Let $a$ and $b$ be coprime (i.e. $a \perp b$). Let $f(a,b,x)$ denotes the number of the primes such that $p=ak+b$ and not greater than $x$. For example $f(4,1,10)= 1$.
Is there an asymptotic formula for $f(a,b,x)$ which is similar to prime number theorem?
There are $\sim\frac{x}{\varphi(a)\log x}$ primes of the form $ak+b$ below x, a consequence of the prime number theorem in arithmetic progressions (as Greg mentioned).