Multipoint Pad\'e Approximation of the Hurwitz Zeta Function and a Riemann-Hilbert Steepest Descent Analysis

Abstract

We study multipoint Pad\'e approximants of type $(n,n)$ for the Hurwitz zeta function $f(a)=\zeta(s,a)$ with $\Re s>1$, constructed at quantile nodes $a_{n,j}=n\alpha_{n,j}$ generated by a real-analytic density $\kappa$ on $[A,B]\Subset(0,\infty)$. Under the determinantal nondegeneracy condition $\mathrm{(ND)}_n$ 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 $O(1/n)$ control. As a consequence, the Pad\'e numerator and denominator admit strong asymptotics uniformly on compact subsets of $\mathbb{C}\setminus[A,B]$, and exhibit Airy scaling in $O(n^{-2/3})$ neighborhoods of the edges.