It is shown that the deterministic infinite trigonometric products
$$begin{aligned} prod _{nin mathbb {N}}left[ 1- p +pcos left( textstyle n^{-s}_{_{}}tright) right] =: {text{ Cl }_{p;s}^{}}(t) end{aligned}$$
with parameters
( pin (0,1] & s>frac{1}{2}), and variable
(tin mathbb {R}), are inverse Fourier transforms of the probability distributions for certain random series
(Omega _{p}^zeta (s)) taking values in the real
(omega ) line; i.e. the
({text{ Cl }_{p;s}^{}}(t)) are characteristic functions of the
(Omega _{p}^zeta (s)). The special case
(p=1=s) yields the familiar random harmonic series, while in general
(Omega _{p}^zeta (s)) is a “random Riemann-
(zeta ) function,” a notion which will be explained and illustrated—and connected to the Riemann hypothesis. It will be shown that
(Omega _{p}^zeta (s)) is a very regular random variable, having a probability density function (PDF) on the
(omega ) 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|)|!le ! exp big (K_{p;s}^{} |t|^{1/(s+1)})), and
(C_{p;s}^{}!:=!-frac{1}{s}int _0^infty ln |{1-p+pcos xi }|frac{1}{xi ^{1+1/s}}mathrm{{d}}xi ), such that
$$begin{aligned} forall ,tin mathbb {R}:quad {text{ Cl }_{p;s}^{}}(t) = exp bigl ({- C_{p;s}^{} ,|t|^{1/s}bigr )F_{p;s}^{}(|t|)}; end{aligned}$$
the regularity of
(Omega _{p}^zeta (s)) follows. Incidentally, this theorem confirms a surmise by Benoit Cloitre, that
(ln {text{ Cl }_{{{1}/{3}};2}^{}}(t) sim -Csqrt{t}; left( trightarrow infty right) ) for
some (C>0). Graphical evidence suggests that
({text{ 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
({text{ Cl }_{{{1}/{3}};2}^{}})), and illustrated by random sampling of the Riemann-
(zeta ) walks, whose branching rules allow the build-up of fractal-like structures.
