A Graphical and Analytical Study of χ(s) and the Localization of Zeros of the Riemann Zeta Function (v3)

This work explores new analytical properties of the auxiliary function χ(s) within the critical strip of the Riemann zeta function. By combining symmetry constraints derived from the functional equation with domain exclusion principles, we identify regions where non-trivial zeros of ζ(s) cannot occur. We employ three complementary tools — graphical visualization, analytical arguments, and numerical computations — leading to two convergent results that map the critical strip and precisely characterize the region where the zeta function may vanish. The systematic elimination of excluded regions highlights the critical line ℜ(s) = 1/2 as the remaining domain of interest. These results provide new insights into the distribution of zeros of the Riemann zeta function and offer a systematic framework that may be applied to related problems in analytic number theory. --- **IMPROVEMENTS IN VERSION 3:** **EDITORIAL OPTIMIZATIONS:** - Reformatting for improved readability - Clarification of technical demonstrations - Improved hierarchical structure for better PDF bookmark navigation - Title modified to better reflect the content **CONCEPTUAL CLARIFICATIONS:** - Corrected several common misinterpretations concerning the zeros of the zeta function within its functional equation (see the Remark in Section 1.3.2) **NEW ANALYTICAL PROOFS ADDED:** - Proof of the strict concavity of g(s) for σ ∈ 0,1 and t > t₀ - Analytic proof of the positivity of g(s) for 0 0 - Proof that g(s) t₃ - Asymptotic analysis for t ≫ 1 **NEW APPROACH (SECTION 6):** A second, more concise demonstration combining: - The strict monotonicity of |χ(s)| for t > t₃ - A single numerical verification using the argument principle on the rectangle 0,1.1×−0.1,7 **NEW SECTION: DIGITAL CERTIFICATION OF RESULTS:** - Complete digital validation methodology using Arb/SageMath interval arithmetic - 4 certification scripts included in the repository - Specification of analysis domains and methods used - Detailed presentation of results The accompanying supplementary materials include all LaTeX source files, 14 figures, and 6 Python/SageMath scripts necessary to reproduce the results.