Order and Chaos in Some Deterministic Infinite Trigonometric Products

It is shown that the deterministic infinite trigonometric products (Formula presented.) with parameters p∈(0,1]&s>12, and variable t∈ R, are inverse Fourier transforms of the probability distributions for certain random series Ωpζ(s) taking values in the real ω line; i.e. the Clp;s(t) are characteristic functions of the Ωpζ(s). The special case p= 1 = s yields the familiar random harmonic series, while in general Ωpζ(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 Ωpζ(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 Kp;s>0, and a function Fp;s(|t|) bounded by (Formula presented.) the regularity of Ωpζ(s) follows. Incidentally, this theorem confirms a surmise by Benoit Cloitre, that lnCl1/3;2(t)∼-Ct(t→∞) for someC> 0. Graphical evidence suggests that Cl1/3;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 Cl1/3;2), and illustrated by random sampling of the Riemann-ζ walks, whose branching rules allow the build-up of fractal-like structures.