# Arithmetical Aspects of the Large Sieve Inequality

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*Olivier Ramaré; D. S. Ramana*

A publication of Hindustan Book Agency

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.

A publication of Hindustan Book Agency; distributed within the Americas by the American Mathematical Society. Maximum discount of 20% for all commercial channels.

#### Readership

Graduate students and research mathematicians interested in number theory.