On the Nonexistence of Zeros off the Critical Line for the Riemann Zeta–Function: A Geometric–Analytic Approach

We present a geometric–analytic approach to the Riemann Hypothesis by assuming, forcontradiction, the existence of a nontrivial zero off the critical line. Let s1 be a zero of ζ(s)in the critical strip with ℜ(s1) ̸= 12 and denote its symmetric counterpart by s2 = 1 − s1.We construct a continuously differentiable path γ connecting s1 and s2 such that the compositemapping Γ = ζ ◦γ is univalent on the open interval (0, 1). Employing techniques from differentialtopology, including transversality and local perturbation methods, we modify Γ to obtain asmooth Jordan curve whose image under ζ encloses the common value corresponding to thezeros. By selecting a point s0 ∈ γ on the critical line and considering vertical segments froms0 to the zeros on the critical line, we show that the ζ–images of these segments must lieentirely within the bounded interior of the Jordan curve. However, since ζ(s) is known tobe unbounded on the critical line, this containment yields a contradiction. Our argument isunconditional, relying solely on classical results from complex analysis and differential topology,thereby confirming that all nontrivial zeros of ζ(s) lie on the critical line.