Order and Chaos in Some Deterministic Infinite Trigonometric Products

Abstract

It is shown that the deterministic infinite trigonometric products

$$\begin{aligned} \prod _{n\in \mathbb {N}}\left[ 1- p +p\cos \left( \textstyle n^{-s}_{_{}}t\right) \right] =: {\text{ Cl }_{p;s}^{}}(t) \end{aligned}$$

with parameters $p\in (0,1]\ \& \ s>\frac{1}{2}$, and variable $t\in \mathbb {R}$, 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 $C_{p;s}^{}\!:=\!-\frac{1}{s}\int _0^\infty \ln |{1-p+p\cos \xi }|\frac{1}{\xi ^{1+1/s}}\mathrm{{d}}\xi$, such that

$$\begin{aligned} \forall \,t\in \mathbb {R}:\quad {\text{ Cl }_{p;s}^{}}(t) = \exp \bigl ({- C_{p;s}^{} \,|t|^{1/s}\bigr )F_{p;s}^{}(|t|)}; \end{aligned}$$

the regularity of $\Omega _{p}^\zeta (s)$ follows. Incidentally, this theorem confirms a surmise by Benoit Cloitre, that $\ln {\text{ Cl }_{{{1}/{3}};2}^{}}(t) \sim -C\sqrt{t}\; \left( t\rightarrow \infty \right)$ 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.

Appendix on Power Walks

If instead of a step size which decreases by the power law $n\mapsto n^{-s}$ one uses an exponentially decreasing step size $n\mapsto s^{-n}$ with $s>1$, the outcome is a “random geometric series” (a sum over powers of 1 / s with random coefficients $R_p^{}(n)\in \{-1,0,1\}$),

$$\begin{aligned} \Omega _{p}^{{\mathrm{pow}}}(s) := {\sum _{n\in \mathbb {N}}^{}} R_p^{}(n)\frac{1}{s^n}, \quad s >1,\quad p\in (0,1]; \end{aligned}$$

(73)

the pertinent walks are called “geometric walks.” With more general random coefficients one simply speaks of “random power series” and their “power walks.”

All these random variables $\Omega _{p}^{{\mathrm{pow}}}(s)$ have characteristic functions with infinite trigonometric product representations obtainable from our (22) by replacing ${}^\zeta \rightarrow {}^{{\mathrm{pow}}}$ and $n^{-s}\rightarrow s^{-n}$. Some of these can be evaluated in terms of elementary functions. We register a few special cases, beginning with three geometric walks and ending with a countable family of more general (but simple) power walks.

• (i) Setting $p=1$ and $s=2$ gives the chacteristic function (see formula (1) of [11])

  $$\begin{aligned} \Phi ^{}_{\Omega _{1}^{{\mathrm{pow}}}(2)}(t) = \prod _{n\in \mathbb {N}} \cos \left( \frac{t}{2^n}\right) \equiv \frac{\sin t}{t}, \end{aligned}$$

  (74)

  an infinite product⁷ representation of the sinc function derived by Euler algebraically by exploiting the trigonometric angle-doubling formulas (see [11]). Recall that sinc$(t)=\int _{-1}^1\frac{1}{2} e^{it\omega }d\omega$ is the (inverse) Fourier transform of the PDF $f_{\Omega ^{{\mathrm{unif}}}}(\omega )$ of the uniform random variable $\Omega ^{{\mathrm{unif}}}$ on $[-1,1]$, i.e. $f_{\Omega ^{{\mathrm{unif}}}}(\omega ) = \frac{1}{2}$ if $\omega \in [-1,1]$, and $f_{\Omega ^{{\mathrm{unif}}}}(\omega ) =0$ otherwise. Indeed, $\Omega _{1}^{{\mathrm{pow}}}(2)$ is a random walk representation of $\Omega ^{{\mathrm{unif}}}$ equivalent to the binary representation of [0, 1]: recalling that any real number $x\in [0,1]$ has a binary representation⁸ $x = 0.b_1b_2b_3\ldots \equiv \sum _{n\in \mathbb {N}} b_n /2^n$ with $b_n\in \{0,1\}$, and noting that if $x\in [0,1]$ then $\omega :=2x-1\in [-1,1]$, it follows that any real number $\omega \in [-1,1]$ has a binary representation $\omega = \sum _{n\in \mathbb {N}} r_2^{}(n) /2^n$ with $r^{}_2(n)\in \{-1,1\}$. It is manifest that any such representation of $\omega$ is an outcome of $\Omega _{1}^{{\mathrm{pow}}}(2)$.

