Multipoint Padé Approximation of the Hurwitz Zeta Function and a Riemann-Hilbert Steepest Descent Analysis
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914315498422272 |
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| author | Kandaian, Artur |
| author_facet | Kandaian, Artur |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_08399 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Multipoint Padé Approximation of the Hurwitz Zeta Function and a Riemann-Hilbert Steepest Descent Analysis Kandaian, Artur Classical Analysis and ODEs 41A21, 41A20, 42C05, 30E10, 31A15, 11M35 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. |
| title | Multipoint Padé Approximation of the Hurwitz Zeta Function and a Riemann-Hilbert Steepest Descent Analysis |
| topic | Classical Analysis and ODEs 41A21, 41A20, 42C05, 30E10, 31A15, 11M35 |
| url | https://arxiv.org/abs/2602.08399 |