Finite Rodriguez-Villegas Approximants to the Riemann $ξ$-Function
We construct a sequence of finite Rodriguez-Villegas transforms converging locally uniformly to the Riemann $ξ$-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.
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