## HRI Lecture Notes Series

Arithmetical Aspects of the Large Sieve Inequality

Oliver Ramare

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.

Contents

IntroductionThe 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

HRI Lecture Notes Series - 1

2009, 9788185931906, 210 pages, paper cover, Rs. 320.00

2009, 9788185931906, 210 pages, paper cover, Rs. 320.00