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 \( \Phi_n(x) = \prod_{d|n} (x^d - 1)^{\mu(n/d)} \), where \( \mu \) is the Möbius function. - For a root of unity \( \zeta = e^{2\pi i k/n} \) with \( \gcd(k,n) = m \), the paper (arXiv:1611.06783v2) derives: \[ \Phi_n(\zeta) = \prod_{d|n} ( \zeta^d - 1 )^{\mu(n/d)} = \prod_{d|n} ( e^{2\pi i k d/n} - 1 )^{\mu(n/d)} \] which simplifies via the identity \( e^{2\pi i a} - 1 = 2i e^{\pi i a} \sin(\pi a) \), yielding a product of sines weighted by \( \mu(n/d) \). - Key constant: \( \Phi_n(1) = p \) if \( n = p^k \) (prime power), else \( \Phi_n(1) = 1 \). At non-primitive roots, the value is a rational integer times a power of a prime — specifically, if \( \zeta \) has order \( d < n \), then \( \Phi_n(\zeta) \) is an in Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com