 If a subset of primes "A" has a natural density, given by the limit of

then it also has a Dirichlet density, and the two densities are the same.However it is usually easier to show that a set of primes has a Dirichlet density, and this is good enough for many purposes.For example, in proving Dirichlet's theorem on arithmetic progressions, it is easy to show that the Dirichlet density of primes
in an arithmetic progression "a" + "nb" (for "a", "b" coprime) has Dirichlet density 1/φ("b"), which is enough to show that there are an infinite number of such primes, but harder to show that this is the natural density.