Dirichlet density of $p \equiv 1 \bmod 8$

Question

I am pretty new to this website although it looks really useful. Could someone help me please? My question is that same as this one: Computing the Dirichlet Density

I can't work out how to calculate the density of the prime numbers which are congruent to $1$ modulo $8$ (I tried to comment but you need reputation). I know the density of primes congruent to $1$ modulo $4$ is $\frac{1}{2}$. Does that help?

Thanks.

Answer 1

It is $1/\varphi(8) = 1/4$ where $\varphi \colon \mathbb{N} \to \mathbb{N}$ is the Euler totient function. You need to take a closer look at the proof of Dirichlet's theorem on primes in arithmetic progressions.