(SAPZ) BMO and VMO Theoretical Foundations of Automorphic L-Functions (v4.2)

This v4.2 update refines the theoretical structure presented in previous versions by offering a clearer, more rigorous development of the BMO and VMO frameworks as applied to automorphic L-functions. Key contributions of this version include:

• A strengthened definition of VMOw(ℝ) with precise decay conditions for oscillation amplitude over expanding intervals.

• Formal derivation of a quantitative lower bound on consecutive nontrivial zeros of Hilbert modular L-functions using vanishing oscillation rate:

  γn+1−γn≥π(Ctot−δ(T))log⁡T,δ(T)=O(T−1/2).\gamma_{n+1} - \gamma_n \geq \frac{\pi}{(C_{\text{tot}} - \delta(T)) \log T}, \quad \delta(T) = O(T^{-1/2}).γn+1−γn≥(Ctot−δ(T))logTπ,δ(T)=O(T−1/2).

• Clarification of the log-derivative structure for Λ(s, f) with decomposition into archimedean and arithmetic components.

• Improved narrative and organization around the transition from local oscillation control to global zero statistics.

• Refined remarks on the universality of the VMO condition across L-functions in the Selberg class.

This version consolidates the theoretical underpinnings for connecting Fourier-based oscillation regularity to the zero statistics of automorphic L-functions, with potential implications for understanding the Riemann Hypothesis in a broader analytical framework.