Published January 29, 2026 | Version v2
Preprint Open

The Modular Spectrum of π: From Prime Channel Structure to Elliptic Supercongruences

Authors/Creators

  • 1. none

Description

Title: The Modular Spectrum of π: From Prime Channel Structure to Elliptic Supercongruences

Abstract This research proposes a unification of two apparently disjoint paradigms in the study of the mathematical constant π: linear analysis based on elementary modular arithmetic (ℤ/6ℤ) and the theory of complex multiplication (CM) modular forms. The study postulates that π is not a monolithic structure but exhibits "Modular Uniformity" across different energy scales.

Key Contributions The work establishes a continuous spectrum connecting classical series with high-performance algorithms through three main findings:

  1. Low Energy Regime (The Arithmetic Substrate): We demonstrate that the Leibniz series can be reformulated via a decomposition into prime channels 6k ± 1. This reveals a filter structure that isolates the distribution of prime numbers, proving that π emerges naturally from the constructive interference between the residue classes C₁ and C₅.

    • Formula: π = 3 Σ (-1)ᵏ [1/(6k+1) + 1/(6k+5)]

  2. High Energy Regime (Elliptic Acceleration): Using experimental mathematics and the PSLQ integer relation algorithm with 200-digit precision, we reconstruct Ramanujan-Sato series of Level 58 associated with the discriminant d = -232. The study validates the coefficients (A=1103, B=26390, C=396⁴) and confirms an exponential convergence rate of ≈ 8 digits per term.

  3. Arithmetic Synthesis (Supercongruences): The research synthesizes these approaches by revealing the phenomenon of Modular Uniformity in finite fields. We identify anomalous arithmetic behavior for inert primes (specifically p=17) in the quadratic field ℚ(√-58), where the truncated series satisfies strict supercongruences modulo p².

Methodology This dataset includes the theoretical manuscript and the computational validation suite (Jupyter Notebooks). The experimental validation utilizes Python libraries (mpmath, scipy) for arbitrary-precision arithmetic to verify the convergence rates, the Spigot algorithm locality, and the algebraic properties of the derived modular series.

Keywords Modular Arithmetic, Ramanujan Series, Supercongruences, PSLQ Algorithm, Number Theory, Pi, Prime Distribution, Experimental Mathematics.

MSC 2020 Classification 11Y60, 11F03, 11A07

Files

Modular_Spectrum_of_π.pdf

Files (264.6 kB)

Name Size Download all
md5:71aa239ee92b4c534660a7dba6237b33
264.6 kB Preview Download

Additional details

Dates

Updated
2026-01-29
English version

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.