Geodesic
Asymptotic structure of eigenvalues and eigenvectors of certain triangular Hankel matrices
Čebyševskij sbornik, Tome 21 (2020) no. 1, pp. 259-272.

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The Hankel matrices considered in the article arose at one reformulation of the Riemann hypothesis proposed earlier by the author. Computer calculations showed that in the case of the Riemann zeta function the eigenvalues and the eigenvectors of such matrices have an interesting structure. The article studies a model situation when instead of the zeta function function one takes a function having a single zero. For this case we indicate the first terms of the asymptotic expansions of the smallest and largest (in absolute value) eigenvalues and the corresponding eigenvectors.
Keywords: Riemann zeta function, Riemann Hypothesis, Hankel matrices, eigenvalues, eigenvectors. 
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Yu. V. Matiyasevich. Asymptotic structure of eigenvalues and eigenvectors of certain triangular Hankel matrices. Čebyševskij sbornik, Tome 21 (2020) no. 1, pp. 259-272. http://geodesic.mathdoc.fr/item/CHEB_2020_21_1_a16/

[1] Beiker Dzh., Greivs-Morris P., Approksimatsii Pade, Mir, M., 1986 ; Baker G. A., Jr., Graves-Morris P. R., Padé Approximations, v. 1, Encyclopedia of mathematics and its applications, 13, Addison Wesley, Reading, Mass., 1981 ; v. 2, 14 | MR | MR

[2] Fonseca C., “The eigenvalues of some anti-tridiagonal Hankel matrices”, Kuwait J. Sci., 45:1, 1–6 | MR | Zbl

[3] Jacobi C. G. J., “Über die Darstellung einer Reihe Gegebner Werthe durch eine Gebrochne Rationale Function”, J. Reine Angew. Math., 30 (1846), 127–156 | MR | Zbl

[4] Matiyasevich Yu., Hidden Life of Riemann's Zeta Function 2. Electrons and Trains, 2007, arXiv: 0709.0028

[5] Matiyasevich Yu., “Riemann's zeta function: Some computations and conjectures”, Proceedings of Conference on Algorithmic Number Theory (Turku, 2007), TUCS General Publication Series, 46, eds. A.-M. Ernvall-Hytönen, M. Jutila, J. Karhumäki, A. Lepsitö, 87–112 http://logic.pdmi.ras.ru/ỹumat/personaljournal/zetahiddenlife/talks/turku2007/Matiyasevich_2007.pdf

[6] Matiyasevich Yu. V., The Riemann hypothesis and eigenvalues of related Hankel matrices. I, Preprint POMI 14-03, 2014 http://www.pdmi.ras.ru/preprint/2014/14-03.html

[7] Matiyasevich Yu. V., Structure of eigenvalues and eigenvectors of certain almost triangular Hankel matrices, Talk given at 7th St.Petersburg Conference in Spectral Theory on July 4 2015

[8] Yu. V. Matiyasevich, “Riemann's hypothesis in terms of the eigenvalues of special Hankel matrices”, Proc. Steklov Inst. Math., 296, suppl. 2 (2017), 78–91 | DOI | DOI | MR

[9] de Montessus de Ballore R., “Sur le fraction continues algébriques”, Bull. Soc. Math. France, 30 (1902), 28–36 | DOI | MR | Zbl

[10] Riemann B., Über die Anzhal der Primzahlen unter einer gegebenen Grösse, Monatsberichter der Berliner Akademie, November 1859 ; Included into: Riemann B., Gesammelte Werke, Teubner, Leipzig, 1892 ; reprinted Dover Books, New York, 1953 http://www.claymath.org/millennium/Riemann_Hypothesis/1859_manuscript/zeta.pdfhttp://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/EZeta.pdf | MR | Zbl | Zbl

[11] Saff E. B., “An extension of Montessus de Ballore's theorem on the convergence of interpolating rational functions”, J. Approximation Theory, 6:1 (1972), 63–67 | DOI | MR | Zbl

[12] Lita da Silva J., Spectral properties of anti-heptadiagonal persymmetric Hankel matrices, arXiv: 1907.00260 | MR