"Daniel" == Daniel Asimov <dasimov@earthlink.net> writes: Gary McGuire writes:
Gary> An example of such a case is Gary> S = {set of primes whose first digit is 1}. It follows easily Gary> from the prime number theorem (according to Serre's "A course Gary> in arithmetic") that the natural density does not exist. Can Gary> anyone explain why this is so?
Daniel> This suprises me about S. The primes become arbitrarily Daniel> sparse, on average, so their natural density should exist and Daniel> = 0. No? Daniel> The # of primes <= x is asymptotic to x/ln(x), so the density Daniel> of primes <= x is asymptotic to 1/ln(x), which -> 0 as x -> Daniel> oo, Daniel> If the primes' density is 0, then it seems that must also be Daniel> true of any subset of the primes, such as the one menitoned Daniel> above. I think that what Serre was talking about here was the relative natural density: # { p < x | p prime and p has leading digit 1}/#{ p < x | p prime} doesn't have a limit as x-->infinity, but, for example it does when 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 first digit phenomenon (first noticed by S. Newcomb 100 years ago) is that a disproportionately large proportion of random numbers begin with the lower numbers. The authors prove the following general theorem: Let $a\sb 1 <a\sb 2<\cdots$ be a suitable subsequence of the integers (which includes the primes) and let $d(A)$ be the density of the sequence $A=\{a\sb 1,\cdots\}$, where $d(A)$ is a generalisation (which the authors call supernatural density) and which includes the logarithmic density and coincides with the ordinary density, if it exists. 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 $\log \sb {10}((k+1)/k)$. Reviewed by P. Erdos