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.

38 pages, 7 figs, corrected typos, added references, minos changes in content

References in corpus (5)

• Exactly solvable Richardson-Gaudin models for many-body quantum systems
• Log-periodic behavior of finite size effects in field theories with RG limit cycles
• Integrability of the russian doll BCS model
• Renormalization group limit-cycles and field theories for elliptic S-matrices
• The elementary excitations of the exactly solvable Russian doll BCS model of superconductivity

Cited by in corpus (10)

• Synthetic Unruh effect in cold atoms
• The logarithmic triplet theory with boundary
• A quantum mechanical model of the Riemann zeros
• The Riemann zeros as energy levels of a Dirac fermion in a potential built from the prime numbers in Rindler spacetime
• The Berry-Keating Hamiltonian and the Local Riemann Hypothesis
• The Riemann zeros as spectrum and the Riemann hypothesis
• General covariant xp models and the Riemann zeros
• Duality between the quantum inverted harmonic oscillator and inverse square potentials
• The Coprime Quantum Chain
• Fractal fits to Riemann zeros