Cojocaru A. An Introduction to Sieve Methods and Their Applications 2005

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Sieve theory has a rich and romantic history. The ancient question of whether there exist infinitely many twin primes (primes p such that p+2 is also prime), and Goldbach's conjecture that every even number can be written as the sum of two prime numbers, have been two of the problems that have inspired the development of the theory. This book provides a motivating introduction to sieve theory. Rather than focus on technical details which can obscure the beauty of the theory, the authors focus on examples and applications, developing the theory in parallel. The text can be used for a senior level undergraduate course or an introductory graduate course in analytic number theory.
Some basic notions
Some elementary Sieves
The normal order method
The Turán Sieve
The Sieve of Eratosthenes
Brun's Sieve
Selberg's Sieve
The Large Sieve
The Bombieri–Vinogradov Theorem
The lower bound Sieve
New directions in Sieve Theory
References
Index