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 D_R.- 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.