Unit-circle location of finite numerator zeros

Determine whether every zero of the reciprocal numerator polynomial U_E(q), constructed from the sampled theta-logistic profile P_\xi for each even positive integer E, lies on the unit circle.

Background

For each even positive integer E, the paper defines the reciprocal polynomial U_E(q)=\sum_{j=0}E(-1)j\binom{E}{j}P_\xi(j/E)qj, where P_\xi is the positive symmetric profile derived from the Riemann theta kernel. The reciprocity relation U_E(q)=qE U_E(q{-1}) is established from the symmetry P_\xi(x)=P_\xi(1-x).

The associated finite Rodríguez-Villegas transforms Z_E satisfy the functional equation Z_E(1-s)=Z_E(s) and converge locally uniformly to the Riemann \xi-function in the critical strip. However, the paper does not establish the stronger unit-circle zero property for U_E; without it, the Rodríguez-Villegas zero theorem cannot be applied to conclude that all zeros of Z_E lie on the critical line.

References

Here reciprocity of U_E follows from the symmetry of the sampled theta profile, but the zeros of U_E are not known to lie on the unit circle.

— Finite Rodriguez-Villegas Approximants to the Riemann $ξ$-Function  (2609.25564 - Silva, 22 Sep 2026) in Section 5, “Concluding remarks and open questions”