Landau's comment on validity of Goldbach's conjecture

Question

Edmund Landau had once put forth the following statement about the validity of the Goldbach conjecture.

"the goldbach conjecture is false for at most 0 % of all even integers ; this at most 0 % does not exclude, however the possibility that there are infinitely many exceptions"

I have not understood the essence of the statement, like for an ordinary novice student like me, the two statements appear contradicting, whilst they appear to make perfect sense for a number theorist. Can anyone please help me with the understanding of the essence of this statement?

Answer 1

Consider this statement: $0\%$ of all natural numbers are powers of $2$. It is true, in this sense: if, for each natural $n$, you compute the proportion $p_n$ of powers of $2$ in $\{1,2,\ldots,n\}$, then $\lim_{n\to\infty}p_n=0$. But that does not mean that no natural number is a power of $2$.

Answer 2

Here's an analogy.

There are infinitely many numbers of the form $10^k$ - a $1$ followed by some number of zeroes. Among the first $10$ numbers there are just $2$. Among the first $100$ numbers there are just three. Among the first $1000$, four. That pattern continues, so the fraction of all numbers of that form might reasonably be said to be $0$, even though there are infinitely many of them.

Whether there are infinitely many counterexamples to Goldbach's conjecture is unknown. We don't even know there's one.