On the Fourier-Walsh Spectrum on the Moebius Function
  We study the Fourier-Walsh spectrum $\{\hat\mu (S); S\subset\{1, ..., n\}\}$ of the Moebius function $\mu$ restricted to $\{0, 1, 2, ..., 2^n-1\}\simeq \{0, 1\}^n$ and prove that it is not captued by levels \{\hat\mu (S)| \, |S|< n^{\frac 23-\epsion}\}. An application to correlation with monotone Boelean functions is given. 
Jean Bourgain
math.NT