Fourier Analysis Unveils Hidden Structure in Cyclotomic Polynomial Zeros — E8 Intelligence Research

FINDING: Cyclotomic polynomials evaluated at non-primitive roots of unity admit an explicit formula via finite Fourier analysis (Möbius inversion on divisor lattices), revealing hidden multiplicative structure in their zero/non-zero pattern. MATH: - Let \( Φ_n(x) = ∏d|n (x^d - 1)μ(n/d) \), where \( μ \) is the Möbius function. - For a root of unity \( ζ = e2π i k/n \) with \( (k,n) = m \), the paper (arXiv:1611.06783v2) derives: \[ Φ_n(ζ) = ∏d|n ( ζ^d - 1 )μ(n/d) = ∏d|n ( e2π i k d/n - 1 )μ(n/d) \] which simplifies via the identity \( e2π i a - 1 = 2i eπ i a sin(π a) \), yielding a product of sines weighted by \( μ(n/d) \). - Key constant: \( Φ_n(1) = p \) if \( n = p^k \) (prime power), else \( Φ_n(1) = 1 \). At non-primitive roots, the value is a rational integer times a power of a prime — specifically, if \( ζ \) has order \( d < n \), then \( Φ_n(ζ) \) is an in Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com