2009 | OriginalPaper | Buchkapitel
Fourier expansion of sieve weights
verfasst von : Olivier Ramaré
Erschienen in: Arithmetical Aspects of the Large Sieve Inequality
Verlag: Hindustan Book Agency
Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.
Wählen Sie Textabschnitte aus um mit Künstlicher Intelligenz passenden Patente zu finden. powered by
Markieren Sie Textabschnitte, um KI-gestützt weitere passende Inhalte zu finden. powered by
The previous chapter contains an expansion of <math display='block'> <msub> <mi>Σ</mi> <mi>d</mi> </msub> <msub> <mi>λ</mi> <mi>d</mi> </msub> <msub> <mn>1</mn> <mi>ℒ</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>β</mi> <mi>κ</mi> </msub> <mrow><mo>(</mo> <mi>n</mi> <mo>)</mo></mrow><mo>=</mo><msup> <mrow> <mrow><mo>(</mo> <mrow> <mstyle displaystyle='true'> <munder> <mo>∑</mo> <mi>d</mi> </munder> <mrow> <msub> <mi>λ</mi> <mi>d</mi> </msub> <msub> <mn>1</mn> <mrow> <msub> <mi>ℒ</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).