Prime-Phase Vector Visualization Tool

Parameter Space Explorer

Automatically scan parameter ranges to discover patterns, find zeros, and visualize the mathematical landscape of the prime-phase vector function.

Heatmap Visualization

Prime Gap Decomposition of ζ(2) = π²/6

By Wessen Getachew, 2025

Theoretical Foundation

Using the Phase Law, ζ(2) can be decomposed into products of prime gap families:

Method

Each P_g is the product over primes in that gap family:

Where the "smallest gap family" is the difference to the next prime. For example:

• Gap 0: Prime 2 only (no next prime with gap 0)
• Gap 2: Twin primes (3,5), (5,7), (11,13), (17,19)...
• Gap 4: Cousin primes (7,11), (13,17), (19,23)...
• Gap 6: Sexy primes (5,11), (7,13), (11,17), (13,19)...

Theoretical Results

Progressive Convergence

Computational Analysis

Run tests to verify the prime gap decomposition with actual prime data:

Run Gap Analysis

OEIS Sequence Integration

Replace prime numbers with any integer sequence from the Online Encyclopedia of Integer Sequences to explore mathematical patterns across different number families.

Available Sequences

The tool includes 18 pre-loaded OEIS sequences for offline use:

How to Use OEIS Sequences

Interpreting Results

When using OEIS sequences instead of primes, the visualization reveals how different mathematical structures interact with modular arithmetic and phase relationships. Key observations:

• Regular sequences (powers of 2, squares) create highly symmetric patterns
• Recursive sequences (Fibonacci, Lucas) show spiral-like growth patterns
• Prime-related sequences (twin primes, Mersenne primes) display irregular but constrained behavior
• Combinatorial sequences (factorials, Catalan) exhibit rapid divergence

Research Applications

OEIS integration enables exploration of:

• Modular distribution patterns across different number families
• Phase interference effects in non-prime sequences
• Convergence behavior of vector sums for various mathematical structures
• Comparative analysis between prime and non-prime number patterns

Mathematical Framework

Parameter Definitions

• M (Modulus Ring): The modular arithmetic base defining the residue class structure. Range: 1 to 10,000. When M=1, all terms map to the same point (complete constructive interference). For M=2, terms alternate between two positions creating maximal structure. Larger M creates denser phase distributions. Special case: M=1 corresponds to gcd(p,1)=1 for all p, placing all vectors at angle 2πγ/1 = 2πγ (a single point on the unit circle).
• t (Critical Line Height): The imaginary component of the Riemann zeta function argument s = 1/2 + it. Range: 0 to 1,000. When t corresponds to a Riemann zero, special cancellation patterns emerge. The tool includes 1,000 precomputed Riemann zeros.
• X (Prime Cutoff): The upper bound for prime summation. Range: 20 to 100,000. Larger values provide more complete vector sums but require more computational resources.
• α (Phase Multiplier): Scaling factor for the critical line phase component. Range: 0 to 100. Standard value is 1, but variations reveal phase sensitivity.
• β (Magnitude Exponent): Power law exponent controlling magnitude decay. Range: 0 to 5. The classical value β = 1/2 corresponds to the critical line of ζ(s).
• γ (Modular Weight): Coefficient controlling the influence of modular phase structure relative to critical line dynamics. Range: 0 to 100.

Vector Construction

For each prime p ≤ X (or term in custom sequence), we construct a complex vector with two phase components:

• Modular Phase: θ_mod = 2πγp/M, creating a discrete rotational structure based on residue classes. For M=1, all terms map to angle 2πγ (complete alignment). For M=2, terms alternate between 0 and πγ. For gcd(p,M)=1, each coprime p creates a unique phase position.
• Critical Phase: θ_crit = -αt·ln(p), encoding the logarithmic growth structure tied to the Riemann zeta function.
• Magnitude: |v_p| = 1/p^β, implementing a power law decay matching the analytic structure of L-functions.

Special Cases:

• M=1: All vectors point in the same direction → maximum constructive interference
• M=2: Binary phase structure → alternating pattern between even/odd residues
• M=p (prime): Creates p-fold rotational symmetry with φ(p)=p-1 coprime positions
• Large M: Approaches uniform phase distribution on the unit circle

Statistical Metrics

The tool computes real-time diagnostic statistics to quantify vector behavior:

• Total Magnitude |P_t|: The norm of the final cumulative vector, measuring net constructive interference.
• Coherence Score: C_t = 1 - (|P_t| / Σ|v_p|), quantifying destructive interference. Values near 1 indicate strong cancellation.
• Path Length: Σ|v_p|, the total distance traversed by the cumulative vector tip through the complex plane.
• Efficiency Ratio: |P_t| / Σ|v_p|, the ratio of direct distance to total path length. Low values indicate circuitous paths with cancellation.
• Maximum Excursion: The furthest distance the cumulative vector reaches from the origin during summation.
• Winding Number: The total angular rotation of the cumulative vector divided by 2π. Non-integer values reveal complex phase dynamics.
• Cancellation Index: 1 - (|P_t| / Σ|v_p|), an alternative measure of interference strength.

Research Applications

This tool enables investigation of several key mathematical questions:

• Riemann Hypothesis Connection: By setting t to known Riemann zeros and exploring parameter space, we can identify modular structures that exhibit enhanced cancellation, potentially revealing algebraic or geometric properties of the zeros.
• Prime Distribution Patterns: The cumulative vector path visualizes how prime residue patterns interact with logarithmic phase spacing, offering geometric insight into number-theoretic phenomena.
• Modular-Analytic Correspondence: The interplay between discrete modular structure (M, γ) and continuous analytic structure (t, α, β) may illuminate connections between algebraic number theory and complex analysis.
• Phase Transition Detection: Monitoring statistics like coherence and winding number as parameters vary can reveal critical thresholds where qualitative behavior changes.

