Wigner Phase-Space Representation, Shifted Cauchy Contours, and Local Saddle-Point Coalescence in the Transverse Wronskian of Riemann's Xi-Function
Year: 2026
Type: preprint
Abstract: We study the phase-space and complex-contour structure of the transverse Wronskian W(d,t) = (1/2) partial_d |Xi(t - i d)|^2 of Riemann's completed zeta function Xi(t - i d) = xi(1/2 + d + i t). First, we express W(d,t) = 16 int_0^infty y sinh(2 d y) I(y,t) dy in terms of the canonical Wigner distribution I(y,t) = 2*pi*W_Phi(y/2, 2t) of the Jacobi-Riemann kernel Phi(u) and show via Hudson's theorem and the zero-marginal identity int_0^infty I(y, gamma_k) dy = 0 that I(y,t) changes sign. Second, we prove holomorphic parity Phi(-u) = Phi(u) on |Im(u)| < pi/8 and double-exponential vertical-wall decay on |Im(z)| < pi/4, showing that horizontal contour shifts preserve I(y,t) by Cauchy's theorem while improving L^1 quadrature conditioning. Third, we derive the local pitchfork normal form for stationary points of ln G_y(z) + 2 i t z away from zeros of Phi(u), locate explicit zeros of Phi(u) inside |Im(u)| < pi/8, and document why this formalism applies equally to functions with off-critical zeros. Includes the complete 10-page AMS-LaTeX manuscript, compiled academic PDF, and 60-decimal mpmath verification suite.
Author: Missiaen, Thijs
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225801821
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