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1: Sidebar 9.SB2: Interference Patterns in Caustics
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►The bright sharp-edged triangle is a caustic, that is a line of focused light.
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2: Sidebar 5.SB1: Gamma & Digamma Phase Plots
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►This pattern is analogous to one that would be seen in fluid flow generated by a semi-infinite line of vortices.
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►The fluid flow analogy in this case involves a line of vortices of alternating sign of circulation, resulting in a near cancellation of flow far from the real axis.
3: Sidebar 9.SB1: Supernumerary Rainbows
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►The faint line below the main colored arc is a ‘supernumerary rainbow’, produced by the interference of different sun-rays traversing a raindrop and emerging in the same direction.
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4: 25.18 Methods of Computation
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§25.18(ii) Zeros
►Most numerical calculations of the Riemann zeta function are concerned with locating zeros of in an effort to prove or disprove the Riemann hypothesis, which states that all nontrivial zeros of lie on the critical line . Calculations to date (2008) have found no nontrivial zeros off the critical line. …5: Karl Dilcher
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►Over the years he authored or coauthored numerous papers on Bernoulli numbers and related topics, and he maintains a large on-line bibliography on the subject.
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6: 25.10 Zeros
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§25.10(i) Distribution
… ►The Riemann hypothesis states that all nontrivial zeros lie on this line. … ►Calculations relating to the zeros on the critical line make use of the real-valued function … ►Calculations based on the Riemann–Siegel formula reveal that the first ten billion zeros of in the critical strip are on the critical line (van de Lune et al. (1986)). More than 41% of all the zeros in the critical strip lie on the critical line (Bui et al. (2011)). …7: 4.3 Graphics
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►Lines parallel to the real axis in the -plane map onto rays in the -plane, and lines parallel to the imaginary axis in the -plane map onto circles centered at the origin in the -plane.
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8: 7.19 Voigt Functions
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7.19.4
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is sometimes called the line broadening function; see, for example, Finn and Mugglestone (1965).
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9: 25.17 Physical Applications
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►Analogies exist between the distribution of the zeros of on the critical line and of semiclassical quantum eigenvalues.
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