An alleged primary motivator for the RH is so that we can bound the error term $|\text{Li}(x) - \pi(x)|$ by a factor of $O(\sqrt{x}\log x)$. However, I also learned about Riemann's explicit formula $R(x)$ that converges, upon addition of the harmonics $C_i(x, \theta_i)$, where $\theta_i$ is the Riemann spectrum. Thus, we basically know that the Zeta function encodes enough information to construct an exact prime counting function and to know the distribution of the primes, $\pi(x)$, exactly. And even if we put the harmonics aside, $R_0(x)$ itself (no harmonics) is a much better approximation to $\pi(x)$ than just $\text{Li}(x)$, so why do we care so much about bounding the error between $\text{Li}(x)$ and $\pi(x)$? I get that $R(x)$ of course is a function of $\text{li}(x)$, but my point is shouldn't $R(x)$ be the fundamental object of study in the study of the distribution of primes, then?
Another, very related, question I have is why a square-root bound is good at all -- the Prime Number Theorem tells us that $\left|\frac{\pi(x)}{\text{Li}(x)}\right| \to 1$, so $\text{Li}(x)$ gets arbitrarily close to $\pi(x)$ for large enough $x$, so what does this (in my mind, very weak) square-root bound tell us that PNT doesn't already? These questions are probably very naïve -- I have taken a course in complex analysis, but not much number theory.
It’s a decent approximation that can be calculated quickly. So it doesn’t help with limits, it helps with actually finding individual values. How many prime numbers approximately between $1.23 \times 10^{24}$ and $1.24 \times 10^{24}$?