A Semidefinite Framework for the Sieve

Abstract: We describe a semidefinite programming framework for proving upper bounds on concrete sifting problems, and show that the Large Sieve can be interpreted as a special case of this framework. With a small tweak, the Larger Sieve also falls into this framework. We compare the semidefinite approach to the linear programming approach (i.e., the general framework of the combinatorial sieve and the Selberg sieve), and show that it has a qualitative advantage in a toy case where the primes are completely independent from each other. No new sieve-theoretic bounds are proved.