Question about the distribution of primes

Question

Let's say that you have two numbers, for example $5,000$ and $10,000$, and we calculate $\pi(5000)$ and $\pi(10000)$. For a number in between them, such as $7,500$, can we do some sort of interpolation to approximate $\pi(7500)$?

Answer 1

Interpolation as if $\pi(x)$ were a linear function will give a fairly good approximation. For example, $$ \pi(7500) = 950, \quad (\pi(10000) + \pi(5000))/2 = 949 $$ and $$ \pi(6000) = 783, \quad \frac{4}{5}\pi(10000) + \frac{1}{5}\pi(5000)= 781 \, . $$ This works well because the function $x \mapsto \frac{x}{\log x}$ appearing in the Prime Number Theorem is increasing and concave down, with derivatives going to zero.