1206.5031 (Michael K. -H. Kiessling)

Order and Chaos in some Trigonometric Series    [PDF]

Michael K. -H. Kiessling

The one-parameter family of deterministic trigonometric series $\pzcS_p: t\mapsto \sum_{n\in\Nset}\sin(n^{-{p}}t)$, $p>1$, is shown to exhibit both order and apparent chaos. 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 $\alpha_p$. A well-motivated conjecture is formulated concerning the seemingly chaotic fluctuations about this overall trend, to the effect that these fluctuations, when properly scaled, converge in distribution to a standard Gaussian when $t\to\infty$, provided that $p$ is irrational; no conjecture has been forthcoming for rational $p$. Moreover, the interesting relationship of the asymptotics of $\pzcS_p(t)$ to properties of the Riemann $\zeta$ function is worked out.

View original: http://arxiv.org/abs/1206.5031