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We know that by the prime number theorem, the number of primes less than $x$ is approximately $\frac{x}{\ln(x)}$. Another good approximation is $\mathrm{li}(x)$. Despite these estimates, we don't know much about the maximal prime gaps. The weaker conjectures, such as Legendre's conjecture, Andrica's conjecture and Opperman's conjecture, imply a gap of $O(\sqrt{p})$. Stronger conjectures even imply a gap of $O(\ln^2(p))$. The Riemann Hypothesis implies a gap of $O(\sqrt{p} \ln(p))$, though proving this is not sufficient to show the RH. The minimal gap is also a subject of research. It has been shown that gaps smaller of equal to 246 occur infintely often. It is conjectured that gaps equal to 2 occur infinitely often. This is know as the twin prime conjecture.
Another subject is the study of number theoretic functions, most notably $\tau(n)$, the number of divisiors of $n$, $\sigma(n)$, the sum of divisiors of $n$ and $\varphi(n)$, the Euler-phi function, the number of numbers smaller then $n$ coprime with $n$.