Selberg sieve

The Selberg sieve constructs an upper bound for a sifting function using a square. Let $P$ be a squarefree integer, choose real weights ${\lambda }_d$ supported on selected divisors of $P$, and impose ${\lambda }_1=1$. ThenWhen $n$ is coprime to $P$, the sum is exactly $1$; otherwise its square is nonnegative, which proves the inequality. Multiplying by nonnegative sequence weights and summing giveswhere $[d,e]$ is the least common multiple and $|{\mathcal{A}}_l|$ is the weight of terms divisible by $l$.

A sieve distribution writes $|{\mathcal{A}}_l|=Xg(l)+r_l$. One then minimizes the main quadratic form ${\sum }_{d,e}{\lambda }_d{\lambda }_{e}g([d,e])$ subject to the support and normalization, while separately controlling the remainder. The Selberg sieve diagonalization explains the minimizing weights. This yields results such as the prime-tuple upper bound from the Selberg sieve, but the nonnegative-square construction by itself supplies only an upper bound.^[ref]