Advanced Browser-Based Computational Testing with Comprehensive Analysis - Up to 100 Million Points
The Riemann Hypothesis, proposed by Bernhard Riemann in 1859, is one of the most important unsolved problems in mathematics. It concerns the distribution of prime numbers and has profound implications for number theory, cryptography, and our understanding of mathematical patterns.
Creator: Wessen Getachew (@7dview)
Mathematical Framework: Based on classical work by Davenport, Halberstam, Montgomery, and modern computational number theory
Implementation: Advanced JavaScript algorithms with comprehensive visualization and interactive analysis for browser-based scientific computing
Special thanks to: The open mathematics community, Andrew Odlyzko's computational work, Xavier Gourdon's large-scale verifications, and all researchers who have contributed to our understanding of the Riemann Hypothesis
Where s = σ + it is a complex number with real part σ and imaginary part t
Series converges absolutely
ζ(2) = π²/6, ζ(4) = π⁴/90
Analytic continuation needed
Functional equation relates ζ(s) to ζ(1-s)
0 < σ < 1
Where non-trivial zeros live
This is why ζ(s) encodes all information about primes! Every prime p contributes a factor to the infinite product.
Where ρ = 1/2 + it are the critical zeros. Each zero ρ contributes an oscillatory term Li(xρ) to the prime counting function!
Our exponential sums S(N,α) = Σ μ(n)e2πinα are related to the zeta zeros through:
• Zeros are behaving "as expected"
• Cancellation is working properly
• RH predictions hold
• Possible resonance effects
• Zero distribution anomalies
• Interesting mathematics!
At negative even integers. These come from the functional equation and are well understood. They don't affect prime distribution significantly.
The stars of the show! These control prime distribution. RH says they ALL lie on the critical line σ = 1/2.
If any non-trivial zeros exist off the line σ = 1/2 but in this strip, RH fails and prime behavior becomes less predictable.
ζ(s) = 2sπs-1sin(πs/2)Γ(1-s)ζ(1-s)
Relates values at s and 1-s
| Height Range | Zeros Computed | All on σ = 1/2? | Computational Effort |
|---|---|---|---|
| 0 < t < 100 | 29 | ✅ Yes | By hand (Riemann era) |
| 0 < t < 10,000 | ~3,000 | ✅ Yes | Early computers |
| 0 < t < 10¹³ | ~10¹³ | ✅ Yes | Modern supercomputers |
The zeros of ζ(s) are like the "harmonics" of the prime number "symphony."
Our sums are directly influenced by zeros with imaginary parts t ≈ α·N/(2π). We're probing the zero distribution!
Small ratios mean the zeros are creating the precise cancellations needed for RH error bounds.
The distribution of our results reflects the statistical properties of zeros predicted by random matrix theory.
Our browser-based Möbius verification connects:
Every test you run explores this magnificent mathematical symphony! 🎼
What we're testing: Under the Riemann Hypothesis, these Möbius exponential sums should satisfy |S(N,α)| = O(√N) uniformly in α. If we find ratios significantly larger than √N, it could indicate interesting mathematical phenomena or even contradict RH predictions.
The Möbius function is defined as:
Examples: μ(1)=1, μ(2)=-1, μ(3)=-1, μ(4)=0, μ(6)=1, μ(12)=0
Prime Number Connection: The hypothesis directly relates to how accurately we can predict the distribution of prime numbers. If true, it provides the best possible bounds for counting primes.
Cryptographic Implications: Many modern encryption systems rely on the difficulty of factoring large numbers. RH predictions help us understand the security foundations of these systems.
Computational Verification: The hypothesis has been verified for the first 1013 zeros, but a general proof remains one of mathematics' greatest challenges.
π(x) = actual count of primes ≤ x | Li(x) = logarithmic integral ≈ x/ln(x)
Historic Insight: In 1792, 15-year-old Gauss predicted π(x) ≈ Li(x). Our RH verification directly validates this prediction!
| x | π(x) actual | Li(x) Gauss | Error | Error % |
|---|---|---|---|---|
| 10³ | 168 | 178 | +10 | 5.9% |
| 10⁶ | 78,498 | 78,628 | +130 | 0.17% |
| 10⁹ | 50,847,534 | 50,849,235 | +1,701 | 0.003% |
RSA security, prime generation, key sizing, factoring difficulty bounds
Primality testing, hash functions, pseudorandom generators, sieve optimization
L-functions, automorphic forms, analytic number theory, additive combinatorics
The exponential sums S(N,α) = Σ μ(n)e^(2πinα) are intimately connected to prime distribution through:
Every successful RH verification means: "Primes are distributed so regularly that a teenager's 1792 intuition predicts their count to within √N accuracy!"
When you see smooth, low-ratio visualizations, you're witnessing computational proof that the universe's prime distribution follows deep, predictable patterns that Gauss intuited over 230 years ago.