Arithmetical aspects of the large sieve inequality

Includes index

Includes bibliography (p. 189-198) and references.

1. The large sieve inequality
2. An extension of the classical arithmetical theory of the large sieve
3. Some general remarks on arithmetical functions
4. A geometrical interpretation
5. Further arithmetical applications
6. The Siegel zero effect
7. A weighted Hermitian inequality
8. A first use of local models
9. Twin primes and local models
10. The three primes theorem
11. The Selberg sieve
12. Fourier expansion of sieve weights
13. The Selberg sieve for sequences
14. An overview
15. Some weighted sequences
16. Small gaps between primes
17. Approximating by a local model
18. Selecting other sets of moduli
19. Sums of two squarefree numbers
20. On a large sieve equality

This book is an elaboration of a series of lectures given at the Harish-Chandra Research Institute. The reader will be taken through a journey on the arithmetical sides of the large sieve inequality when applied to the Farey dissection. This will reveal connections between this inequality, the Selberg sieve and other less used notions like pseudo-characters and the $\Lambda_Q$-function, as well as extend these theories. One of the leading themes of these notes is the notion of so-called\emph{local models} that throws a unifying light on the subject. As examples and applications, the authors present, among other things, an extension of the Brun-Tichmarsh Theorem, a new proof of Linnik's Theorem on quadratic residues and an equally novel one of the Vinogradov three primes Theorem; the authors also consider the problem of small prime gaps, of sums of two squarefree numbers and several other ones, some of them being new, like a sharp upper bound for the number of twin primes $p$ that are such that $p+1$ is squarefree. In the end the problem of equality in the large sieve inequality is considered and several results in this area are also proved.