The Riemann zeros and the cyclic Renormalization Group

Abstract

We propose a consistent quantization of the Berry-Keating Hamiltonian x p, which is currently discussed in connection with the non trivial zeros of the Riemann zeta function. The smooth part of the Riemann counting formula of the zeros is reproduced exactly. The zeros appear, not as eigenstates, but as missing states in the spectrum, in agreement with Connes adelic approach to the Riemann hypothesis. The model is exactly solvable and renormalizable, with a cyclic Renormalization Group. These results are obtained by mapping the Berry-Keating model into the Russian doll model of superconductivity. Finally, we propose a generalization of these models in an attempt to explain the oscillatory part of the Riemann's formula.