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

摘要

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 S_p: t {mapping} Σ_(n∈?)sin(n~(-pt)), p > 1, exhibits both order and apparent chaos, and how this has prompted some professionals to offer their expert insights. As to order, an elementary (undergraduate) proof is given that ?t∈?, with explicitly computed constant αp. As to chaos, the seemingly erratic fluctuations about this overall trend are discussed. Experts' commentaries are reproduced as to why the fluctuations of Sp(t) - αpsign(t) {pipe}t{pipe}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 ?t~1/~((p+1))?-th partial sum of S_p(t), when properly scaled, do converge in distribution to a standard Gaussian when t~(→∞), though-provided that p is chosen so that the frequencies {n~-p}n∈? are rationally linear independent; no conjecture has been forthcoming for rationally dependent {n~-p}_(n∈?). Moreover, following other experts' tip-offs, the interesting relationship of the asymptotics of Sp(t) to properties of the Riemann ζ function is exhibited using the Mellin transform.