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 $\pzcS_p(t) = α_p\rm{sign}(t)|t|^{1/{p}}+O(|t|^{1/{(p+1)}})\;\forall\;t\in\Rset$, with explicitly computed constant $α_p$. Experts' commentaries are reproduced stating the fluctuations of $\pzcS_p(t) - α_p{\rm{sign}}(t)|t|^{1/{p}}$ 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 $ζ$ function is exhibited using the Mellin transform.

Based on the invited lecture with the same title delivered by the author on Dec.19, 2011 at the 106th Statistical Mechanics Meeting at Rutgers University in honor of Michael Fisher, Jerry Percus, and Ben Widom. (19 figures, colors online). Comments of three referees included. Conjecture 1 revised. Accepted for publication in J. Stat. Phys