Asymptotic formula of Modular Prime Counting Function

Question

Is there a simple known asymptotic formula for $\pi_{a,b}(x)=|\{n \leq x: n\equiv a \pmod b \space n:prime\}|$ which is analogous to the prime number theorem? (Here, $a$ and $b$ are coprime.)

My guess would be $\pi_{a,b}(x) \sim \frac{x}{b \log x}.$

Answer 1

The formula depends on $a$ and $b$. If $\gcd(a,b)>1$, then there are only a finite number of primes $p$ such that $p\equiv a \pmod b$, so $\pi_{a,b}(x)$ is finite.

If $\gcd(a,b)=1$, then Dirichlet's theorem on arithmetic progressions is that

$$\pi_{a,b}(x) \sim \frac{x}{\varphi(b) \log x}.$$

where $\varphi(b)$ is Euler function that counts the number of $a$ such that $0<a<b$ and $\gcd(a,b)=1$.