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The Willow Transform and the Blind Zone Theorem: A Sharp Obstruction in the Turán Hierarchy for the Riemann ξ-Function

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Contributed by Changxia Lin ↗

Version 1 / Oct 18, 2026 / CC BY 4.0

Abstract

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 resulting hierarchy is not discriminating. We show that this failure of discrimination persists far beyond level 2, and we determine the range over which it does so: every level r≤43 is non-discriminating. Parametrising the zeros of the generating function G(z)=Σ_{k≥0}c_k z^k by β_j=-1/z_j, the dual Jacobi--Trudi identity yields the exact identity D(n,r)=c_0^r s_{(r^n)}(β), where s_{(r^n)} denotes the Schur function of the rectangular partition (r^n) in the alphabet β. A zero of ζ off the critical line contributes to β a conjugate pair ρ e^{±iθ} whose argument θ coincides with the vertical angle of the corresponding violating pair, whereas zeros on the critical line contribute positive real variables. For an alphabet of cardinality two we prove the closed form s_{(a,b)}(ρ e^{iθ}, ρ e^{-iθ}) = ρ^{a+b} sin((a+1-b)θ)/sin θ = ρ^{a+b} U_{a-b}(cos θ), where U_k is the Chebyshev polynomial of the second kind; the sign is therefore governed by the criterion U_k(cos θ)>0 if and only if (k+1)θ<π. Through the coproduct expansion s_λ(X⊔Y)=Σ_μ s_μ(X)s_{λ/μ}(Y), the Littlewood--Richardson expansion s_{λ/μ}=Σ_ν c^λ_{μν}s_ν of each skew factor, and the positivity of Schur functions on positive real alphabets, the sign analysis reduces to two-variable constituents. Since c^λ_{μν}>0 forces ν⊆λ=(r^n), the index k=ν_1-ν_2 of each constituent is bounded by the number of columns r and is independent of the shift n. All constituents then have the same sign, so no cancellation occurs. The same bound persists for an alphabet containing several conjugate pairs: iterating the coproduct over the pairs and using the fact that the entry 1 occurs at most once in each column of a semistandard tableau bounds every constituent index by ν_1≤r, independently of the number of pairs. This yields the following result. Blind Zone Theorem. Let the alphabet contain any number of violating pairs, each subtending a vertical angle at most θ_max, and suppose θ_max<π/(r+1). Then every minor D(n,r') with r'≤r is strictly positive, notwithstanding the presence of non-real zeros. The critical strip together with γ_1=14.134725141... forces θ≤θ_max=2 arctan(1/(2γ_1))=0.07071826... radians, whence π/θ_max=44.424...; consequently every order r≤43 is non-discriminating: for all n≥0 and all r≤43 the minor D(n,r) is positive if RH fails and non-negative if RH holds, so no minor of order r≤43 can certify either the truth or the falsity of RH. The bound is on the order r and not on the shift n; the corresponding n-based assertion is false, and the mechanism responsible is exhibited in the remarks section. The threshold is established for an arbitrary number of violating pairs; the case of several pairs is settled in the coproduct section and requires no separation condition on the moduli and no bound on the number of pairs. No claim of a proof of the Riemann hypothesis is made.
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Version history

  1. v1Submitted by Changxia LinInitial depositCurrentOct 18, 2026