Density of Primes in an interval

Question

Is it possible to characterize all or at least some $k \in \mathbb{N}$ such that there are more than $k$ primes between $k^2$ and $\lfloor k^2/2\rfloor$? Or is it possible to characterize all or at least some $k \in \mathbb{N}$ such that there are at most $k$ primes between $k^2$ and $\lfloor k^2/2\rfloor$?

Answer 1

Let $k \geq 16$. By Wikipedia bounds on the prime-counting function, $\pi(k^2/2) <1.25506\frac{k^2}{4\ln{k}-2\ln{2}} \leq 1.25506\frac{k^2}{3.5\ln{k}}$, and $\pi(k^2) \geq \frac{k^2}{2\ln{k}}$. Thus $\pi(k^2)-\pi(k^2/2) \geq \frac{k^2}{\ln{k}}(0.5-1.25506/3.5)$. So as soon as $\frac{k}{\ln{k}} (0.5-1.25506/3.5) > 1$, your statement holds.

Using Wolfram, we can thus see that the statement holds for all $k \geq 22$, and the smaller numbers can be tested manually.