# Distribution of primes?

Do primes become more or less frequent as you go further out on the number line? That is, are there more or fewer primes between $1$ and $1,000,000$ than between $1,000,000$ and $2,000,000$?

A proof or pointer to a proof would be appreciated.

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Yes (as has been answered): the heuristic reason is that we can list the primes $p_1, p_2, \dots$. Each time you apply the sieve of Eratosnese you exclude the numbers divisible by $p_n$ and weed out "$(1-1/p_n)$" of the natural numbers (i.e., asymptotic density). So in the end you are left with $\prod ( 1 - 1/p_n)$ fraction of the natural numbers; since the sum $\sum \frac{1}{p_n} = \infty$, this is the proportion zero. So we should expect the primes to have density zero. –  Akhil Mathew Jul 21 '10 at 0:23
Check out the Riemann-zeta function if you want something a bit more technical. It gives a pretty precise estimate of the distribution of primes, indirectly. –  Noldorin Jul 21 '10 at 7:38