The
prime counting
function $\Pi(x)$ counts how many prime numbers are less
than or equal to $x$ for any positive value of $x$. Since the
primes start $2, 3, 5, 7, 11, 13, \dots$ we therefore have, for
example, $\Pi(11) = 5$ and $\Pi(8) = 4$.
It is believed by mathematicians that $\frac{x}{\ln(x)}$ is a good
approximation to $\Pi(x)$. It is believed to get progressively
better as $x$ increases to very, very large numbers. How well does
it work for lower values of $x$ (up to the 100,000th prime)
Use the following interactivity to examine the percentage accuracy
of this approximation for these values.
This text is usually replaced by the Flash movie.
Use a few sensible values / choice of axes to try to create a
useful graphical representation of $\ln(\Pi(x))$ against $\ln(x)$
for $x$ taking values up to about a million. Use your curve to try
to predict $\Pi(x)$ for a few values of $x$ away from your data
points. How close are your estimates for whole number multiples of
$100,000$?
Use your judgement to try to extrapolate your curve to make
approximations as to
$$
\Pi(10^7)\quad\quad \Pi(10^8)\quad\quad \Pi(10^9)
$$