Order and Chaos in some Trigonometric Series: Curious Adventures of a Statistical Mechanic

Abstract

This paper tells the story how a MAPLE-assisted quest for an interesting undergraduate problem in trigonometric series led some "amateurs" to the discovery that the one-parameter family of deterministic trigonometric series $\pzcS_p: t\mapsto \sum_{n\in\Nset}\sin(n^{-{p}}t)$, $p>1$, exhibits both order and apparent chaos, and how this has prompted some professionals to offer their expert insights. It is proved that $\pzcS_p(t) = \alpha_p\rm{sign}(t)|t|^{1/{p}}+O(|t|^{1/{(p+1)}})\;\forall\;t\in\Rset$, with explicitly computed constant $\alpha_p$. Experts' commentaries are reproduced stating the fluctuations of $\pzcS_p(t) - \alpha_p{\rm{sign}}(t)|t|^{1/{p}}$ are presumably not Gaussian. Inspired by a central limit type theorem of Marc Kac, a well-motivated conjecture is formulated to the effect that the fluctuations of the $\lceil t^{1/(p+1)}\rceil$-th partial sum of $\pzcS_p(t)$, when properly scaled, do converge in distribution to a standard Gaussian when $t\to\infty$, though --- provided that $p$ is chosen so that the frequencies $\{n^{-p}\}_{n\in\Nset}$ are rationally linear independent; no conjecture has been forthcoming for rationally dependent $\{n^{-p}\}_{n\in\Nset}$. Moreover, following other experts' tip-offs, the interesting relationship of the asymptotics of $\pzcS_p(t)$ to properties of the Riemann $\zeta$ function is exhibited using the Mellin transform.