HRI Lecture Notes Series

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 which, when applied to the Farey dissection, will reveal connections between this inequality, the Selberg sieve and other less used notions such as 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 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's 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 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.
Introduction
The large sieve inequality
An extension of the classical arithmetical theory of the large sieve
Some general remarks on arithmetical functions
A geometric interpretation
Further arithmetical applications
The Siegel zero effect
A weighted hermitian inequality
A first use of local models
Twin primes and local models
The Selberg sieve
Fourier expansion of sieve weights
The Selberg sieve for sequences
An overview
Some weighted sequences
Small gaps between primes
Approximating by a local model
Selecting other sets of moduli
Sums of two squarefree numbers
On a large sieve equality
Appendix
Notations
References
Index