Multipoint Padé Approximation of the Hurwitz Zeta Function and a Riemann-Hilbert Steepest Descent Analysis
Artur Kandaian
TL;DR
The paper develops a rigorous Deift–Zhou steepest-descent analysis for multipoint Padé approximants of type $(n,n)$ to the Hurwitz zeta function $\zeta(s,a)$ with $\Re s>1$, using quantile nodes generated by a positive analytic density on $[A,B]$. By formulating the approximation as a $2\times2$ Riemann–Hilbert problem with poles and applying pole removal, contour compression, a $g$-transformation, and lens opening, the authors construct an outer parametrix and Airy-type endpoint parametrices. In the regular one-cut soft-edge regime and under the nondegeneracy condition $\mathrm{(ND)}_n$, they obtain uniform $O(1/n)$ control for the final ratio problem, yielding strong asymptotics for the Padé numerator and denominator on $\mathbb{C}\setminus[A,B]$ and Airy scaling in $O(n^{-2/3})$ neighborhoods of the endpoints. The approach ties discrete orthogonality, equilibrium potential theory, and Riemann–Hilbert analysis to provide a precise asymptotic description of multipoint Padé approximants to the Hurwitz zeta function, with potential implications for related Hermite–Padé and discrete orthogonality problems. The results establish a robust framework for sharp asymptotics in a nontrivial special-function setting, demonstrating the power of integrable systems techniques in approximation theory.
Abstract
We study multipoint Padé approximants of type $(n,n)$ for the Hurwitz zeta function $f(a)=ζ(s,a)$ with $\Re s>1$, constructed at quantile nodes $a_{n,j}=nα_{n,j}$ generated by a real-analytic density $κ$ 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 Padé numerator and denominator admit strong asymptotics uniformly on compact subsets of $\mathbb{C}\setminus[A,B]$, and exhibit Airy scaling in $O(n^{-2/3})$ neighborhoods of the edges.