Typical-value anchors and weighted polynomial approximation of the Riemann xi-function

Abstract

For F(z)=ξ((1+iz)/2)/4, we approximate growing families of localized targets by ordinary polynomial multiples of F in the original derivative-plus-absolute-potential norm. A selection of simple real zeros is equipped simultaneously with a harmonic pole bound, a global weighted graph budget, and a typical-value anchor furnished by Selberg's theorem. Transport through a root-free complex corridor and a centered Fourier tail estimate cancel the height factor in the original graph norm. The resulting sufficient log degree is O(log T log log T) for on the order of T log T targets with a fixed whole-span error. Consequently each parity and every sufficiently large degree N admit two rational-basis spaces of dimension at least C exp(c log N/log log N)(log N/log log N), with opposite correction signs and strictly positive original square form on both entire spans. This strengthens the earlier square-root-logarithmic scale. The proof retains all roots, global tails and mixed terms, and also bounds the true graph-norm Ritz residual. We also construct a fixed-width receiver for prescribed entire targets formed from F, a critical cosh factor and a bandlimited multiplier, measured in the original strip derivative energy and real-axis absolute square-form weight. A higher-derivative graph selection then gives the same rational channel spaces, of the stated dimension order, simultaneously for 0 ≤ y ≤ c(log N/log log N)^(−K) with a fixed finite K. The width tends to zero in the stated construction.