Order and Chaos in Some Deterministic Infinite Trigonometric Products

Abstract

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.

Desain Pembangkit Kunci Block Cipher Berbasis CSPRNG Chaos Menggunakan Fungsi Trigonometri

“…Ke-enam nisbah trigonometri yaitu sin x, cos x, tan x, sec x, csc x, dan cot x dijadikan sebagai generator untuk menghasilkan bilangan acak berbasis CSPRNG chaos. Penelitian [7]- [9] sebelumnya juga menggunakan fungsi trigonemetri sebagai pembangkit bilangan acak. Tetapi pendekatan dalam menggunakan fungsi berbeda, penelitian ini menggunakan fungsi pemotongan untuk mengambil bilangan bulat dari mantisa hasil iterasi.…”