Spectral and Arithmetic Perspectives on Generalized Zeta-like Functions: Selberg Class, Modular Forms, and Quantum Chaos

This second release is a substantial revision of the original preprint (v1).It strengthens the mathematical foundations and clarifies the spectral framework introduced for generalized zeta-like functions. Major updates Refined space definitionsFinite space G\mathcal{G}G and Infinite space G∞\mathcal{G}_{\infty}G∞ are now defined with explicit absolute-convergence and frequency-ordering conditions.A new notion of spectral discreteness is introduced. Vertical-line Bounded Mean Oscillation (BMO) Formal definition adapted to log⁡∣G(σ+it)∣\log |G(\sigma+it)|log∣G(σ+it)∣ and use of the John–Nirenberg inequality. Concrete verification of the BMO hypothesis for the Riemann zeta function and Dirichlet/automorphic LLL-functions. Strengthened main theoremsTheorem 5.1 (infinitely many critical zeros) and Theorem 5.2 (linear zero density) now include rigorous proof sketches based on Paley–Wiener theory, Hadamard factorisation and Jensen’s formula under BMO control. Re-ordered arithmetic–spectral–quantum triadSections are re-sequenced “Selberg Class → Mellin transform of modular forms → Selberg trace formula,” yielding a clearer narrative from number theory to quantum chaos. Expanded future-work sectionProposes research directions on GUE statistics under BMO, large zero gaps, and preservation of spectral properties under Rankin–Selberg convolutions. Bibliography and numbering cleaned upAdded Conrey (2003), Iwaniec–Kowalski (2004), Stein (1993); unified labels and equation numbers by section. File list spectral_zeta_proposal_v2.pdf — revised PDF spectral_zeta_proposal_v2.tex — full LaTeX source