The Willow Transform and the Blind Zone Theorem: A Sharp Information-Theoretic Obstruction in the Turán Hierarchy for the Riemann xi-Function
Let Xi(t) = xi(1/2 + it) be the Riemann Xi-function, let Xi(t) = sum_{k>=0} (-1)^k c_k t^{2k} be its even Taylor expansion at the origin, and let D(n,r) = det[c_{n+i-j}] (i,j = 0..r-1) be the Toeplitz minors whose positivity for all n >= 0, r >= 1 is equivalent to the Riemann Hypothesis (RH). We prove a sharp and quantitative obstruction in the resulting hierarchy of criteria. Parametrising the zeros of G(z) = sum c_k z^k by beta_j = -1/z_j, we establish by the dual Jacobi-Trudi identity the exact identity D(n,r) = s_{(r^n)}(beta), the Schur function of the rectangular partition (r^n) in the…
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