Hamiltonian for the Zeros of the Riemann Zeta Function

Area of Science:

• Quantum mechanics
• Number theory
• Mathematical physics

Background:

• The Riemann hypothesis, a Millennium Prize Problem, concerns the distribution of prime numbers.
• The Berry-Keating conjecture proposes a quantum mechanical system whose energy levels correspond to the nontrivial zeros of the Riemann zeta function.

Purpose of the Study:

• To construct a Hamiltonian operator whose eigenvalues are related to the nontrivial zeros of the Riemann zeta function.
• To explore the connection between quantum mechanics and the Riemann hypothesis.

Main Methods:

• Construction of a specific Hamiltonian operator H.
• Analysis of the classical limit of H, showing consistency with the Berry-Keating conjecture.
• Investigation of the PT symmetry properties of iH.
• Heuristic analysis for constructing a metric operator to define a Hermitian inner-product space.

Main Results:

• The constructed Hamiltonian operator H, under specific boundary conditions, yields eigenvalues corresponding to the nontrivial zeros of the Riemann zeta function.
• The classical limit of H is 2xp, aligning with the Berry-Keating conjecture.
• The operator iH exhibits broken PT symmetry, suggesting real eigenvalues for H.
• A heuristic method for defining a Hermitian Hamiltonian is proposed.

Conclusions:

• The study presents a novel Hamiltonian operator with potential implications for the Riemann hypothesis.
• If the Hamiltonian can be rigorously shown to be self-adjoint, it would imply the truth of the Riemann hypothesis.
• The work bridges quantum mechanics and number theory through the Berry-Keating conjecture.