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1: 25.10 Zeros
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►More than 41% of all the zeros in the critical strip lie on the critical line (Bui et al. (2011)).
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2: Bibliography B
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Applied Combinatorial Mathematics.
Robert E. Krieger, Malabar, FL.
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Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model.
Ann. of Math. (2) 150 (1), pp. 185–266.
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Occurrence of periodic Lamé functions at bifurcations in chaotic Hamiltonian systems.
J. Phys. A 34 (40), pp. 8199–8220.
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More than 41% of the zeros of the zeta function are on the critical line.
Acta Arith. 150 (1), pp. 35–64.
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Euler functions with complex and applications.
In Approximation, probability, and related fields (Santa Barbara,
CA, 1993), G. Anastassiou and S. T. Rachev (Eds.),
pp. 127–150.
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3: Errata
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Subsection 25.10(ii)
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In the paragraph immediately below (25.10.4), it was originally stated that “more than one-third of all zeros in the critical strip lie on the critical line.” which referred to Levinson (1974). This sentence has been updated with “one-third” being replaced with “41%” now referring to Bui et al. (2011) (suggested by Gergő Nemes on 2021-08-23).
4: 30.7 Graphics
5: Bibliography
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Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics.
Comput. Phys. Comm. 150 (1), pp. 1–20.
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The -Matrix Method. Developments and Applications.
Springer-Verlag.
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Frequency response characteristics of the multiport planar elliptic patch.
IEEE Trans. Microwave Theory Tech. 40 (8), pp. 1726–1730.
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Sturm-Liouville Theory.
Birkhäuser Verlag, Basel.
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Special Functions: Group Theoretical Aspects and Applications.
D. Reidel Publishing Co., Dordrecht.
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6: Bibliography T
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Computational aspects of incomplete gamma functions with large complex parameters.
In Approximation and Computation. A Festschrift in Honor
of Walter Gautschi, R. V. M. Zahar (Ed.),
International Series of Numerical Mathematics, Vol. 119, pp. 551–562.
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The universal Askey-Wilson algebra and DAHA of type
.
SIGMA 9, pp. Paper 047, 40 pp..
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Representation Theory and Harmonic Analysis.
Contemporary Mathematics, Vol. 191, American Mathematical Society, Providence, RI.
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Sul comportamento asintotico dell’-esimo polinomio di Laguerre nell’intorno dell’ascissa
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Comment. Math. Helv. 22, pp. 150–167.
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Funzioni Ellittiche.
2nd edition, Nicola Zanichelli Editore, Bologna (Italian).
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7: Bibliography K
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Periodic Orbits, Spectral Statistics, and the Riemann Zeros.
In Supersymmetry and Trace Formulae: Chaos and Disorder, J. P. Keating, D. E. Khmelnitskii, and I. V. Lerner (Eds.),
pp. 1–15.
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Elliptic asymptotics of the first and second Painlevé transcendents.
Uspekhi Mat. Nauk 49 (1(295)), pp. 77–140 (Russian).
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On the complex zeros of for real or complex order.
J. Comput. Appl. Math. 40 (3), pp. 337–344.
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Rays and caustics as physical objects.
In Progress in Optics, E. Wolf (Ed.),
Vol. 26, pp. 227–348.
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Algebras of Virasoro type, the energy-momentum tensor, and operator expansions on Riemann surfaces.
Funktsional. Anal. i Prilozhen. 23 (1), pp. 24–40 (Russian).
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8: Bibliography D
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Plancherel-Rotach asymptotic expansion for some polynomials from indeterminate moment problems.
Constr. Approx. 40 (1), pp. 61–104.
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Pólya’s Theory of Counting.
In Applied Combinatorial Mathematics, E. F. Beckenbach (Ed.),
pp. 144–184.
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Sur les zéros imaginaires des polynômes de Bernoulli.
C. R. Acad. Sci. Paris Sér. I Math. 304 (6), pp. 147–150 (French).
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Special Functions.
World Scientific Publishing Co., Inc., River Edge, NJ.
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Coulomb wave functions with complex values of the variable and the parameters.
J. Math. Phys. 40 (12), pp. 6145–6166.