The Moebius trigonometric polynomial and the Siegel zero

This talk presents some news on bilinear decompositions of the Moebius
function. In particular, we will exhibit a family of such decompositions
inherited from Motohashi's proof of the Hoheisel and Linnik Theorem that
leads to

\sum_{n\le X, (n,q)=1}\mu(n) e(na/q) << X /\sqrt{q}

for q\<=X^{1/9} and any a prime to q. Such an estimate has
consequences for the value of the derivative of the L-function at the
exceptional zero.