Sieves in Number Theory

Slightly more than 25 years ago, the first text devoted entirely to sieve meth­ ods made its appearance, rapidly to become a standard source and reference in the subject. The book of H. Halberstam and H.-E. Richert had actually been conceived in the mid-1960's. The initial stimulus had been provided by the paper of W. B. Jurkat and Richert, which determined the sifting limit for the linear sieve, using a combination of the ,A2 method of A. Selberg with combinatorial ideas which were in themselves of great importance and in­ terest. One of the declared objectives in writing their book was to place on record the sharpest form of what they called Selberg sieve theory available at the time. At the same time combinatorial methods were not neglected, and Halber­ stam and Richert included an account of the purely combinatorial method of Brun, which they believed to be worthy of further examination. Necessar­ ily they included only a briefer mention of the development of these ideas due