Abstract
The objects of the number theory arise naturally in the scattering theory for automorphic functions on the Lobachevsky–Poincaré plane. Although the best-known results on the distribution of primes are obtained by the method of trigonometric sums, recent progress in the harmonic analysis of automorphic functions has given a new interpretation of the important properties of the ζ-function and hence has created new hopes for mathematicians in this challenging area. The chapter provides a brief review of the relevant facts of the spectral theory of automorphic functions. The reflection coefficient defined above in terms of a stationary problem from the asymptotics of scattered waves at infinity may be interpreted in terms of a nonstationay problem. A well-known idea of Hilbert suggests that the Riemann hypothesis can be proved by working out a self-adjoint operator, A, such that the spectrum of1/2 +iA coincides with the set of the ζ-zeroes.
Published Version
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