Landau's problem in sieve theory

Question

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?

Answer 1

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.

Answer 2

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.