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

Let α be an irrational real number; the behaviour of the sum S_N (α):= (−1)^[α] +(−1)^[2α] + ··· +(−1)^[Nα] 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)2log log(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)log log(N))1188.