Order and Chaos in some deterministic infinite trigonometric products

Abstract

This paper discusses some infinite trigonometric products which are characteristic functions of simple random walks on the real line; in fact, these define "random Riemann-$\zeta$ functions," a notion which is explained. The concept of typicality for random Riemann $\zeta$ functions is explained and connected to the Riemann hypothesis. Then it is shown that the distribution functions of these random walks are Schwarz functions. It is also shown that their characteristic function factors into the characteristic function of a Levy stable random variable and a subdominant fluctuating factor. As a corollary it also follows that amateur mathematician Benoit Cloitre's infinite trigonometric product $\prod_{n=1}^\infty \left[\frac23+\frac13\cos\left(\frac{x}{n^{2}}\right)\right] = e^{- C \,\sqrt{|x|} +\varepsilon(|x|)},$ with $|\varepsilon(|x|)| \leq K |x|^{1/3}$ for some $K>0$, and with $ C= \int\frac{\sin\xi^2}{2+\cos\xi^2}{\rm{d}}\xi;$ numerically, $C = 0.319905585... \sqrt{\pi}$. This confirms a surmise of Benoit Cloitre. The $O\big(|x|^{1/3}\big)$ error bound is empirically found to be accurate for moderately sized $|x|$ but not for larger $|x|$. This difference $\varepsilon(|x|)$ between Cloitre's $\log \prod_{n\geq1}\left[\frac23 +\frac13\cos\left(\frac{x}{n^{2}}\right)\right]$ and its regular trend $-C\sqrt{|x|}$, although deterministic, appears to be an "empirically unpredictable" function. Our probabilistic investigation of this phenomenon connects the fluctuations to the "random Riemann-$\zeta$ function with argument 2."