Abel-Renormalized Continuous Dual Hahn Expansions for the Riemann Zeta Function

This preprint develops an Abel-renormalized continuous dual Hahn expansion for the Riemann zeta function, starting from the centered Taylor coefficients of the Jacobi theta constant introduced by Romik. A specific continuous dual Hahn polynomial family is combined with these arithmetic coefficients to form an Abel-regularized series. As the regulator approaches the boundary, the divergence is shown to be carried by two explicit boundary modes. After subtracting these modes, the remaining finite part is identified exactly with a gamma-normalized form of the Riemann zeta function. The resulting renormalized family extends to an entire-function-valued regulator flow. The paper establishes a full boundary expansion to all orders, exact evaluations on a distinguished reciprocal-gamma lattice, and a decomposition of the lattice values into an algebraic boundary contribution and an exponentially small modular correction. The regulator jets satisfy a closed finite-difference recursion, reducing every higher-order coefficient to finitely many shifted values of the limiting zeta-related function. For a natural arithmetic choice of the regulator, the two boundary modes become the familiar dual pair associated with the functional equation, while the renormalized continuous dual Hahn series admits an all-orders expansion in inverse even powers of the cutoff parameter. The same specialization produces a precise hierarchy at the reciprocal-gamma lattice with exponentially small errors. The limiting entire function is also related explicitly to the Riemann Xi function and Hardy’s Z-function. Compact-open convergence of the regulator family yields convergence of zero counts on bounded domains and, near every simple limiting zero, an all-orders asymptotic expansion for the corresponding zeros of the finite-regulator approximants. The first displacement is expressed explicitly through shifted normalized zeta values. The paper also shows that the boundary-subtracted real-space kernel has a fixed sign throughout the regulator flow, leading to a positive-definite normalized Fourier transform. All principal results are proved analytically. Supplementary files provide high-precision computations for independent verification of selected identities and normalizations.