Gary McGuire <gmg@maths.nuim.ie> wrote:
To repeat the question, let S be the set of primes with first digit 1. Define the natural density of S in the primes to be the limit of
number of elements in S that are <= n -------------------------------------- as n --> infinity. number of primes that are <= n
This does not exist (stated in Serre "A course in arithmetic"). Why so?
S is as far ahead as possible when x=2*10^n, after it's just gotten a whole bunch of primes starting with 1. At that point the ratio gets up to nearly 1/2, since there are about as many primes in [x,2x] as in [0,x]. (Of course, only "about" in the limit -- the ratio is close to 2( ln x / (ln x + ln 2) ) - 1, and for big enough x, that's 1ish.) But S is as far behind as possible when x=10^n, when there have been no primes starting with 1 for a long time. By similar logic, by then the ratio is down to about 1/9. Since the ratio keeps getting near those two far-apart marks, it can't possibly tend to a limit. --Michael Kleber -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.