Riemann zeros as quantized energies of scattering with impurities

We construct an integrable physical model of a single particle scattering with impurities spread on a circle. The $S$-matrices of the scattering with the impurities are such that the quantized energies of this system, coming from the Bethe Ansatz equations, correspond to the imaginary parts of the non-trivial zeros of the the Riemann $\zeta(s) $ function along the axis $\Re ( s )= \half$ of the complex $s$-plane. A simple and natural generalization of the original scattering problem leads instead to Bethe Ansatz equations whose solutions are the non-trivial zeros of the Dirichlet $L$-functions again along the axis $\Re (s) = \half$. The conjecture that all the non-trivial zeros of these functions are aligned along this axis of the complex $s$-plane is known as the Generalised Riemann Hypothesis (GRH). In the language of the scattering problem analysed in this paper the validity of the GRH is equivalent to the completeness of the Bethe Ansatz equations. Moreover the idea that the validity of the GRH requires both the duality equation (i.e. the mapping $s \rightarrow 1 - s$) and the Euler product representation of the Dirichlet $L$-functions finds additional and novel support from the physical scattering model analysed in this paper. This is further illustrated by an explicit counterexample provided by the solutions of the Bethe Ansatz equations which employ the Davenport-Heilbronn function $\CD (s) $, i.e. a function whose completion satisfies the duality equation $\chi(s) = \chi(1-s)$ but that does not have an Euler product representation. In this case, even though there are infinitely many solutions of the Bethe Ansatz equations along the axis $\Re (s) = \half$, there are also infinitely many pairs of solutions away from this axis and symmetrically placed with respect to it.