Is there a known elementary function bound in terms of $a,b,n$ for the $n$-th prime equal to $b$ modulo $a$ (coprime to $b$)?


Bounds on Linnik's constant answer this for the first prime in each progression. Is there a known analogue for an $n$-th prime in a progression? And I found some references on an error term for the prime number theorem for arithmetic progressions. But I don't see how to turn these into a construction for arbitrary $a,b,n$.



How constructive is Dirichlet on primes in progressions?
Corollary 18.8 in Iwaniec-Kowalski's Analytic number theory shows the existence of an explicitly computable $L>3/2$ such that for $x>a^L$ and $(a,b)=1$, the number of primes less than $x$ and congruent to $b$ modulo $a$ is at least a constant times $\frac{x}{\varphi(a)\sqrt{a}\log x}$. So the $n$-th prime congruent to $b$ modulo $a$ is less than a constant times $a^L n\log (2n)$.


