H = xp model revisited and the Riemann zeros

Area of Science:

• Mathematical Physics
• Quantum Mechanics
• Number Theory

Background:

• The Berry-Keating conjecture proposes a link between the classical Hamiltonian H = xp and Riemann zeta function zeros.
• Previous models showed average semiclassical energies matching Riemann zeros but lacked closed classical trajectories, indicating an incomplete model.

Purpose of the Study:

• To investigate a modified Hamiltonian that incorporates closed periodic orbits.
• To demonstrate the spectral coincidence of this new Hamiltonian with average Riemann zeros.
• To generalize findings to Dirichlet L functions and explore experimental connections.

Main Methods:

• Introduced a new Hamiltonian: H = x(p + ℓ(p)²/p).
• Analyzed the properties of classical trajectories for the modified Hamiltonian.
• Examined the energy spectrum of the Hamiltonian and its relation to Riemann zeros.
• Utilized self-adjoint extensions for generalization to Dirichlet L functions.

Main Results:

• The modified Hamiltonian H = x(p + ℓ(p)²/p) possesses closed periodic orbits.
• The energy spectrum of this Hamiltonian aligns with the average Riemann zeros.
• The results were successfully generalized to Dirichlet L functions.

Conclusions:

• The study provides a more complete model connecting classical mechanics and Riemann zeros through a Hamiltonian with closed orbits.
• The findings offer a potential pathway for experimental verification using the Landau model.
• The work deepens the understanding of the interplay between quantum mechanics, classical dynamics, and number theory.