Interactive Features

• Real-Time Visualization: All three canvases (main vector space, phase space, magnitude history) update dynamically as parameters change.
• Riemann Zero Presets: One-click navigation to the first 1,000 non-trivial Riemann zeros for immediate investigation of zero-correlated behavior.
• Animation Mode: Automated scanning through t-parameter space with adjustable speed, enabling observation of continuous transitions.
• Adaptive Legends: Each canvas includes a dynamic legend displaying current parameters and key statistics without visual overlap.
• Zoom and Pan: Mouse wheel zoom and click-drag panning for detailed examination of vector structures.
• Color Customization: Full control over vector path, grid, result vector, and background colors for optimal visual clarity and presentation.
• High-Resolution Export: PNG export at 4K resolution (3840×2160) for publication-quality figures, plus SVG for vector graphics and CSV for numerical data.
• OEIS Integration: Load custom integer sequences from the Online Encyclopedia of Integer Sequences to replace prime numbers with other mathematically significant sequences.

Credits & Attribution

Primary Developer

Wessen Getachew (@7dview)

• Twitter/X: @7dview
• GitHub: wessengetachew.github.io

Independent mathematical researcher and software developer specializing in interactive visualization tools for advanced number theory concepts. Self-taught in complex mathematical domains including Riemann Hypothesis exploration, prime distribution theory, and modular arithmetic structures.

Mathematical Foundations

Riemann Zeta Function

The Riemann zeta function ζ(s) and its connection to prime numbers forms the theoretical foundation of this tool. The non-trivial zeros of ζ(s) lie on the critical line Re(s) = 1/2, a conjecture known as the Riemann Hypothesis.

• Riemann, Bernhard (1859): "Über die Anzahl der Primzahlen unter einer gegebenen Größe" - The foundational paper introducing the zeta function and its connection to primes.
• Zero Data Source: The first 1,000 non-trivial zeros are precomputed from established mathematical tables and verified computational sources.

Modular Arithmetic & Number Theory

• Farey Sequences: Connections to rational approximations and continued fractions.
• Dirichlet Characters: Modular structures related to L-functions and prime distribution in arithmetic progressions.
• Euler's Totient Function: φ(n) counting coprime integers, fundamental to modular arithmetic analysis.

Complex Analysis

Vector summation in the complex plane provides geometric interpretation of analytic number theory:

• Euler Product Formula: Connection between primes and zeta function values.
• Phase Space Analysis: Geometric representation of complex function behavior.
• Winding Number Theory: Topological invariants in complex analysis.

Technical Implementation

Algorithms & Computation

• Sieve of Eratosthenes: Classical algorithm for efficient prime number generation up to 100,000.
• Complex Vector Summation: Real-time computation of cumulative vector paths with phase and magnitude tracking.
• Statistical Analysis: Coherence metrics, winding numbers, and efficiency ratios computed on-the-fly.

Visualization Technologies

• HTML5 Canvas: High-performance 2D graphics rendering for complex visualizations.
• JavaScript ES6+: Modern web technologies enabling real-time parameter manipulation.
• Adaptive Legends: Dynamic positioning system to prevent visual overlap with data.
• Export Capabilities: PNG (4K resolution), SVG (vector graphics), and CSV (numerical data).

OEIS Integration & Custom Sequences

The tool supports multiple ways to load integer sequences:

• Built-in Sequences: 18 hardcoded sequences available offline, including primes, Fibonacci, powers, polygonal numbers, factorials, and Catalan numbers. These work without internet connection.
• Custom File Upload: Upload your own sequences in three formats:
  • Text file (.txt): One number per line or space-separated
  • CSV file (.csv): Comma-separated values
  • JSON file (.json): Array format: [1, 2, 3, 5, 8, ...]
• Example sequence files you could create:
  • Your own research data
  • Experimental measurements
  • Custom mathematical sequences
  • Filtered prime subsets
  • Any positive integer sequence up to 10,000 terms

When uploading custom sequences, the filename becomes the sequence name displayed in visualizations. Ensure all values are positive integers.

Research Context

Related Projects

This tool is part of a broader research program exploring geometric and algebraic approaches to classical number theory problems:

• Farey Triangle & Cayley Transform: Hyperbolic geometry and conformal mappings in number theory.
• Modular Rings Visualization: Geometric representation of residue class structures.
• Interactive Modular Lifting Rings: Exploring Riemann Hypothesis connections through modular arithmetic.
• Modular Sieve Calculator: Computing π and ζ(2n) using Euler product decompositions.

Educational Mission

These visualization tools serve dual purposes: advancing mathematical research through computational exploration and making complex concepts accessible to students and researchers. By bridging abstract theory with intuitive geometric representation, the tools facilitate deeper understanding of fundamental mathematical structures.

License & Usage

This visualization tool is created for mathematical research and educational purposes. Users are encouraged to:

• Explore parameter spaces to discover novel patterns
• Export high-resolution visualizations for academic presentations
• Share findings with the mathematical community
• Provide feedback and suggestions for improvements

Citation: When using this tool in academic work, please credit Wessen Getachew (@7dview) and reference the tool's GitHub repository.

Acknowledgments

Development of this tool benefited from:

• The global mathematics research community and publicly available mathematical datasets
• Open-source software libraries and web standards enabling sophisticated browser-based computation
• Historical mathematical work spanning centuries, from Euler to Riemann to contemporary researchers
• The Online Encyclopedia of Integer Sequences (OEIS) for providing comprehensive sequence data

Version: 1.0 | Last Updated: 2025
For questions, collaborations, or bug reports, contact via Twitter/X @7dview