This book develops a unified, modern framework for the large sieve and the Selberg sieve, built on a common system of Hermitian inequalities. It connects classical arguments with more recent methods and applications in analytic and algebraic number theory. The exposition includes improved Brun–Titchmarsh-type bounds, large sieve extensions for sifted sequences, and a development of the enveloping sieve arising from the Selberg sieve. An appendix gathers the relevant mean-value estimates and technical tools into a coherent reference.

Several parts are written with particular audiences in mind. In particular, the chapters on enveloping sieves for the squares and on prime powers and number fields, are aimed, respectively, at harmonic analysts and algebraic number theorists. This second edition refreshes and extends the bibliography and updates the 2005 lecture notes in three main directions. It adds new theoretical material, including Nazarov’s extremal examples, a new Hilbertian basis, Huxley’s lower bound, and explicit smoothings of Holt and Vaaler. Moreover, it expands applications of the Brun–Titchmarsh inequality to products of three primes, an oscillatory sum, and norms in number fields, and it develops further the Selberg sieve as an enveloping sieve, incorporating ideas in the spirit of Green–Tao. This monograph is aimed at researchers and advanced graduate students, offering a unified perspective on classical methods and modern developments in mathematical research.

D. S. Ramana is with the Harish-Chandra Research Institute (HRI), Prayagraj, Uttar Pradesh, India, and was formerly its Acting Registrar. He works in combinatorial and analytic number theory. In particular, he has been interested in problems that apply the large sieve, exponential sums, and ideas from additive combinatorics.  He has supervised several doctoral students, organized and lectured in numerous instructional programmes at various institutions across India on various topics in analytic number theory.

Olivier Ramaré is Senior Researcher at the CNRS, France, in position at Aix-Marseille University, as well as Scientific Officer for the CIMPA. He first became known for his work on the Golbach problem by proving that any even integer can be written as sum of six prime numbers. He worked on the sieve, where from his enveloping sieve stems, on L-functions and on primes. He helped to structure the field of explicit estimates in multiplicative number theory and obtained several results like a particularly simple proof that any arithmetic progression without fixed divisor contains a product of three small primes. He is also involved in the study of mathematical games like solitaire. His latest book, Excursions in Multiplicative Number Theory, has been warmly welcomed.