Convolution Smoothing in the Turán Hierarchy: The Blind Zone Extends from Order 43 to Order 61, and Empirically to Order 68
Let Xi(t)=xi(1/2+it) be the Riemann xi-function on the critical line, Xi(t)=sum_{k>=0}(-1)^k c_k t^{2k} its even Taylor expansion, and D(n,r)=det[c_{n+i-j}]_{i,j=0}^{r-1} the associated Toeplitz minors. Writing G(z)=sum_{k>=0}c_k z^k and beta_j=-1/z_j for the zeros of G, the dual Jacobi-Trudi identity gives D(n,r)=c_0^r s_{(r^n)}(beta), the Schur function of the rectangular partition (r^n) in the alphabet beta. Positivity of D(n,r) for all n>=0 and all r>=1 is equivalent to the Riemann hypothesis (RH) by the Schoenberg-Edrei characterisation of Polya frequency sequences; for each fixed r,…
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