Fourier expansion of sieve weights

Abstract

The previous chapter contains an expansion of ${\Sigma _d}{\lambda _d}{1_\mathcal{L}}_{_d}\left( n \right)$ as a linear combination of additive characters, simply by combining (11.30) and (11.33). The theme of the present chapter is to expand similarly the sieve weights

$${\beta _\kappa }\left( n \right) = {\left( {\sum\limits_d {{\lambda _d}{1_{{\mathcal{L}_d}}}\left( n \right)} } \right)^2}.$$

((12.1))

This is indeed what is done in the case of primes in (Ramaré, 1995) and what is rapidly presented in a general context in (Ramaré & Ruzsa, 2001), equation (4.1.21). Such a material is used in (Green & Tao, 2006).