On the Riemann Hypothesis, Area Quantization, Dirac Operators, Modularity, and Renormalization Group

Abstract

Two methods to prove the Riemann Hypothesis are presented. One is based on the modular properties of Θ (theta) functions and the other on the Hilbert–Polya proposal to find an operator whose spectrum reproduces the ordinates ρn (imaginary parts) of the zeta zeros in the critical line: sn = ½ + iρn. A detailed analysis of a one-dimensional Dirac-like operator with a potential V(x) is given that reproduces the spectrum of energy levels En = ρn, when the boundary conditions ΨE (x = -∞) = ± ΨE (x = +∞) are imposed. Such potential V(x) is derived implicitly from the relation [Formula: see text], where the functional form of [Formula: see text] is given by the full-fledged Riemann–von Mangoldt counting function of the zeta zeros, including the fluctuating as well as the [Formula: see text] terms. The construction is also extended to self-adjoint Schroedinger operators. Crucial is the introduction of an energy-dependent cut-off function Λ(E). Finally, the natural quantization of the phase space areas (associated to nonperiodic crystal-like structures) in integer multiples of π follows from the Bohr–Sommerfeld quantization conditions of Quantum Mechanics. It allows to find a physical reasoning why the average density of the primes distribution for very large [Formula: see text] has a one-to-one correspondence with the asymptotic limit of the inverse average density of the zeta zeros in the critical line suggesting intriguing connections to the renormalization group program.

The search for a Hamiltonian whose energy spectrum coincides with the Riemann zeta zeroes

“…The Riemann Hypothesis (RH) has also been studied from the point of view of mathematics and physics by [14,24,8,13,25,26,10,16,28,30,32] among many others. We refer to the website devoted to the interplay of Number Theory and Physics [22] for an extensive list of papers related to the RH.…”

“…In [28], we studied a modified Dirac operator involving a potential related to the number counting function of zeta zeroes and left the Schroedinger operator case for a future project that we undertake in this work. After this introduction, in Sec.…”