What question did this study set out to answer?

This work aims to refine the representation of the Riemann zeta function through a fractal-holographic operator.

April 1, 2026Open Access

A Fractal-Holographic Representation of the Riemann Zeta Function via a Self-Similar Operator (Version 4.4 – Final)

Key Points

This work aims to refine the representation of the Riemann zeta function through a fractal-holographic operator.
Introduced a linear functional \(\Lambda : \mathcal{B} \to \mathbb{C}\) that extracts \(\zeta(s)\) via Möbius inversion.
Formulated the central conjecture linking non-trivial zeros to the Fredholm determinant.
Developed a fractal-holographic operator \(\mathcal{F}_s\) acting on a Banach space of holomorphic functions.
Validated the pole at \(s=1\) due to operator norm divergence.
Provided rigorous justification for the analytic continuation of the zeta function.
Demonstrated a connection between non-trivial zeros and the zeros of the Fredholm determinant.

Abstract

Update in Version 4. 4: Major refinement of the analytic continuation. Introduced an explicit linear functional \ (: B C\) that rigorously extracts \ ( (s) = (fₛ^*) \) via Möbius inversion and the boundary condition of the attractor \ (\). Full justification of the pole at \ (s=1\) arising solely from the divergence of the operator norm. Honest formulation of the central conjecture linking the non-trivial zeros to the zeros of the Fredholm determinant \ ( (I - Fₛ) \). This article introduces a fractal-holographic operator \ (Fₛ\) that reformulates the Riemann zeta function \ ( (s) \) as the fixed point of a compact operator acting on a Banach space of holomorphic functions. . . . We are deeply grateful to Prof. Michel L. Lapidus for his insightful and constructive feedback, which greatly strengthened the functional-analytic rigor and overall presentation of this manuscript.

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