# Prime Number Theorem

May 29, 2012 Leave a comment

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!