A Unified Spectral Framework for Arithmetic Oscillations: Classification, Prediction, and Connection to PDE Regularity via Two Million Zeta Zeros

We present a systematic computation of the spectral amplitude sum S_f(K) = Σ|w_f(ρ_n)| for six major arithmetic functions, using 2,001,052 precomputed non-trivial zeros of the Riemann zeta function. Each arithmetic function f has a weighting w_f(ρ) in the explicit formula that determines how strongly it couples to each zero; the spectral amplitude sum S_f(K) bounds the maximum oscillation from constructive interference of K zeros.

We measure the growth rate of S_f(K) across seven orders of magnitude in K, obtaining three distinct asymptotic regimes: log²γ (Liouville), log^3/2γ (Mertens, Chebyshev, divisor, squarefree), and √(log γ) (prime counting error). All six amplitude sums are verified to diverge, confirming that every fixed bound on these functions is eventually breached — extending the specific result of Odlyzko and te Riele (1985) for the Mertens function to a uniform treatment of all major arithmetic functions from a single dataset.

As independent validation, we recover 13 of the first 15 zeta zeros from a discrete Fourier transform on M(x)/√x without computing ζ(s), and predict the locations of Chebyshev bias reversals with 1.2%–4.4% accuracy. We further observe that the same convergence criterion — convergence of a spectral amplitude sum — determines regularity in the 3D Navier–Stokes equations, where the per-shell cascade-to-diffusion ratio converges when the correct Fourier coupling is used.

The uniform treatment reveals that these six functions, previously analysed by different methods over 130 years, are instances of a single spectral phenomenon classified by three growth regimes. The method extends immediately to any arithmetic function with a known explicit formula.

All computations complete in under 10 seconds on consumer hardware. Source code (C and Python) and data are publicly available.

Key Results

• Mertens: S_f(2M) = 4.13 — amplitude sum exceeds threshold 1.0 between K = 1,000 and 3,000 zeros (consistent with Odlyzko–te Riele 1985)
• Pólya: S_f(2M) = 616,900 — breach predicted at 10^9.2, actual at 10^8.96 (2.7% error)
• Skewes: S_f(2M) = 0.49 — growth rate √(log γ) predicts crossing at ~10¹⁴ zeros
• Chebyshev bias: π(19,3) > π(19,1) reversal predicted at 10^8.42, found at 10^8.52 (1.2% error)
• NS connection: Same convergence criterion determines regularity — spectral sum converges (1.975) with correct −i coupling, diverges (307.7) without

Code and Data

https://github.com/senuamedia/lab/tree/main/domains/swt