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Uniform Distribution Theory
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Journal

Uniform Distribution Theory (UDT)


Volume & Issue

UDT. Volume 15. Issue 2


Published on

December, 2020


Pages

1-8


DOI

https://doi.org/10.2478/udt-2020-0007

Article

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


Authors

Francesco Amoroso and Moubinool Omarjee


Abstract

Let α be an irrational real number; the behaviour of the sum SN (α):= (−1)[α] +(−1)[2α] + ··· +(−1)[Nα] depends on the continued fraction expansion of α/2. Since the continued fraction expansion of 2/2 has bounded partial quotients, SN(2)=O(log(N)) 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)2log 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 SN(e)=O(log(N)log log(N)).


Citation

Amoroso, F. & Omarjee, M. (2020). A curiosity About (−1)[e] +(−1)[2e] + ··· +(−1)[Ne]. Uniform distribution theory, 15(2), 1-8. https://doi.org/10.2478/udt-2020-0007