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.
