A curiosity about $(-1)^{[e]}+(-1)^{[2e]}+\cdots+(-1)^{[Ne]}$. - Archive ouverte HAL
Pré-Publication, Document De Travail Année : 2020

A curiosity about $(-1)^{[e]}+(-1)^{[2e]}+\cdots+(-1)^{[Ne]}$.

Francesco Amoroso

Résumé

Let $\alpha$ be an irrational real number; the behavior of the sum $S_N(\alpha):=(-1)^{[\alpha]}+(-1)^{[2\alpha]}+\cdots+(-1)^{[N\alpha]}$ depends on the continued fraction expansion of $\alpha/2$. Since the continued fraction expansion of $\sqrt{2}/2$ has bounded partial quotients, $S_N(\sqrt{2})=O(\log(N))$ and this bound is best possible. The partial quotients of the continued fraction expansion of $e$ grow slowly and thus $S_N(2e)=O(\frac{\log(N)^2}{\log\log(N)^2})$, again best possible. The partial quotients of the continued fraction expansion of $e/2$ behave similarly as those of $e$. Surprisingly enough $S_N(e)=O(\frac{\log(N)}{\log\log(N)})$.
Fichier principal
Vignette du fichier
E8.pdf (296.49 Ko) Télécharger le fichier
Origine Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-02872415 , version 1 (17-06-2020)

Identifiants

  • HAL Id : hal-02872415 , version 1

Citer

Francesco Amoroso, Moubinool Omarjee. A curiosity about $(-1)^{[e]}+(-1)^{[2e]}+\cdots+(-1)^{[Ne]}$.. 2020. ⟨hal-02872415⟩
249 Consultations
151 Téléchargements

Partager

More