The Willow Transform and the Blind Zone Theorem: A Sharp Obstruction in the Turán Hierarchy for the Riemann ξ-Function
Let Ξ(t)=ξ(1/2+it) denote the Riemann ξ-function in the critical-line parametrisation, let Ξ(t)=Σ_{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 associated Toeplitz minors. By the Schoenberg--Edrei characterisation of Pólya frequency sequences, positivity of D(n,r) for all n≥0 and all r≥1 is equivalent to the Riemann hypothesis (RH); for each fixed r the inequalities D(n,r)≥0 supply a necessary condition for RH, and for r=2 they were proved unconditionally by Csordas, Norfolk and Varga, so that the second level of the…
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