Page 11's Franel–Landau sum Σ|F_k − k/|F_N|| measures how evenly the Farey fractions sit compared to a perfectly even spacing — and its growth rate is equivalent to the Riemann Hypothesis. This page grades that same sum by sector, using the fact that a fraction's sector index is exactly its first continued-fraction digit, to see how the discrepancy is distributed across sectors rather than only its total.
For a Farey fraction x = a/b in lowest terms, the sector containing it is S_s = (1/(s+1), 1/s] with n = ⌊b/a⌋. Writing out the continued fraction x = [0; a₁, a₂, …], the first partial quotient a₁ is also ⌊b/a⌋ — the two definitions coincide exactly, for every fraction. Sector membership is literally "what does the continued fraction expansion of x start with?" Try it on any pair:
F(s,N) ∝ 1/(s(s+1)) is the plain Euclidean width of sector S_s — how much of the interval (0,1] it occupies. A second, unrelated quantity lives on the same partition: the Gauss–Kuzmin distribution P(a₁=n) = log₂(1+1/(n(n+2))), the natural invariant measure of the Gauss map (the dynamical system behind continued fractions). Both decay like 1/n² for large n — the same power law — but they are not the same measure: their ratio converges to 1/ln 2 ≈ 1.4427, not 1. One counts length, the other counts ergodic frequency under repeated iteration of the Gauss map. Plotted as n²·(value), each curve should flatten to its own horizontal line:
Builds the Farey sequence F_N, computes δ_k = F_k − k/|F_N| for each fraction, and buckets |δ_k| by the sector (equivalently, first CF digit) of F_k. Large-n sectors contain very few points at any finite N, so their bars are naturally noisier — that's a real feature of finite-N sampling, not something to smooth away.
This page decomposes a known RH-equivalent quantity by an existing structural index; it doesn't argue toward a proof of anything. The continued-fraction / Gauss-map route connects, via the transfer-operator literature (Mayer's theorem), to the Selberg zeta function of the modular surface — a real and beautiful correspondence, but a different zeta function from ζ(s) itself. For the spectral check against actual zeta-zero heights, see Page 11, which runs that comparison on the unsectored sum.