Band-Pass Detection of Local Hecke Data for the Ramanujan ∆-Form

We present a band-pass detector that isolates local Hecke data of Ramanujan's Δ-form from windowed averages of ℜ log L. In unitary normalization L_u(s) = ∑{n≥1} λ(n)/n^s with λ(n) = τ(n)/n^{11/2}, we study J{a,p₀}(σ,T) = ∫ φ_T(t) cos(at log p₀) ℜ log L_u(σ+it) dt, where φ_T(t) = max(0,1−|t|/T) is the Fejér (triangular) window, σ > 1, a ≥ 1, and p₀ is prime. Because the Fejér spectrum is non-negative, the "main line" at frequency a log p₀ is positive and dominates the residual terms. We prove the uniform decomposition J_{a,p₀}(σ,T) = (T/(a·p₀^{aσ}))·c_{a,p₀}(T)·λ(p₀^a) + O(1), with explicit c_{a,p₀}(T) ∈ [1/2,1] and an absolute O(1) remainder independent of T (uniform in p₀, a, T for fixed σ > 1). Hence, for T ≫ p₀^{aσ} the sign of J_{a,p₀} stabilizes and equals sign λ(p₀^a); a one-step Richardson combination across T and 2T yields a stable estimate of λ(p₀^a). We also provide a fast weighted Euler-sum implementation (no time integration) enabling fully reproducible runs. The construction extends with minor changes to higher powers a, newforms of general weight/level (with standard changes at bad primes), Dirichlet twists, higher-order Fejér/B-spline windows, and (in outline) symmetric powers.

• Keywords: Ramanujan tau, Hecke eigenvalues, unitary L-function;
Re log L, Fejér window, band-pass detector, Chebyshev Um, Sato–Tate,
local Euler factors, Richardson extrapolation

• MSC 2020: 11F30. Secondary: 11F11, 11F66, 11M41, 11Y35.

• Code availability: The paper refers to a public GitHub.