Multipoint Padé Approximation of the Hurwitz Zeta Function and a Riemann-Hilbert Steepest Descent Analysis

We study multipoint Padé approximants of type $(n,n)$ for the Hurwitz zeta function $f(a)=\zeta (s,a)$ with $\mathrm{\Re }s>1$, constructed at quantile nodes $a_{n,j}=n{\alpha }_{n,j}$ generated by a real-analytic density $\kappa$ on $[A,B]⋐(0,\mathrm{\infty })$. Under the determinantal nondegeneracy condition ${(\mathrm{N}\mathrm{D})}_n$ for large $n$ and in the regular one-cut soft-edge regime of the associated constrained equilibrium problem, we formulate the approximation as a matrix Riemann–Hilbert problem with poles and carry out a Deift–Zhou nonlinear steepest descent analysis. We construct an explicit outer parametrix together with Airy-type local parametrices at the endpoints and reduce the problem to a small-norm Riemann–Hilbert problem with uniform $O(1\mathrm{/}n)$ control. As a consequence, the Padé numerator and denominator admit strong asymptotics uniformly on compact subsets of $\mathbb{C}\setminus [A,B]$, and exhibit Airy scaling in $O(n^{-2\mathrm{/}3})$ neighborhoods of the edges.