On the number of primes between prime $p_n$ and $p_{n}^2$

Question

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.

Answer 1

There is nothing special about the difference $\pi(p_n^2) - \pi(p_n)$; we can take the input to be any large positive number. That is, it makes sense to consider

$$\displaystyle \pi(x^2) - \pi(x), x \gg 1.$$

By the prime number theorem we have

$$\displaystyle \pi(y) \sim \frac{y}{\log y},$$

and if we use the logarithmic integral we have the more precise form

$$\displaystyle \pi(y) = \int_2^y \frac{dt}{\log t} + O \left(y \exp \left( - A (\log y)^{\frac{3}{5}} \right) \right)$$

for some $A > 0$. Therefore, we see that

$$\displaystyle \pi(x^2) \sim \pi(x^2) - \pi(x)$$

for all $x$ sufficiently large. Assuming the Riemann hypothesis, one can expect that the fluctuations of

$$\displaystyle \pi(x^2) = \int_2^{x^2} \frac{dt}{\log t} + O_\varepsilon \left(x^{1 + \varepsilon} \right)$$

will overwhelm the main term of $\pi(x) \sim x (\log x)^{-1}$.