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

TL;DR

This paper explores the intriguing order and chaos in a family of deterministic trigonometric series, combining analytical proofs, expert insights, and conjectures about their fluctuation behavior and connections to the Riemann zeta function.

Contribution

It provides a detailed asymptotic analysis of the series, explicitly computes constants, and formulates a new conjecture on the Gaussian nature of fluctuations, linking to number theory and statistical mechanics.

Findings

Proved asymptotic form of the series with explicit constants

Conjectured Gaussian fluctuations under certain independence conditions

Linked series properties to the Riemann zeta function via Mellin transform

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.