The Spectral Power of the Digit Function
The bridge
The previous papers left a gap: the autocorrelation R(ℓ) determines the eigenvalue spectrum, but R(ℓ) itself was computed empirically, not expressed in closed form. This paper closes the gap.
The Dirichlet kernel formula
The digit function δ(r) = ⌊br/p⌋ partitions remainders into contiguous bins. Because the bins are intervals, their Fourier transforms are Dirichlet kernels. The spectral power at frequency k is:
$$\Phi(k) = \sum_{d=0}^{b-1} \left|\frac{\sin(\pi k n_d / p)}{\sin(\pi k / p)}\right|^2$$
This is exact. No approximation. The formula depends only on the bin sizes n_d, which are explicitly computable.
The unification
The spectral power Φ(k) is the single object from which all previously defined invariants are derived:
- Paper 3 (alignment limit): Φ(0) = S(p,b) = Σ n_d²
- Paper 5 (coherence decomposition): α, σ, F follow from Φ(0) and the orbit structure
- Paper 7 (eigenvalues): the DFT of R(ℓ)/L, where R is determined by Φ
- Paper 2 (digit-partitioning): DP iff Φ(k) = p − 1 for all k (flat spectrum)
The zeroth mode is the bin-sum from paper 3. The higher modes extend the theory to the full eigenvalue spectrum.
Additive vs multiplicative characters
The Legendre symbol (multiplicative character) was tested and found insufficient to determine R(ℓ). The spectral power uses additive characters instead, which succeed because the bin partition is defined by contiguous intervals. The digit function bridges multiplicative structure (the orbit of b) with additive structure (the bin intervals), and the spectral power is the natural invariant of this bridge.
The paper
The Spectral Power of the Digit Function (PDF)
Verified computationally:
./nfield spectral 17 # show Phi(k) for p=17
./nfield verify8 # verify all paper 8 claims
.:.