Polynomial synthesis and zero ideals for the Riemann xi-function

Abstract

Let F(z)=xi((1+iz)/2)/4. On every fixed horizontal strip of half-width y>1, we approximate arbitrary holomorphic zero-compatible Dirichlet targets by ordinary polynomial multiples of F in the same boundary norm, without assuming a larger-strip extension. For all positive widths we also prove synthesis with a strict strip margin. Convergence retains the derivative energy of the entire product, an anchored value, and the original real-axis absolute-potential weights. A paired-product minimum-modulus estimate controls initially large quotients; Gaussian regularization, Fourier truncation and a Meixner–Pollaczek expansion then give a constructive return to polynomials. The zero-free collars outside the unconditional zero strip permit a same-width Sobolev Fourier passage. In one fixed Hilbert space at each y>1, and in the compatible projective space over all finite strips, the closure of the polynomial multiples is exactly the full zero ideal. The potential weights themselves force cancellation at real zeros. The remaining quotient has complex dimension equal to total nonreal zero multiplicity when that number is finite, and infinite dimension otherwise. This gives an F-dependent density characterization equivalent to the Riemann hypothesis. A singular-inner countermodel explains why fixed-width decay and interior zero jets alone do not imply a general synthesis theorem.