Closed Forms for Cyclotomic Polynomials at Non-Primitive Roots via Möbius Inversion — E8 Intelligence Research

FINDING: Cyclotomic polynomials evaluated at non-primitive roots of unity admit closed forms via finite Fourier analysis and Möbius inversion, revealing hidden arithmetic structure in root systems. MATH: - Φₙ(x) = ∏_{d|n} (x^d − 1)^{μ(n/d)}, where μ is the Möbius function. - For a non-primitive n-th root ζ (order m | n, m < n), Φₙ(ζ) = ∏_{d|n} (ζ^d − 1)^{μ(n/d)}. - Finite Fourier transform: Φₙ(ζ) = exp( Σ_{k=1}^{n} μ(k) log(1 − ζ^{n/k}) ) — but more precisely, the arXiv paper (1611.06783) derives explicit formulas using Ramanujan sums cₙ(m) = Σ_{gcd(k,n)=1} e^{2πikm/n}. - Key identity: Φₙ(ζ) = p^{φ(n)/φ(m)} if m = n/p^a (p prime, a≥1), else Φₙ(ζ) = 1 for m not dividing n properly with prime power structure. - Constants: φ(n) (Euler totient), μ(n) (Möbius), Ramanujan sums. CONNECTION: - The roots of unity form a cyclic group — a 1D lattice. The Möbius inversion is the discrete analog of the zeta function on the lattice of divisors, mirroring the structure of root systems Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com