This suprises me about S. The primes become arbitrarily sparse, on average, so their natural density should exist and = 0. No?
The # of primes <= x is asymptotic to x/ln(x), so the density of primes <= x is asymptotic to 1/ln(x), which -> 0 as x -> oo,
If the primes' density is 0, then it seems that must also be true of any subset of the primes, such as the one menitoned above.
Apologies - I meant to say the natural density in the set of primes! (and not the density in the positive integers) 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? Gary McGuire