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.