A Potential Geometric–Analytic Roadmap Toward the Riemann Hypothesis

In this paper we assume for contradiction the existence of two distinct nontrivial zeross1, s2 = 1 − s1,of the Riemann zeta function ζ(s) in the critical strip with s1 off the critical line. We thenconstruct a continuously differentiable path γ joining s1 and s2 such that the ζ–imageΓ = ζ ◦ γis univalent on the open interval (0, 1) (with Γ(0) = Γ(1) = 0). Next, by selecting a points0 ∈ γ([0, 1]) with ℜ(s0) = 12 and considering vertical segments from s0 to the zeros on thecritical line, we show that the corresponding ζ–images are enclosed by a suitable Jordan curveclose to Γ. Finally, using the classical result that ζ(s) is unbounded on the critical line, wederive a contradiction with the boundedness of the enclosed region. This contradiction forcesthe non–existence of zeros off the critical line.