does anyone knows if there are studies on the number of primes between prime $p_n$ and $p_{n}^2$, where $p_n$ is the $n$-th prime?

I am studying it through the following formula:

\begin{align} \pi(p^2_n)-\pi(p_n) \simeq (p_{n}^2-1) \prod_{i=1}^{n} \frac{(p_i-1)}{p_i}, \end{align}

where $p_n$ is the $n$-th prime number, $p_i$ the $i$-th primer number as well, and $\pi(n)$, for positive integer $n$, is the prime counting function.

I have seen a related question here:

The values of $n$ which satisfy an inequality about prime numbers

**EDIT:**

Since there are questions about the interest of this, let me explain why it is interesting for me:

All the integers up to $p_n^2$ have only prime factors in $\{2,3,\ldots,p_n\}$. So if we exclude the even numbers (*i.e.* $1/2$), then the multiples of $3$ (*i.e.* $2/3$), then the multiples of $5$ (*i.e.* $4/5$) and so on ... up to $p_n$ we should obtain exactly the prime numbers in the interval $(p_n,p_n^2)$. However it seems not to be as easy as it appears.