The Modular Spectrum of π: From Prime Channel Structure to Elliptic Supercongruences
Description
Executive Summary
This work establishes a unified paradigm for understanding π through its modular spectrum, connecting elementary arithmetic with high-level modular forms.
Key Results
🔷 Modular Representation Theorem
π = 3 × SUM[k=0 to ∞] of (-1)^k × [1/(6k+1) + 1/(6k+5)]
This reveals the "prime channels" structure 6k±1 as a fundamental arithmetic filter, showing that π emerges from the interaction between two prime number progressions.
🔷 Experimental Reconstruction of Ramanujan-Sato Series (Level 58)
1/π = (2√2)/9801 × SUM[k=0 to ∞] of [(4k)!/(k!)^4] × (1103 + 26390k)/396^(4k)
Validated with 8 digits per term convergence using PSLQ algorithm with 200-digit precision.
🔷 Modular Uniformity Theory
We demonstrate that arithmetic supercongruences for inert primes (e.g., p=17 where S(17) ≡ 246 mod 289) and Spigot algorithm locality emerge from the same underlying structure.
🔷 Hybrid Computational Architecture
A paradigm combining parallel processing (prime channels) with local access (Spigot algorithms).
Mathematical Framework
- Prime Channels: Decomposition of natural numbers into classes modulo 6
- Hilbert Space V₆: Formal vector space over Z/6Z with basis |1⟩, |5⟩
- Level 58 Modular Forms: Connection to imaginary quadratic field Q(√-58)
- Supercongruences: Anomalous modular behavior for inert primes
Impact and Applications
🎯 Theoretical
Unification of elementary modular arithmetic (Z/6Z) with complex multiplication theory
🎯 Computational
New hybrid algorithms for constant calculation combining parallel and local access
🎯 Mathematical Physics
Connections with spectral theory and fundamental constants like fine-structure constant
Available Resources
- Source Code: Espectro-Modular-Pi
- Complete Implementation: BBP algorithms, PSLQ, convergence analysis
- Numerical Validation: 200-digit precision experiments
Future Research Directions
- Extension to other mathematical constants (ζ(3), Catalan's constant)
- Spectral theory of modular operators
- Connections with arithmetic string theory
- Physical interpretations of modular spectrum
Files
ESPECTRO_MODULAR_π.pdf
Additional details
Dates
- Created
-
2025-11-22
Software
- Repository URL
- https://github.com/NachoPeinador/Espectro-Modular-Pi
- Programming language
- Python
- Development Status
- Active
References
- Bailey, D. H., Borwein, P. B. and Plouffe, S.: On the rapid computation of various polylogarithmic constants. Math. Comp. 66 (1997), 903–913.
- Borwein, J. M. and Borwein, P. B.: Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity. Wiley, New York, 1987.
- Ferguson, H. R. P., Bailey, D. H. and Arno, S.: Analysis of PSLQ, an integer relation finding algorithm. Math. Comp. 68 (1999), 351–369.
- Guo, V. J. W.: Proof of a supercongruence modulo 𝑝 2𝑟 . Bull. London Math. Soc. 57 (2025).
- Ramanujan, S.: Modular equations and approximations to 𝜋. Quart. J. Math. 45 (1914), 350– 372.