The project has three main objectives, all situated a unified program for high-dimensional sieves. The first one is to establish the bilinear remainder term in the DHR sieve. The second objective is to explore new sieve weights and apply them to study the bounded gaps between primes. The third objective is to establish an upper bound and lower bound for the sifting function and to extend (and improve) J. Brüdern and É. Fouvry’s results which are derived from vector sieve.