This repository presents an independent research project focused on operator-theoretic approaches to the Riemann Hypothesis.
The work explores the construction of spectral operators whose properties may correspond to the non-trivial zeros of the Riemann zeta function, following ideas inspired by the Hilbert–Pólya conjecture, Alain Connes’ noncommutative geometry program, and the Berry–Keating model.
The main objective is to investigate whether it is possible to construct an operator framework capable of enforcing the critical line condition:
Re(ρ) = 1/2
for all non-trivial zeros of the Riemann zeta function.
- Riemann zeta function
- Non-trivial zeros and critical line
- Hilbert–Pólya conjecture
- Spectral theory and self-adjoint operators
- Trace formulas
- Noncommutative geometry (Connes program)
- Berry–Keating Hamiltonian (H = xp)
paper/
├── riemann-operator-approach-en.pdf # Main version (English)
└── riemann-operator-approach-pt.pdf # Portuguese version
notes/ # Additional notes (future work) experiments/ # Computational experiments (future work)
- 🇺🇸 English version (main)
- 🇧🇷 Portuguese version
The English version is the primary reference for international readers.
The Portuguese version is provided for accessibility.
This research combines:
- Structural analysis of operator-based frameworks
- Study of existing approaches (Connes, Berry–Keating, spectral methods)
- Investigation of positivity conditions and trace formulas
- Conceptual modeling of operators aligned with the Riemann Hypothesis
- Preliminary computational reasoning for testing candidate structures
This is an ongoing research project.
The work does not claim to provide a complete proof of the Riemann Hypothesis, but instead explores structural directions and possible frameworks that may contribute toward a solution.
The Riemann Hypothesis remains an open problem in mathematics.
This project aims to explore theoretical directions and does not claim a definitive resolution.
Raphael Soares dos Santos
Independent Researcher
Feedback, suggestions, and discussions are welcome.
If you are working in number theory, spectral theory, or related areas, feel free to contribute or share insights.
- Development of computational experiments
- Construction of explicit operator models
- Numerical analysis of spectral distributions
- Exploration of connections with random matrix theory
- Refinement of positivity conditions in operator frameworks
This project is shared for research and educational purposes.
(You may add a formal license later if needed.)