Landau's problem in sieve theory In Tao's blog, one of Landau's problems is interpreted in the setting of sieve theory. More precisely, the twin prime conjecture leads to considering the following:

*

*$A$ the set of prime numbers on $[x/2, x)$

*$E_p$ the set of residue classes $0$ ans $-2$ mod $p$, for all prime $p$
and we are interested in the cardinality of the sifted set
$$A \backslash \bigcup_{p \leq \sqrt{x}} E_p.$$
The blog post claims that there are analogous formulations "sieve shape" for the other Landau's problem, but I do not find anything matching (the sieving sets $E_p$ can for instance rule out divisors, hence selecting primes, but I don't see how it can select sums of primes or so...).
What are sieve statements of these Landau problems?
 A: I hope this is right:
If
\[ A=\{ n(N-n)|n\leq N\} \]
and
\[ E_p=\{ dp|d\in \mathbb N\} \]
then
\[ A-\bigcup _{p\leq \sqrt N}E_p\]
leaves us with the elements of $A$ that are coprime to all $p\leq \sqrt N$, in other words they are composed of primes $>\sqrt N$.  An element $n(N-n)$ of $A$ can be composed of such primes only if $n$ and $N-n$ are themselves prime.  If $n$ and $N-n$ is prime then $N$ is a sum of two primes, so that's the Goldbach problem.
The other problems have similar set ups.
A: The answer provided by @tomos is essentially how mathematicians study the representation of even numbers as a sum of two almost primes. That is, by applying various analytic methods, they obtain lower bound for the cardinality of the following set
$$
\mathcal A\setminus\bigcup_{p\le N^{1/u}}\mathcal A_p
$$
where $A=\{n(N-n):n\le N\}$ and $\mathcal A_d=\{a\in\mathcal A:d|a\}$.
When $u$ is a positive integer, this quantity would provide lower bound for the number of ways to express $N$ as a sum of two almost primes of order $u-1$ (i.e. product consisting of at most $u$ prime factors).
It was not until the 1940s that Rényi successfully showed that every large even integer can be expressed as a sum of a prime and an almost prime by analyzing the asymptotic behavior of the cardinality of the following set:
$$
\mathcal B\setminus\bigcup_{p\le N^{1/u}}\mathcal B_p
$$
where $\mathcal B=\{N-p:p\le N\}$ and $\mathcal B_d=\{b\in\mathcal B:d|b\}$.
In brief, I would say that both $\mathcal A$ and $\mathcal B$ would be possible to characterize the Goldbach's problem.
