1. bookVolumen 15 (2020): Edición 2 (December 2020)
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eISSN
2309-5377
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30 Dec 2013
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access type Acceso abierto

A curiosity About (1)[e] +(1)[2e] + ··· +(1)[Ne]

Publicado en línea: 25 Dec 2020
Volumen & Edición: Volumen 15 (2020) - Edición 2 (December 2020)
Páginas: 1 - 8
Recibido: 05 Jun 2020
Aceptado: 17 Jun 2020
Detalles de la revista
License
Formato
Revista
eISSN
2309-5377
Primera edición
30 Dec 2013
Calendario de la edición
2 veces al año
Idiomas
Inglés
Abstract

Let α be an irrational real number; the behaviour of the sum SN (α):= (1)[α] +(1)[2α] + ··· +(1)[] depends on the continued fraction expansion of α/2. Since the continued fraction expansion of 2/2\sqrt 2 /2 has bounded partial quotients, SN(2)=O(log(N)){S_N}\left( {\sqrt 2 } \right) = O\left( {\log \left( N \right)} \right) and this bound is best possible. The partial quotients of the continued fraction expansion of e grow slowly and thus SN(2e)=O(log(N)2loglog(N)2){S_N}\left( {2e} \right) = O\left( {{{\log {{\left( N \right)}^2}} \over {\log \,\log {{\left( N \right)}^2}}}} \right), again best possible. The partial quotients of the continued fraction expansion of e/2 behave similarly as those of e. Surprisingly enough SN(e)=O(log(N)loglog(N))1188.

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MSC 2010

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