Upper-bound sieve

An upper-bound sieve bounds the size of a sifted set from above. In a dimension-one problem with one forbidden class modulo each relevant prime $p$, its main density factor is comparable to ${\prod }_p(1-1/p)$.

If three nonproportional linear forms exclude three distinct residue classes modulo every sufficiently large prime, the upper-bound sieve has dimension three and gives an upper bound of order $x/(\log x{)}^3$.