Arithmetic, Harmonic Analysis and the Riemann Hypothesis

This reference develops a connected mathematical description of the Riemann hypothesis from arith- metic sources, harmonic analysis and operator-valued response. Its basic arithmetic data are the integers, prime valuations, the Möbius function and the von Mangoldt function. Its principal an- alytic objects are the completed explicit formula, compensated theta sums, directional Gaussian observations and the simultaneous response of all prime-power channels. Gaussian frames, Mellin transforms and dual tori provide exact representations of these data. They are used with their metrics, domains and source maps specified.