Let $\{a_n\}_{n\in\mathbb{N}}$ be defined as $a_n = a + bn$ for some $a, b >0,(a, b) = 1$. Are there good bounds on the minimal $k$ s.t. $a_k$ is prime. It is well known that there are infinitely many primes in this series,

This is Linnik's theorem, and the best known bound is $O(b^5)$ due to Xylouris. (This is in the Wikipedia page, and as I admitted in this year's JMM I added it to the Wikipedia page. It's in his thesis but otherwise unpublished as far as I know.) 

