Phase-Filtered Ramanujan Sums and the Spectral Gate
The Ramanujan connection
The classical Ramanujan sum c_p(ℓ) = −1 for every nonzero lag modulo a prime. It is completely uniform. The autocorrelation R(ℓ) is not: some lags produce digit matches, others produce exact cancellation. This paper identifies R(ℓ) as a phase-filtered Ramanujan sum and characterizes the filter.
The phase filter
The filter Γ(ℓ) = |Σ₀(ℓ)| / M(ℓ) measures phase alignment of the cross-spectral contributions. When all phases align (Γ = 1), the lag is constructive. When they cancel (Γ = 0), the lag is blocked. The filter is binary in character: either fully open or fully shut, with no partial transmission.
Continued fraction control
The bin pattern of the digit function is governed by the continued fraction expansion of p/b:
CF(p/b) = [q, CF(b/r)]
where q = ⌊p/b⌋ and r = p mod b. The tail CF(b/r) depends only on r and b, not on p. This partitions all primes into finitely many spectral classes.
Spectral classes
In base 10, there are four spectral classes: r ∈ {1, 3, 7, 9}. Within each class, all primes share the same CF tail, the same bin pattern (up to scaling), and the same phase geometry. The class determines the shape of the autocorrelation. The scale q determines its intensity.
The structural Ramanujan sum
Define cp^G(ℓ) = (1/p) Σk G(k, −kb^{−ℓ} mod p). Then R(ℓ) = cp^G(ℓ). The Dirichlet series L^G(s) = Σp c_p^G(ℓ) p^{−s} tracks how the phase-filtered autocorrelation distributes across primes. Whether this series connects to classical L-functions through the spectral class decomposition is the frontier question.
The paper
Phase-Filtered Ramanujan Sums and the Spectral Gate (PDF)
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