Why don't prime counting approximations work for Brocard's conjecture?

Question

So the number of primes $\le n$ is like $n/\log n$, and so the $n$th prime is like $n \log n$.

Great, so $p_{n}^2 = n^2 \log^2 n$, and so the number of primes smaller would then be

$$\frac{n^2 \log^2 n}{\log(n^2 \log^2 n)}$$

Therefore, we can say the number of primes inbetween $p_n^2$ and $p_{n+1}^2$ would be like

$$f(n)=\frac{(n+1)^2 \log^2 (n+1)}{\log((n+1)^2 \log^2 (n+1))} -\frac{n^2 \log^2 n}{\log(n^2 \log^2 n)}$$

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As you can see, $f(n)>4$ whenever $n>n_0$ for some $n_0$, and the limit of $f$ as $n\to \infty$ clearly is infinity.

So my question is, first of all, all I've done is extremely obvious and every mathematician who tried this problem likely started here, or eventually ended up here one way or another, so this line of thinking has probably been studied somewhere. I'm curious where specifically these approximations fail. It's not like these are bad approximations. You know, we can use similar things with another term in each to upper and lower bound specifically, and try to squeeze somehow. But I don't even bother trying, because I know that any other mathematician would have done it already because what I wanted to try is tedious but overall trivial.

So where are the resources for where tries of this problem have been worked out? Where do these approximations fall apart when you try to use similar lower and upper bounds to constitute a proof. I mean, all we need to show is that the true $f$ has $f > 4$ for large enough $n$. So, y'know, where does this go wrong, when the approximation looks to be increasing as $n\to\infty$? This problem looks very easy but in actuality there is no way to tie things up? So unbelievably frustrating. But I want to see when you have a mathematician try this with various lower and upper bounds where specifically they all go wrong in showing that $f>4$, thank you.

Answer 1

Hint:$$\lim_{n\to\infty}\frac{(n+1)^2 \log^2 (n+1)}{\log((n+1)^2 \log^2 (n+1))} -\frac{(1+\epsilon)n^2 \log^2 n}{\log(n^2 \log^2 n)} = -\infty$$ Whenever $\epsilon>0$. Therefore, it is not enough even that $x/\log x < \pi(x) < \frac{4x}{3\log(x)}$, where $\pi(x)$ is the number of primes less than $x$. You would need that limit above to go to $+\infty$ for $\epsilon \in [0, 1/3]$, which is simply not true.

Answer 2

Estimates for the number of primes up to n are reasonably precise. But they are smooth real valued functions, while the exact number of primes is integer valued, is constant most of the time and jumps by one from time to time.

If you want to estimate the number of primes from n to m, you could take f(m) - f(n-1) where f is any estimate. But if the difference between n and m is much smaller than m, the relative error will be much larger. You can estimate the number of primes up to n = 10^18 and up to n + 1,000,000 quite precisely, compared to n / ln n, but the error will be large compared to 1,000,000 or 1,000,000 / ln 1,000,000.