Gary McGuire writes: << ... Number theorists tend to work with another type of density, called Dirichlet density. This is often easier to compute. When both densities exist, they are equal. Sometimes the Dirichlet density exists when the above natural density does not. An example of such a case is S = {set of primes whose first digit is 1}. It follows easily from the prime number theorem (according to Serre's "A course in arithmetic") that the natural density does not exist. Can anyone explain why this is so?
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. --Dan Asimov
"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
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