• (ii) $\Omega _{1}^{{\mathrm{pow}}}(3)$ is the random variable for which the characteristic function

  $$\begin{aligned} \Phi ^{}_{\Omega _{1}^{{\mathrm{pow}}}(3)}(t) = \prod _{n\in \mathbb {N}} \cos \left( \frac{t}{3^n}\right) =:\Phi ^{}_{\Omega ^{{\mathrm{Cantor}}}}(t/2) \end{aligned}$$

  (75)

  is a trigonometric product discussed in [11]. Morrison explains that $\Phi ^{}_{\Omega ^{{\mathrm{Cantor}}}}(t)$ is the characteristic function of a random variable ${\Omega ^{{\mathrm{Cantor}}}}$ that is uniformly distributed over the Cantor set constructed from $[-1,1]$ by removing middle thirds ad infinitum. For uniform distributions on other Cantor sets, see [3]. We remark that this is a nice example of a random walk whose endpoints are distributed by a singular distribution, in the sense that the Cantor set obtained from $[-1,1]$ has Lebesgue measure 0. As pointed out to us by an anonymous referee, the distribution of $\Omega _{1}^{{\mathrm{pow}}}(s)$ is singular and concentrated on some Cantor set for all $s>2$, while for $1< s < 2$ the story is more complicated: Solomyak [20] proved that the distribution of $\Omega _{1}^{{\mathrm{pow}}}(s)$ is absolutely continuous (i.e., it is equivalent to a PDF, an integrable function) for almost every $s \in (1, 2)$; see also [13]. However, the distribution of $\Omega _{1}^{{\mathrm{pow}}}(s)$ is not absolutely continuous for all $s\in (1,2)$—in 1939 Erdős found values of $s\in (1,2)$ for which the distribution is singular; these are still the only ones known. See [14] for further reading.

• (iii) Setting $p=\frac{2}{3}$ and $s=3$ yields

  $$\begin{aligned} \Phi ^{}_{\Omega _{2/3}^{{\mathrm{pow}}}(3)}(t) = \prod _{n\in \mathbb {N}}\left[ \frac{1}{3} +\frac{2}{3}\cos \left( \frac{t}{3^{n}}\right) \right] \equiv \frac{\sin (t/2)}{t/2} , \end{aligned}$$

  (76)

  which becomes formula (9) of [11] under the rescaling $t\mapsto 2t$ (see also Exercise 3 on page 11 of [8].) Recalling our discussion of example (i), we conclude that (76) is the characteristic function of the uniform random variable on the interval $[-\frac{1}{2},\frac{1}{2}]$, expressed as a random walk equivalent to the ternary representation of the real numbers in [0, 1], shifted to the left by $-\frac{1}{2}$.

• (iv) The sinc representations (74) and (76) (after rescaling $t\mapsto 2t$) are merely the first two members of a countable family of infinite trigonometric product representations of $\sin t / t$ derived by Kent Morrison [11], and given by

  $$\begin{aligned} \frac{\sin t}{t} = \prod _{n\in \mathbb {N}}\sum _{m=1-s}^{s-1}\frac{1-(-1)^{s+m}}{2s} \cos \left( \frac{m}{s^n}t\right) ,\; 1<s\in \mathbb {N}; \end{aligned}$$

  (77)

  s even in (77) is formula (12) in [11], s odd in (77) is formula (13) in [11]. These representations of the characteristic function of the uniform random variable over $[-1,1]$ are obtained by considering random walks that enter with equal likelihood into any one of s branches which “s-furkate” off of every vertex of a symmetric tree centered at 0, equivalent to the usual “s-ary” representation of the real numbers in [0, 1] (shifted to the left by $-\frac{1}{2}$ and scaled up by a factor 2). When $s>3$ these are no longer random geometric series, but still simple random power series.