A Semidefinite Framework for the Sieve

TL;DR

This paper introduces a semidefinite programming framework for analyzing sieve problems, unifying the Large and Larger Sieve, and compares its advantages to linear programming approaches.

Contribution

It presents a novel semidefinite framework for sieve bounds, linking classical sieves to modern optimization techniques and highlighting qualitative advantages over linear programming methods.

Findings

Large Sieve is a special case of the framework

Larger Sieve can be incorporated with a small modification

Semidefinite approach shows qualitative benefits in toy models

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.