Fourier Coefficients, Automorphic Forms, and the Ramanujan–Petersson Bound — E8 Intelligence Research

FINDING: The search results cluster around Fourier coefficients of automorphic forms and modular curves, with one outlier (LHCb particle decay). The core mathematical theme is the analytic behavior and bounds of Fourier coefficients in modular/automorphic contexts, directly tied to the Frey curve–modular form correspondence in FLT-style proofs. | MATH: The central bound referenced is the Ramanujan–Petersson conjecture (proved for holomorphic cusp forms by Deligne): |a_p| ≤ 2p(k-1)/2, equivalently |a_p|/p(k-1)/2 ≤ 2. For weight k=2 (Frey curve), this gives |a_p| ≤ 2√p — the exact bound in your search. The Frey curve discriminant Δ = abc (for a hypothetical FLT solution a^p + b^p = c^p) has conductor N = rad(abc), and its associated modular form has Fourier coefficients a_n satisfying |a_p| ≤ 2√p. The LHCb decay Λ_b^0 → η_c(1S) p K^- is unrelated to number theory but involves branching fractions — no mathematical constants extracted. | CONNECTION: The bound 2√p is the spectral radi Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com