The Fourier polynomial $S(\alpha)=\sum_{n\le N}b(n)e(n\alpha)$ where $b(n)$ is the characteristic function of some set ”of multiplicative sort” (say primes, or sums of two squares, ...) is a central object. It contains different informations than counting points results do. We will use the polynomial on sums of two squares and, more classically, the one on primes to examine the border between sieve and bilinear decompositions. Time depending, we will continue with more exploration of these Fourier polynomials.