Cosine Products, Fourier Transforms, and Random Sums

Morrison, Kent E.

doi:10.1080/00029890.1995.12004647

Cited by 9 publications

(10 citation statements)

References 2 publications

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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%

A New Class of Infinite Products Generalizing Viète's Product Formula for π

Levin

2005

Ramanujan J

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We show how functions F(z) which satisfy an identity of the form F(αz) = g(F(z)) for some complex number α and some function g(z) give rise to infinite product formulas that generalize Viète's product formula for π . Specifically, using elliptic and trigonometric functions we derive closed form expressions for some of these infinite products. By evaluating the expressions at certain points we obtain formulas expressing infinite products involving nested radicals in terms of well-known constants. In particular, simple infinite products for π and the lemniscate constant are obtained.

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Section: Introductionmentioning

confidence: 99%

A New Class of Infinite Products Generalizing Viète's Product Formula for π

Levin

2005

Ramanujan J

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“…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%

Small values of signed harmonic sums

Bettin

Molteni

Sanna

2018

Comptes Rendus. Mathématique

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“…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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Order and Chaos in Some Deterministic Infinite Trigonometric Products

Albert

Kiessling

2017

J Stat Phys

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It is shown that the deterministic infinite trigonometric products n∈N 1 − p + p cos n −s t =: Cl p;s (t) with parameters p ∈ (0, 1] & s > 1 2 , and variable t ∈ R, are inverse Fourier transforms of the probability distributions for certain random series Ω ζ p (s) taking values in the real ω line; i.e. the Cl p;s (t) are characteristic functions of the Ω ζ p (s). The special case p = 1 = s yields the familiar random harmonic series, while in general Ω ζ p (s) is a "random Riemann-ζ function," a notion which will be explained and illustrated -and connected to the Riemann hypothesis. It will be shown that Ω ζ p (s) is a very regular random variable, having a probability density function (PDF) on the ω line which is a Schwartz function. More precisely, an elementary proof is given that there exists some K p;s > 0, and a function F p;s (|t|) bounded by |F p;s (|t|)| ≤ exp K p;s |t| 1/(s+1) ), and C p;the regularity of Ω ζ p (s) follows. Incidentally, this theorem confirms a surmise by Benoit Cloitre, that ln Cl 1/3;2 (t) ∼ −C √ t (t → ∞) for some C > 0. Graphical evidence suggests that Cl 1/3;2 (t) is an empirically unpredictable (chaotic) function of t. This is reflected in the rich structure of the pertinent PDF (the Fourier transform of Cl 1/3;2 ), and illustrated by random sampling of the Riemann-ζ walks, whose branching rules allow the build-up of fractal-like structures.

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Cosine Products, Fourier Transforms, and Random Sums

Cited by 9 publications

References 2 publications

A New Class of Infinite Products Generalizing Viète's Product Formula for π

A New Class of Infinite Products Generalizing Viète's Product Formula for π

Small values of signed harmonic sums

Order and Chaos in Some Deterministic Infinite Trigonometric Products

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