - R^k/ζ(k) for every single radius value. The chart shows how actual counts differ from predictions.
Computation Info:
- k=2: Exact computation up to R=200, then uses prediction
- k=3: Exact computation up to R=40, then uses prediction
- k=4: Exact computation up to R=15, then uses prediction
- k=5: Uses prediction (computational limit)
Note: Computes for every single R value (no skipping)
Computing error analysis for all R values...
Click on chart points or use "Analyze Specific R" button to see detailed analysis
Error Bound Theory:
The error term O(R^(k-1)) arises from boundary effects. For k=2, the error is O(R), corresponding to points near the circle boundary. As k increases, the error term becomes relatively smaller compared to the main term R^k.
Relative Error: Shows |Predicted - Actual| / Predicted as a percentage. This decreases as R grows, confirming the asymptotic accuracy.
Running Average: The cumulative average of absolute errors up to each R. Shows convergence behavior.
Boundary Term: The theoretical O(R^(k-1)) term, showing the expected growth rate of the error.
Key Feature: This analysis computes for every single R value from start to max (no skipping), giving you the complete error landscape.
Computational verification of the theoretical predictions for dimension k=2:
| R | N(R) Actual | R²/ζ(2) Predicted | Δ(R) Error | Δ(R)/(R log R) |
|---|---|---|---|---|
| 10 | 63 | 60.79 | 2.21 | 0.096 |
| 50 | 1,519 | 1,519.7 | -0.7 | -0.008 |
| 100 | 6,087 | 6,079.3 | 7.7 | 0.017 |
| 500 | 151,983 | 151,982.5 | 0.5 | 0.0002 |
| 1000 | 607,926 | 607,927.0 | -1.0 | -0.0001 |
Observations:
- The error Δ(R) oscillates around zero, confirming the prediction is unbiased
- The normalized error Δ(R)/(R log R) decreases as R increases
- Even at R=1000, the error is less than 0.0002% of the predicted value
- This validates the asymptotic formula N(R) = R²/ζ(2) + O(R log R) for k=2
Explore the complete collection of interactive number theory and lattice point visualizations by Wessen Getachew. Each project focuses on a specific mathematical concept with deep interactive exploration.
Analysis of arithmetic lattice residues and the Möbius inversion principle.
Visit ProjectComplete discovery engine for GCD patterns and boundary cancellation principles.
Visit ProjectInteractive exploration of Goldbach's conjecture and twin prime patterns.
Visit ProjectVisualization of Farey sequences, Ford circles, and Stern-Brocot trees.
Visit ProjectComplete analysis of Pythagorean triples and right triangle geometry.
Visit ProjectAncient multiplication algorithms and binary number patterns.
Visit ProjectAll visualizations are created by Wessen Getachew (@7dview) as part of an ongoing research initiative into the geometric and visual aspects of number theory.
Each project combines:
- Rigorous mathematical theory
- Interactive web-based visualizations
- High-performance computational algorithms
- Educational accessibility
- Professional-grade export capabilities
These tools are designed for mathematicians, researchers, educators, and anyone curious about the hidden patterns in numbers.
Jump to any visualization category:
Create animations by sweeping through radii, dimensions, or moduli. Perfect for presentations and pattern exploration.
Animation Types:
• Radius Sweep: Watch primitive lattice points grow as radius increases from start to end
• Dimension Sweep: See how density approaches 100% as dimension k increases
• Modular Pattern: Observe how residue class distributions change with modulus
Export high-resolution images with custom titles, subtitles, and legends. Choose resolution and canvas to export.
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Export Features:
- High-resolution output (up to 8K: 7680×4320)
- Custom titles, subtitles, and legends
- Automatic parameter documentation
- Clean layout without overlap
- Professional formatting for publications
- Timestamp and attribution options
- Configurable decimal precision (2-17 places)
- Full precision data preservation
CSV Export Features:
- Descriptive headers with metadata
- ISO 8601 timestamps
- Full attribution and licensing info
- Up to 17 decimal places for high-k dimensions
- Comment lines (# prefix) for documentation
- Automatic precision selection based on dimension
The Farey sequence F_n contains all reduced fractions p/q where 0 ≤ p ≤ q ≤ n. These correspond to primitive lattice points and visible points from the origin.
Farey Sequence Properties:
• F_n = {p/q : 0 ≤ p ≤ q ≤ n, gcd(p,q) = 1}
• |F_n| = 1 + Σ_{k=1}^n φ(k) where φ is Euler's totient
• Mediant property: If p/q and r/s are consecutive in F_n, then (p+r)/(q+s) appears in F_{q+s}
• Connection to continued fractions and best rational approximations
• Ford circles: Circle at p/q with radius 1/(2q²) - consecutive circles are tangent
Save your current parameters and visualization state to continue your research later or share configurations with others.
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What Gets Saved:
• All radius and dimension parameters
• Color modes and visualization settings
• Active tab and display preferences
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• Label and zoom settings
Note: Sessions are stored in your browser's local storage and persist between visits.
Features from the roadmap have been implemented throughout the platform. See the Farey Sequences tab and Save/Load Session tab for newly added functionality.
Start with all lattice points in a k-dimensional sphere of radius R:
Asymptotically, this equals the volume: L_k(R) ~ V_k R^k
Partition points by their greatest common divisor d:
Where N_k(R/d) counts primitive points in a sphere of radius R/d.
Inverting the previous relation using Möbius function:
Substituting L_k(R/d) ~ V_k (R/d)^k:
The sum equals the reciprocal of the zeta function:
This elegant formula connects geometry (V_k), analysis (ζ(k)), and number theory (primitive points)!
Watch the derivation unfold step-by-step, showing how Möbius inversion leads to the zeta function connection.
Step 1 of 7: Introduction
We begin by counting all lattice points (x,y) where x² + y² ≤ R². This gives us L_k(R), the total count including both primitive and non-primitive points.
Hardy, G. H., & Wright, E. M. (1979). An Introduction to the Theory of Numbers. Oxford University Press.
Classical proofs for 6/π² density and fundamental results in coprimality.
Dirichlet, G. L. (1849). Über die Bestimmung der mittleren Werthe in der Zahlentheorie.
Foundational work on lattice point problems and average values in number theory.
Mertens, F. (1874). Über einige asymptotische Gesetze der Zahlentheorie.
Refinement of error bounds in arithmetic sums and the Mertens function.
Titchmarsh, E. C. (1986). The Theory of the Riemann Zeta-Function. Oxford Science Publications.
Connection between error terms and the Riemann Hypothesis.
Cesàro, E. (1883). Probabilité de certains faits arithmétiques.
Early work on the probability interpretation of 6/π².
Pillai, S. S., & Chowla, S. (1930). On the Error Terms in Some Asymptotic Formulae.
Analysis of k=2 error bounds and refinements.
van der Corput, J. G. (1923). Zahlentheoretische Abschätzungen.
Classical O(R^(2/3)) bound for the Gauss circle problem.
Further Reading:
- Apostol, T. M. (1976). Introduction to Analytic Number Theory. Springer-Verlag.
- Nathanson, M. B. (1996). Additive Number Theory: The Classical Bases. Springer.
- Iwaniec, H., & Kowalski, E. (2004). Analytic Number Theory. American Mathematical Society.