Zeta on a Klein Bottle: An Annulus/Klein–Surface Model of the Critical Strip

We develop an annulus/Klein–surface model for the completed zeta function ξ(s)on the critical strip and show how the functional equation can be interpreted asa geometric involution on an annulus that descends to a di–analytic map on aKlein surface. Transporting by the exponential change of variables w = exp(2π(s − 1/2)), f(w) := ξ(1/2 + (log w)/(2π)), the line Re s = 1/2 corresponds to the unit circle |w| = 1, while Re s = 0 andRe s = 1 map to the boundary circles of an annulus. The functional equationξ(s) = ξ(1 − s) becomes f(w) = f(1/w), which we realize as the angular-reversinggluing of annular boundaries, producing a Klein surface on which f is welldefined. On this platform we prove Poisson–Szego and Jensen-type identitiestailored to annuli, and we recast the Riemann Hypothesis as the triviality ofthe annular Blaschke factor (equivalently, equality of boundary means oflog|f|). We complement the analysis with numerical illustrations on the annulusand its Klein-bottle quotient.