# Prime Number Theorem

So, I lied. I decided not to give a proof of the prime number theorem. Here’s a brief overview of what it says (without proof). If $\pi(x)$ denotes the number of primes less than or equal to $x$, then the prime number theorem says

$\pi(x)\sim \frac x{\log x}$

In other words, if I asked you to estimate the number of primes less than $10^6$, a good guess would be $10^6/\log(10^6)\approx 72382.4$. In actuality, the number is $78498$, so we’d be off by about six-thousand. This may seem like a lot, but compared to the one-million, it’s not that bad. Only $0.6\%$ error. In fact, here’s a table that might be of interest:

When we say $\pi(x)\sim\frac x{\log x}$, we essentially mean that as $x$ gets big, the percent error drops to zero (which we see numerically in the table). More technically, we mean that

$\displaystyle\lim_{x\to\infty}\frac{\pi(x)\log x}x=1$.

So that about wraps up what I want to say about primes. I have no idea what’s coming next, but stay tuned!