Distribution of primes?

From the Wikipedia article about the prime number theorem:

Roughly speaking, the prime number theorem states that if a random number nearby some large number N is selected, the chance of it being prime is about 1 / ln(N), where ln(N) denotes the natural logarithm of N. For example, near N = 10,000, about one in nine numbers is prime, whereas near N = 1,000,000,000, only one in every 21 numbers is prime. In other words, the average gap between prime numbers near N is roughly ln(N).

The Sieve of Eratosthenes is a very intuitive visual representation of why the frequency of prime numbers goes down as you go further out on the number line.