Coprime Lattice & Riemann Hypothesis Explorer

Welcome to the Coprime Lattice Explorer

This tool provides an interactive visualization of one of the most beautiful connections in mathematics: the relationship between coprime integer pairs (primitive lattice vectors) and the Riemann Hypothesis, one of the most important unsolved problems in mathematics.

This explorer focuses on primitive lattice vectors - points (a,b) where gcd(a,b) = 1, also known as "visible points" from the origin. These points form the building blocks of the integer lattice and have profound connections to Gaussian integers, Fermat's Two-Square Theorem, and the distribution of prime numbers.

What Are Coprime Numbers?

Two integers a and b are coprime (or relatively prime) if their greatest common divisor (GCD) is 1. In other words, they share no common factors except 1. These pairs form primitive lattice vectors that are "visible" from the origin when you imagine standing at (0,0) and looking through the lattice.

Gaussian Integers & Norm

Each lattice point (a,b) corresponds to a Gaussian integer z = a + bi. The norm is N(z) = a² + b², which measures the squared distance from the origin. Fermat's Two-Square Theorem states that a prime p can be written as a sum of two squares (p = a² + b²) if and only if p ≡ 1 (mod 4) or p = 2.

A fundamental result in number theory states that the probability that two randomly chosen integers are coprime is 6/π² ≈ 0.607927. This remarkable constant connects the geometry of circles (π) with the distribution of prime numbers!

The Riemann Zeta Function

The Riemann zeta function ζ(s) is defined for complex numbers s with Re(s) > 1 as:

This function has a deep connection to prime numbers through the Euler product formula:

The reciprocal of ζ(s) can be expressed using the Möbius function μ(n):

where μ(n) = 1 if n is a product of an even number of distinct primes, -1 if n is a product of an odd number of distinct primes, and 0 if n has a squared prime factor.

The Riemann Hypothesis

The Riemann Hypothesis (RH) is one of the seven Millennium Prize Problems, with a $1 million prize for its proof. It states:

In other words, if ζ(s) = 0 and s is not a negative even integer, then s = 1/2 + it for some real number t.

Connection to Coprime Lattice Points

Let V(R) be the number of coprime lattice points (a, b) inside a disc of radius R centered at the origin. The Gauss Circle Problem and related theory tell us:

where E(R) is the error term. The exponent in the error bound is directly related to the zeros of ζ(s):

This means that how coprime lattice points distribute in a circular region is fundamentally connected to where the zeros of ζ(s) lie in the complex plane.

Tool Features

How to Use This Tool

1. Basic Exploration

• Click the "Explorer" tab above to access the interactive tool
• Start with a moderate grid size (500×500) and click "Generate Visualization"
• Green/white points are coprime pairs; other colors indicate non-coprime pairs
• Observe the density and distribution patterns

2. Analyzing the Riemann Hypothesis Connection

• Enable "Show Radius Circle" in the Visualization section
• Adjust the circle radius R to different values
• Check the "Disc Analysis" statistics to see V(R), E(R), and |E(R)|/R^(1/2)
• The ratio |E(R)|/R^(1/2) should remain bounded if RH is true

3. Detailed Point Inspection

• Click on any point in the visualization
• The "Point Inspector" panel shows: coordinates, GCD, polar form, modular properties
• Compare coprime vs non-coprime points to understand the patterns

4. Zooming and Labeling

• Use the Zoom slider to magnify regions (up to 20x)
• Pan using the X and Y offset sliders (or drag the canvas)
• Enable "Show Point Labels" and choose a label type (GCD, fractions, polar, norm)
• Choose label style: Overlay (on points), Replace (instead of points), or Offset (above points)
• Adjust font size and threshold for optimal visibility
• Enable "Label Coprime Only" to focus on primitive vectors

5. Statistical Analysis

• Check "Global Statistics" for overall coprime density (should approach 6/π² ≈ 0.6079)
• Review "Möbius Function Analysis" to see Σμ(n)/n² converging to 6/π²
• Export statistics as JSON for further analysis
• Comprehensive exports: CSV includes 23 properties per point (polar, Gaussian, Farey, modular)

Mathematical Background

Why 6/π²?

The probability that gcd(a,b) = 1 equals 6/π² because:

Key Mathematical Results

The Error Term and RH

If we count coprime points in a disc, the main term is (6/π²)πR² = 6R²/π. The error E(R) measures the deviation from this expectation. Better bounds on E(R) are equivalent to information about ζ(s) zeros:

• E(R) = O(R) is easy to prove (trivial bound)
• E(R) = O(R^(2/3)) can be proven unconditionally
• E(R) = O(R^(1/2 + ε)) is equivalent to RH
• E(R) = O(R^(1/2)) would be the best possible bound

Möbius Function and Prime Numbers

The Möbius function μ(n) encodes information about the prime factorization of n:

• μ(n) = 1 if n is a product of an even number of distinct primes
• μ(n) = -1 if n is a product of an odd number of distinct primes
• μ(n) = 0 if n has a squared prime factor

Its summatory properties are intimately connected to the distribution of primes and the zeros of ζ(s). The sum M(N) = Σμ(n) oscillates around zero, and RH implies |M(N)| = O(N^(1/2 + ε)).

Euler's Totient Function φ(m)

The totient function φ(m) counts how many integers from 1 to m are coprime to m. The ratio φ(m)/m represents the "coprime density" for a specific modulus m. Averaging these ratios over all moduli converges to 6/π².

Note: While primes have φ(p)/p = (p-1)/p → 1, which is higher than 6/π² ≈ 0.608, primes become increasingly sparse, so composite numbers with lower ratios dominate the average.

Ready to Explore?

Click the "Explorer" tab above to begin your journey into the beautiful world of coprime lattice points and their connection to one of mathematics' deepest mysteries!