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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