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

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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 ₚ: t ∑n∈(n-pt), p>1, exhibits both order and apparent chaos, and how this has prompted some professionals to offer their expert insights. It is proved that ₚ(t) = αₚsign(t)|t|1/p+O(|t|1/(p+1))∀t∈, with explicitly computed constant αₚ. Experts' commentaries are reproduced stating the fluctuations of ₚ(t) - αₚsign(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 t¹/⁽ᵖ⁺¹⁾-th partial sum of 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 ₚ(t) to properties of the Riemann ζ function is exhibited using the Mellin transform.