Landau's problem in sieve theory

Solution 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.

Solution 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.