A Graphical and Analytical Study of χ(s) and the Localization of Zeros of the Riemann Zeta Function

This deposit contains the supplementary materials (LaTeX source, figures, Python/SageMath scripts, README) for the preprint: "A Graphical and Analytical Study of χ (s) and the Localization of Zeros of the Riemann Zeta Function (v5) " Author: Mohammed Djamal CHEBBAH --- Scientific summary --- The manuscript investigates the auxiliary function χ (s) defined by ζ (s) = χ (s) ζ (1-s). A detailed analytical study of |χ (s) | in the critical strip is carried out, establishing monotonicity, asymptotic behavior, and the existence of three critical thresholds t₁, t₂, t₃ near t ≈ 2π. The main result is conditional: assuming Conjecture 3. 1 (that |χ (s₀) | = 1 for every non‑trivial zero s₀ of ζ), all non‑trivial zeros lie on the critical line Re (s) =1/2. Hence the Riemann Hypothesis follows from Conjecture 3. 1. The conjecture is stated explicitly and remains unproven. --- Changes in version 5 (compared to v4, DOI 10. 5281/zenodo. 19817104 - Conjecture 3. 1 is retained (no credible proof found despite extensive efforts). - A new transition section clarifies the link between the study of |χ (s) | and the numerical certification. - A new figure (Figure 13) illustrates the symmetric values of |ζ (s) | in the strip DR. - The subsection on symmetries of the ξ function (Figure 1) has been restored (was missing in v4). - The README has been updated. --- Contents --- - LaTeX source file- Compiled PDF- Bibliography (. bbl) - Figures (16 files) - Python scripts (11) - SageMath scripts (4) - README- CC-BY license All numerical experiments are reproducible. License: CC-BY-4. 0.