Legendre's conjecture says that there is always a prime between $n^2$ and $(n+1)^2$ for every positive integer.
See : https://en.wikipedia.org/wiki/Legendre%27s_conjecture
Now I wonder why people believe it and what their arguments are.
I wonder about that because I know the reasons for similar conjectures like the prime twin conjecture. Many conjectures in number theory (such as the prime twins) have arguments that are based on modular arithmetic and some kind of equidistribution.
But I do not see that here much. No convincing modular argument imo. Nothing related to zeta functions either.
Note, I am not talking about for sufficiently large $n$ but for all positive integer $n$. I am also not looking for checking the first 1000 integers, or assuming stronger conjectures like Cramer. Although Cramer has an argument of equidistribution.
I would like to point out that using $\frac{x}{\ln(x)}$ is not such a good estimate for the density of primes. Even $\pi(x) = Li(x) + O(\ln^3(x) \sqrt x)$ is stronger than what we can prove and yet weaker than the RH. Since the error term is larger than the square root, it is clear this (alone) is insufficient. Hence this estimate $\frac{x}{\ln(x)}$ does not make such a good argument.
So are there modular arguments for Legendre's conjecture ? Or any kind of sieve ?