On the theory of prime producing sieves

We develop the foundations of a general framework for producing optimal upper and lower bounds on the sum $\sum_p a_p$ over primes $p$, where $(a_n)_{x/2<n\le x}$ is an arbitrary non-negative sequence satisfying Type I and Type II estimates. Our lower bounds on $\sum_p a_p$ depend on a new sieve method, which is non-iterative and uses all of the Type I and Type II information at once. We also give a complementary general procedure for constructing sequences $(a_n)$ satisfying the Type I and Type II estimates, which in many cases proves that our lower bounds on $\sum_p a_p$ are best possible. A key role in both the sieve method and the construction method is played by the geometry of special subsets of $\mathbb{R}^k$. This allows us to determine precisely the ranges of Type I and Type II estimates for which an asymptotic for $\sum_p a_p$ is guaranteed, that a substantial Type II range is always necessary to guarantee a non-trivial lower bound for $\sum_p a_p$, and to determine the optimal bounds in some naturally occurring families of parameters from the literature. We also demonstrate that the optimal upper and lower bounds for $\sum_p a_p$ exhibit many discontinuities with respect to the Type I and Type II ranges, ruling out the possibility of a particularly simple characterization.