Coordinate System Layout
Current System:
Traditional polar coordinates with concentric rings
Traditional polar coordinates with concentric rings
Global Scale Control
Scales entire system:
• Ring radii and spacing
• Point sizes and line thickness
• Labels and annotations
• Prime-phase field vectors
Use for better visibility or to fit more rings on screen
• Ring radii and spacing
• Point sizes and line thickness
• Labels and annotations
• Prime-phase field vectors
Use for better visibility or to fit more rings on screen
Riemann Hypothesis Research Hub
Enable Alignment Analysis
Enable Multi-Point Measurement
Farey order N:
Show Farey Discrepancy Plot
Show Franel-Landau Test
Franel-Landau Theorem: RH is equivalent to
D(N) = O(N^(1/2+ε))
where D(N) measures discrepancy from uniform spacing
D(N) = O(N^(1/2+ε))
where D(N) measures discrepancy from uniform spacing
Enable RH Predictions
Show Dirichlet Character Support
Visualize L-function Critical Line
Test Prime Equidistribution
Critical Line Analysis
t-range: to
Character Support Domains
Max rings:
Prime Distribution
Uses active sieve data
Zeta Function Zeros
Show zeros:
Prime Counting π(x)
π(x) limit:
Residue Heatmap
Uses active sieve data
Chi-Squared Test
Significance:
Modulus Comparison
Display:
L-Function Magnitude
L-Function Parameters:
Type:
Character χ mod:
index:
Critical line σ=1/2, t: to
Dirichlet series terms:
More terms = better accuracy, slower computation
More terms = better accuracy, slower computation
Current: L(s,χ₀) with χ₀ mod 30 (principal character)
Test Range:
to
About Conjecture Testing:
Automatically verify famous unsolved problems up to computational limits. Tests generate verification certificates with evidence strength and statistical analysis.
Automatically verify famous unsolved problems up to computational limits. Tests generate verification certificates with evidence strength and statistical analysis.
Test modulus:
Critical Line t-range: to
Zeta Zeros Count:
π(x) limit:
Search up to:
Using modulus:
Mathematical Interpretation:
Uniform spacing places rings at equal intervals regardless of modulus value.
Uniform spacing places rings at equal intervals regardless of modulus value.
Gap Analysis
Max Gap:
Add Consecutive Range:
From: To:
From: To:
Modular Ring System
Multi-Select Moduli:
Single Modulus:
Consecutive Range:
From: To:
From: To:
By Mathematical Property:
Up to:
Up to:
Quick Selection Examples:
• Primes: 2,3,5,7,11,13,17,19,23,29
• Powers of 2: 2,4,8,16,32,64
• Fibonacci: 1,2,3,5,8,13,21,34
• Custom: Any combination you want!
• Primes: 2,3,5,7,11,13,17,19,23,29
• Powers of 2: 2,4,8,16,32,64
• Fibonacci: 1,2,3,5,8,13,21,34
• Custom: Any combination you want!
Visual Customization
Smith Chart Transform (Cayley Map)
Enable Smith Chart Transform
3D Rotation Panel
Or enter exact degree:
Enable Perspective Projection
Rotation Controls:
• Main Degree: Overall rotation angle
• X-axis: Pitch (tilt forward/backward)
• Y-axis: Yaw (turn left/right)
• Z-axis: Roll (rotate clockwise/CCW)
• Each ring can rotate independently
• Main Degree: Overall rotation angle
• X-axis: Pitch (tilt forward/backward)
• Y-axis: Yaw (turn left/right)
• Z-axis: Roll (rotate clockwise/CCW)
• Each ring can rotate independently
Advanced Animation Engine
Enable Individual Ring Rotation
Invert Colors (Dark/Light Mode)
Reverse Rotation Direction
Enable Nesting Inversion
Enable Ring Inversion
Label Controls
Show Residue Labels
Label Unit Circle
Label GCD=1 Residues
Label GCD≠1 Residues
Show gcd(r,M) value
Show ✓/✗ coprime indicator
Show position in φ(M) sequence
Label Specific Moduli:
Moduli:
Moduli:
Label Consecutive Range:
From M: To M:
From M: To M:
Min Ring Radius:
Show Prime Counts (when sieve active)
GCD Analysis & Visualization
Highlight GCD=1 boundary
Show GCD structure connections
Animate GCD=1 residues
Show φ(M) count per ring
Show GCD distribution chart
Compare totient densities
Lift Controls
Direct Lifts (r → r, consecutive only)
Modular Lifts (r → r + M×2ⁿ, consecutive only)
Skip-Level Direct Lifts (r → r, all combinations)
Skip-Level Modular Lifts (r → r + M×2ⁿ, all combinations)
Display Controls
Highlight Unit Circle
Dirichlet Character Controls
Show χ(r) ≠ 0 (Character Support)
Show χ(r) = 0 (Character Vanishing)
Highlight Character Support Difference
Mathematical Note: gcd(r,M) = 1 ⟺ χ(r) ≠ 0
Character support determines L-function behavior
Character support determines L-function behavior
Ready
View Controls
Standard
4K
Black Background
White Background
Include Title & Subtitle
Include Legend
WebM (High Quality)
MP4 (Compatible)
GIF (Optimized)
Navigation:
• Click & drag to pan
• Mouse wheel to zoom
• Double-click to reset center
• Click & drag to pan
• Mouse wheel to zoom
• Double-click to reset center
Riemann Hypothesis Visual Analysis
Critical Line Analysis
Re(s) = 1/2 visualization for L(s,χ)
Character Support Domains
χ(r) ≠ 0 regions across moduli
Prime Equidistribution
GRH uniformity predictions
Farey Discrepancy
Franel-Landau RH equivalence
Franel-Landau Test
D(N) = O(N^(1/2+ε)) verification
Zeta Function Zeros
First non-trivial zeros on critical line
Prime Counting Function π(x)
π(x) vs Li(x) comparison
Residue Class Heatmap
Prime density visualization
Chi-Squared Test
Uniformity statistical test
Multi-Modulus Comparison
φ(M) across selected rings
L-Function Magnitude |L(s,χ)|
Along critical line approximation