Fourier expansion of sieve weights

The previous chapter contains an expansion of <math display='block'> <msub> <mi>&#x03A3;</mi> <mi>d</mi> </msub> <msub> <mi>&#x03BB;</mi> <mi>d</mi> </msub> <msub> <mn>1</mn> <mi>&#x2112;</mi> </msub> <msub> <mrow></mrow> <mrow> <msub> <mrow></mrow> <mi>d</mi> </msub> </mrow> </msub> <mrow><mo>(</mo> <mi>n</mi> <mo>)</mo></mrow> </math> $${\Sigma _d}{\lambda _d}{1_\mathcal{L}}_{_d}\left( n \right) $$ 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 (12.1)<math display='block'> <mrow> <msub> <mi>&#x03B2;</mi> <mi>&#x03BA;</mi> </msub> <mrow><mo>(</mo> <mi>n</mi> <mo>)</mo></mrow><mo>=</mo><msup> <mrow> <mrow><mo>(</mo> <mrow> <mstyle displaystyle='true'> <munder> <mo>&#x2211;</mo> <mi>d</mi> </munder> <mrow> <msub> <mi>&#x03BB;</mi> <mi>d</mi> </msub> <msub> <mn>1</mn> <mrow> <msub> <mi>&#x2112;</mi> <mi>d</mi> </msub> </mrow> </msub> <mrow><mo>(</mo> <mi>n</mi> <mo>)</mo></mrow> </mrow> </mstyle> </mrow> <mo>)</mo></mrow> </mrow> <mn>2</mn> </msup> <mo>.</mo> </mrow> </math> $${\beta _\kappa }\left( n \right) = {\left( {\sum\limits_d {{\lambda _d}{1_{{\mathcal{L}_d}}}\left( n \right)} } \right)^2}.$$ 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).