Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Order and Chaos in some Trigonometric Series: Curious Adventures of a
  Statistical Mechanic   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.

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The categories of this paper are: astro-ph
33
The categories of this paper are: plasm-ph
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The categories of this paper are: funct-an
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The categories of this paper are: nucl-th
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The categories of this paper are: econ
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The categories of this paper are: chem-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: q-fin
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The categories of this paper are: solv-int
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The categories of this paper are: ao-sci
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The categories of this paper are: alg-geom
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The categories of this paper are: atom-ph
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The categories of this paper are: cs
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The categories of this paper are: q-bio
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The categories of this paper are: supr-con
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The categories of this paper are: bayes-an