Victor Miller wrote:

one looks at relative harmonic density. There is a nice paper by DIA Cohen and Talbot Katz that explains this and other generalizations.

MR0746863 (85j:11014) Cohen, Daniel I. A.(1-RCF); Katz, Talbot M.(1-RCF) Prime numbers and the first digit phenomenon. J. Number Theory 18 (1984), no. 3, 261--268. 11A63 (11B05) .... The authors prove that for a fairly general class of sequences $A$ the relative supernatural density of the subsequence of $A$ whose first digit (in the decimal expansion) is $k$ is $\log \sb {10}((k+1)/k)$.

So the set of primes with first digit 1 has no natural density, but has supernatural/Dirichlet density log_{10} (2) ~= 0.3, the primes with first digit 2 have (supernatural) density log_{10} (3/2) ~=0.176, .... and the primes with first digit 9 have density log_{10} (10/9) ~= 0.046. This would seem to explain the first digit phenomenon all right. Nice that sum_{k=1}^9 log_{10} (k+1)/k = 1. Gary McGuire