The Autocorrelation Formula
The missing formula
Paper 7 proved that eigenvalues are the DFT of the autocorrelation R(ℓ). Paper 8 gave the spectral power Φ(k) in closed form. But R(ℓ) itself was still computed by brute-force digit matching. This paper derives the exact formula.
The cross-spectral function
Define G(k, k') = Σ_d δ̂_d(k) δ̂_d(k'), where δ̂_d is the Fourier transform of the indicator function of bin B_d. This is the cross-spectral function of the digit partition. It encodes how the bins interact across frequency pairs.
The autocorrelation is:
$$R(\ell) = \frac{1}{p} \sum_{k=0}^{p-1} G(k,; -k b^{-\ell} \bmod p)$$
The sum evaluates G along the multiplicative orbit of b. The orbit determines which frequency pairs contribute.
The vanishing sum
A key structural result: the total cross-spectral sum vanishes.
$$\sum_{k'} G(k, k') = 0 \quad \text{for } k \neq 0$$
This forces R(ℓ) to be selective. The constructive and destructive contributions must balance globally, so the autocorrelation discriminates among lags rather than treating them uniformly.
Connection to Ramanujan sums
The classical Ramanujan sum c_p(ℓ) = −1 for all nonzero lags. It treats every lag equally. The autocorrelation R(ℓ) is a Ramanujan-type sum with a non-uniform weight: the weight is G, evaluated along the orbit. The digit-bin geometry controls which lags survive and which cancel. Paper 10 develops this connection fully.
The paper
The Autocorrelation Formula (PDF)
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