Number of even integers not satisfying Goldbach's conjecture from Vinogradov. Infinity of numbers not satisfying the Goldbach's conjecture.

Question

If $A(x)$ is the number of even integers less than $x$ that don't write as a sum of two (odd) primes, then $$ \lim_{x\to \infty} \frac{A(x)}{x} = 0$$

That is what is written in my book (Elementary Number Theory from Burton) in the chapter dedicated to the Goldbach conjecture. It is a result from Vinogradov.

It then says that "almost all" even integers satisfy the conjecture.

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I don't understand. If $A(x)\sim _{\infty} \sqrt{x}$ (for example) then the Vinogradov relation would be satisfied and yet there'd be plenty of integers not satisfying the conjecture. Is my example in accordance with the affirmation that "almost all" even integers satisfy the conjecture? If $A(x)\sim _{\infty} \sqrt{x}$ were true, how would "almost all" integers satisfying the conjecture fit with the infinity (what I called "plenty") of integers not satisfying the conjecture?

EDIT: Here is what I understand so far:

Let $B(x)$ be the number of even integers less than $x$ that satisfy the Goldbach conjecture. We have that $B(x) = x/2 - A(x)$

Hence, $\lim_{x\to \infty} \frac{B(x)}{x} = 1/2$

Thanks a lot.

Answer 1

See HERE

In number theory, "almost all positive integers" can mean "the positive integers in a set whose natural density is 1".