Probability that n is a prime in arithmetic progressions

Question

I read that a result that followed the PNT is the probability n is prime is $\frac{1}{\log(n)}$. Does is follow from the PNT on arithmetic progressions that the probability n is prime and is q mod a ( like 1 mod 4 or something ) is $\frac{1}{\phi(a) * \log(n) }$? Thank you

Answer 1

It follows from the PNT that when using the uniform distribution on $1 \ldots n$ letting $p_n$ be the probability that $X$ is prime then $\lim_{n \to \infty} p_n \log n = 1$.

The probability that $X$ is prime and $\equiv a \bmod q$ is $p_{n,a,q}$ and (by the PNT in arthmetic progressions) $\lim_{n \to \infty} p_{n,a,q} \log n$ doesn't depend on $a,gcd(a,q)=1$ thus $ \lim_{n \to \infty} p_{n,a,q} \phi(q) \log n = 1$.