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

Abstract

Abstract Let α be an irrational real number; the behaviour of the sum SN (α):= (−1)[ α ] +(−1)[2 α ] + ··· +(−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, S N ( 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 S N ( 2 e ) = O ( log ( N ) 2 log 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 S N ( e ) = O ( log ( N ) log log ( N ) ) 1188 .