What question did this study set out to answer?

The research aims to establish a comprehensive proof of the Riemann zeta function's properties using fractal holography.

April 1, 2026Open Access

A Fractal-Holographic Representation of the Riemann Zeta Function via a Self-Similar Operator — Part II: Full Hard-Analysis Proofs, Fredholm Determinant Study, High-Resolution Spectral Diagonalization and Variational Principle

Key Points

The research aims to establish a comprehensive proof of the Riemann zeta function's properties using fractal holography.
Conducted hard-analysis proofs to establish mathematical foundations.
Performed a Fredholm determinant study to analyze eigenvalues.
Utilized high-resolution spectral diagonalization on a large matrix.
Applied the variational principle to find energy minimizers.
Proved that d_H(φ) = 1/2 is the unique energy minimizer of the fractal energy functional.
Achieved spectral diagonalization with an error margin below 10^{-14}.

Abstract

Version 5. 2 (Final Companion Paper to V4. 4) — 31 March 2026 Full hard-analysis proofs, Fredholm determinant study, high-resolution spectral diagonalization (N=4096, error < 10^-14) and variational principle proving that dH (φ) = 1/2 is the unique energy minimizer of the natural fractal energy functional. Companion to Version 4. 4: https: //zenodo. org/records/19325293 (DOI 10. 5281/zenodo. 19325293) This record contains: - Full PDF and LaTeX source of V5. 2 - Python script to regenerate all figures - All high-resolution figures (V4. 4 + energy landscape) - V4. 4. tex for completeness

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