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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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Cusped rainbows and incoherence effects in the rippling-mirror model for particle scattering from surfaces.
J. Phys. A 8 (4), pp. 566–584.
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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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3: About MathML
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►In rare cases, a browser lacks both MathML support and a robust enough javascript implementation capable of running MathJax; you may wish to visit the Customize DLMF page and choose the HTML+images document format.
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►Most modern browsers support ‘Web Fonts’, fonts that are effectively included with a web site.
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4: Tom M. Apostol
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►He was a visiting professor at the University of Patras in Greece in 1978, and was elected a Corresponding Member of the Academy of Athens in 2001 (where he delivered his inaugural lecture in Greek).
►He was internationally known for his textbooks on calculus, analysis, and analytic number theory, which have been translated into five languages, and for creating Project
MATHEMATICS!, a series of video programs that bring mathematics to life with computer animation, live action, music, and special effects.
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►He additionally served as a visiting lecturer for the MAA, and as a member of the MAA Board of Governors.
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5: 25.17 Physical Applications
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►See Armitage (1989), Berry and Keating (1998, 1999), Keating (1993, 1999), and Sarnak (1999).
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►Quantum field theory often encounters formally divergent sums that need to be evaluated by a process of regularization: for example, the energy of the electromagnetic vacuum in a confined space (Casimir–Polder effect).
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6: Sidebar 7.SB1: Diffraction from a Straightedge
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►The faint circular patterns are additional diffraction effects due to imperfections in the edge.
7: 15.18 Physical Applications
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►The hypergeometric function has allowed the development of “solvable” models for one-dimensional quantum scattering through and over barriers (Eckart (1930), Bhattacharjie and Sudarshan (1962)), and generalized to include position-dependent effective masses (Dekar et al. (1999)).
►More varied applications include photon scattering from atoms (Gavrila (1967)), energy distributions of particles in plasmas (Mace and Hellberg (1995)), conformal field theory of critical phenomena (Burkhardt and Xue (1991)), quantum chromo-dynamics (Atkinson and Johnson (1988)), and general parametrization of the effective potentials of interaction between atoms in diatomic molecules (Herrick and O’Connor (1998)).
8: 5.21 Methods of Computation
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►An effective way of computing in the right half-plane is backward recurrence, beginning with a value generated from the asymptotic expansion (5.11.3).
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9: 27.17 Other Applications
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►Schroeder (2006) describes many of these applications, including the design of concert hall ceilings to scatter sound into broad lateral patterns for improved acoustic quality, precise measurements of delays of radar echoes from Venus and Mercury to confirm one of the relativistic effects predicted by Einstein’s theory of general relativity, and the use of primes in creating artistic graphical designs.
10: 7.21 Physical Applications
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►Voigt functions , , can be regarded as the convolution of a Gaussian and a Lorentzian, and appear when the analysis of light (or particulate) absorption (or emission) involves thermal motion effects.
…Dawson’s integral appears in de-convolving even more complex motional effects; see Pratt (2007).
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