ABSTRACT
This chapter explains the properties of the Riemann zeta function in some detail to unfold its connection to the subsequent developments on the dynamical theory. It focuses onto scattering theory, and to the inclusion of diffractive orbits to the trace formula. The chapter provides an elementary account of quantum scattering theory for a plane wave in two dimensions incident on a single scatterer. It show that the scattering cross section for small angles is very different from its classical behavior even in the limit of the de Broglie wave length of the particle being much smaller than the radius of the disc. If one devises a method of locating the zeros with precision, it will then be a pointer to finding a rule for quantization of a Hamiltonian that is classically chaotic. This is a central problem in the understanding of the quantum-classical link of nonintegrable systems.
