Six Routes Towards the Riemann Hypothesis

We revisit the one-scale version of the Nyman–Beurling criterion, in which the dilations are a geometric progression \(\lambda ^{k}\) and the Gram matrix is a Toeplitz form whose symbol \(w_{\lambda }\) is the periodization of \(\vert{}\zeta(1/2 + it)\vert{}^2 / (1/4 + t^2)\). In this setting, the Riemann Hypothesis (RH) is equivalent to \(J_\lambda \to 0\) as \(\lambda \to 1^+\), where \(J_{\lambda }\) is an explicit Szegő-type functional. We propose six routes that connect this model to classical theories not previously exploited in this context, and for each one we state what is proved, what is observed numerically, the precise missing lemma, and its consequence. Proved results include: (i) an exact two-scale (duplication) identity for the symbol and an antipodal integral formula for the increment \(D(\epsilon) = J_{\lambda^2} - J_\lambda\); (ii) the identification of the one-scale problem with orthogonal polynomials on the unit circle, whence, by Nevai–Totik, \(\lim \sup_m \vert{}\alpha_m\vert{}^{1/m} = e^{-\epsilon \alpha_{\min}(\epsilon)}\) for the Verblunsky parameters; (iii) a theorem showing that a pair of zeros off the critical line produces a double zero of the exterior model function pinned at the mirror point, for every \(\lambda \); (iv) a frozen-prime lemma and a block-Toeplitz (matrix Szegő) extremal formula for adding a new prime to a set of smooth dilations; (v) an unconditional \(\ell ^{1}\)–\(L^{p}\) uncertainty inequality for Nyman–Beurling approximants. Numerically, the zeros of \(\zeta \) are recovered from the Verblunsky parameters by a matrix-pencil analysis; the increment \(D(\epsilon)\) is localized on the zeros of height \(\lesssim \pi/\epsilon\); and the Davenport–Heilbronn function provides a negative control in which the model exhibits a pinned double zero of winding index 2. None of the six routes proves RH. The common bottleneck — information at the scale of an individual zero, i.e., the size of \(\vert{}\zeta'(\rho)\vert{}\) and of small gaps — is identified precisely in each case.