Logarithmic Dirac Distributions and Operator Self-Duality in the Spectral Formulation of the Riemann Zeta Function

This paper develops a distribution-theoretic and operator-theoretic framework for the Riemann zeta function based on representing integers as Dirac delta distributions. In logarithmic variables, multiplicative structure becomes additive convolution structure, and the zeta function appears naturally as a Fourier/Mellin transform of a logarithmic Dirac distribution. The same framework also yields regularized prime-sum identities of explicit-formula type. Version 2 further asks what would be needed in order to interpret the Riemann Hypothesis as arising from a genuine dynamical or spectral system. It argues that boundary conditions alone cannot explain the arithmetic structure of the zeta function, the functional equation, or the special role of the critical line. Instead, the analysis suggests that any successful formulation should involve an intrinsic self-duality of the relevant operator. The paper therefore identifies operator self-duality, rather than boundary conditions, as the more plausible structural principle behind a future spectral realization of the nontrivial zeros.