The parity problem in sieve theory

Let $A$ be a set whose elements are all products of an odd number of primes, or all products of an even number of primes. Parity problem. Without injecting additional ingredients, sieve theory is unable to provide non-trivial lower bounds on the size of $A$; moreover, any upper bounds must be off from the truth by a factor of at least $2$. This is presented as a fundamental unresolved obstruction in sieve theory, preventing proofs of problems such as the twin prime and Goldbach conjectures.