Finite Rodriguez-Villegas Approximants to the Riemann $\xi$-Function

Abstract

We construct a sequence of finite Rodriguez-Villegas transforms converging locally uniformly to the Riemann $\xi$-function in the critical strip. The input is a positive symmetric profile on the unit interval obtained from the Riemann theta kernel through convolution with the hyperbolic-secant kernel and the logistic coordinate. The profile is a Stieltjes function of $x(1-x)$. Its Bernstein polynomials produce reciprocal numerators and exact finite functional equations. The same numerators admit an exact realization as fermionic supertraces, while the Bernstein polynomials are normalized Gibbs traces.