1995
DOI: 10.1080/00029890.1995.12004647
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Cosine Products, Fourier Transforms, and Random Sums

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Cited by 9 publications
(10 citation statements)
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“…Given the simplicity, elegance, and age of Viète's product, it is surprising that there seems to have been few attempts at finding similar formulas. One interesting possibility is given in [9], where the relationship between Viète's product and probability and Fourier transforms is explored. An elementary generalization combining Viète's product and Wallis's famous product has also been given in [10].…”
Section: Introductionmentioning
confidence: 99%
“…Given the simplicity, elegance, and age of Viète's product, it is surprising that there seems to have been few attempts at finding similar formulas. One interesting possibility is given in [9], where the relationship between Viète's product and probability and Fourier transforms is explored. An elementary generalization combining Viète's product and Wallis's famous product has also been given in [10].…”
Section: Introductionmentioning
confidence: 99%
“…On the other hand, the alternating signs harmonic number H n = 1 − 1 2 + 1 3 − · · · + (−1) n+1 n converges to log 2 as n → +∞. Building on earlier work by Morrison [8,9], Schmuland [16] proved that the random harmonic series…”
Section: Introductionmentioning
confidence: 88%
“…Starting at ω = 0, he keeps carrying out these random steps ad infinitum. We call this a "random Riemann-ζ walk," and its outcome (whenever it converges) is a "random Riemann-ζ function" evaluated at s. Absolute convergence is guaranteed for each and every such walk when s > 1 (because the series (1) for ζ(s) converges absolutely for s > 1), and by a famous result of Rademacher conditional convergence holds with probability 1 when s > 1 2 , see [Kac59], [Mor95], and [Sch03]. Since the harmonic series diverges logarithmically, the outcome of the random walks with 1 2 < s ≤ 1 is distributed over the whole real line; see [Sch03] for s = 1.…”
Section: Random Riemann-ζ Walks and Their Kinmentioning
confidence: 99%
“…random coefficients, with Prob(R p (n) = 0) = 1 − p and Prob(R p (n) = 1) = p/2 = Prob(R p (n) = −1). We draw heavily on the probabilistically themed publications by Kac [Kac59], Morrison [Mor95], and Schmuland [Sch03], in which the random harmonic series Ω harm ≡ Ω ζ 1 (1) is explored; in [Sch03] also the special case Ω ζ 1 (2) is explored. We register that p = 1 and s = 1 in Cl p;s (t) yields (cf.…”
Section: Introductionmentioning
confidence: 99%
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