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 
denotes the number of primes less than or equal to 
, then the prime number theorem says
In other words, if I asked you to estimate the number of primes less than 
, a good guess would be 
. In actuality, the number is 
, 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 
error. In fact, here’s a table that might be of interest:
When we say 
, we essentially mean that as gets big, the percent error drops to zero (which we see numerically in the table). More technically, we mean that
.
So that about wraps up what I want to say about primes. I have no idea what’s coming next, but stay tuned!
