Zeta on a Klein Bottle: An Annulus/Klein–Surface Model of the Critical Strip
Year: 2025
Type: article
Abstract: The map w = exp(2π(s − 1/2)) sends the critical strip onto the annulus e^(−π) < |w| < e^π, the critical line onto the unit circle, and s ↦ 1 − s onto w ↦ 1/w. Since ξ has no vertical period, the transported function is single-valued only on the logarithmic cover, where it is the even entire function F(z) = ξ(1/2 + z/2π). The quotient of the annulus by the inversion is a disk with two cone points, the Klein bottle arises from gluing the boundary circles alone, and F descends to a Möbius band covering it, not to the Klein bottle. The fibrewise periodization Σ_k ξ(s + ik) is zero-free and constant to within 10^(−391000) on the annulus. Circular means of a branch satisfy m_T″ = Z_T + E_T, Littlewood's lemma in distributional form, with an explicit positive seam term; hence |ξ(σ + it)| decreases strictly in t on [0, √(γ₁² − 1)] for all σ ∈ [0, 1]. On the strip, ξ is a Blaschke product times an outer function.
Author: Ismail, Mesut
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7084600874
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