  Boolean functions on the space $F_{2}^m$ are not only important in the theory
of error-correcting codes, but also in cryptography, where they occur in
private key systems. In these two cases, the nonlinearity of these function is
a main concept. In this article, I show that the spectral amplitude of boolean
functions, which is linked to their nonlinearity, is of the order of
$2^{m/2}\sqrt{m}$ in mean, whereas its range is bounded by $2^{m/2}$ and $2^m$.
Moreover I examine a conjecture of Patterson and Wiedemann saying that the
minimum of this spectral amplitude is as close as desired to $2^{m/2}$. I also
study a weaker conjecture about the moments of order 4 of their Fourier
transform. This article is inspired by works of Salem, Zygmund, Kahane and
others about the related problem of real polynomials with random coefficients.
