Introduction to sieve methods in number theory. Paper from the 29th Brazilian mathematics colloquium -- \(29^{\text o}\) Colóquio Brasileiro de Matemática, Rio de Janeiro, Brazil, July 22 -- August 2, 2013 (Q2848164)

In this short book, the author introduces various aspects of sieve methods in number theory. After a short chapter on Chebyshev's bounds and the Abel summation formula the author presents a chapter on elementary sieves like Gallager's sieves and Heath-Brown's perfect squares sieve. This is followed by the Hardy-Ramanujan's inequalities and Turán's normal order theorem for the number of distinct prime factors of a positive integer. Chapter 4 is devoted to Turán basic sieve with applications to counting square values of polynomials. Chapters 5, 6, 7 and 8 present with quite a few details the classical sieves of Erathostenes, Brun, Selberg and the large sieve, respectively. An additional chapter is devoted to the Bombieri-Vinogradov theorem on primes in progressions. Each chapter contains a few classical exercises meant to test the reader's understanding of the material just covered. While invariably in such a project many things would have to be stated without proofs (like the prime number theorem for example), the book gives the basic ideas and enough details of each of the sieves themselves. The book is a good introduction to the basic ideas of sieve methods and could be used by students during a summer course, independent study or as a complement to a beginning course in analytic number theory.