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

TL;DR

This paper develops a detailed asymptotic analysis of multipoint Padé approximants for the Hurwitz zeta function using Riemann-Hilbert techniques, revealing precise behavior and scaling near the function's branch points.

Contribution

It introduces a Riemann-Hilbert framework for analyzing multipoint Padé approximants of the Hurwitz zeta function, including explicit parametrices and asymptotics in the complex plane.

Findings

Strong uniform asymptotics for numerator and denominator of Padé approximants

Airy-type local behavior at the edges of the approximation domain

Reduction to a small-norm Riemann-Hilbert problem with O(1/n) control

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.