Erdos conjectured that any set $ A $ of positive integers such that $ \sum_{n\in A}\dfrac{1}{n} $ diverges contains arbitrary long arithmetic progressions. The celebrated Green-Tao theorem is a special case of this conjecture, where $ A $ is the set of primes.

I would like to have references on this conjecture, and also to know if considering a set $ S $ of L-functions whose elements $ F : s\mapsto\sum_{n>0}\dfrac{a_{n}}{n^{s}} $ can help shed a light on it through 'twisting' $ F $ by the function $ 1_{A} : n\mapsto 1 $ iff $ n\in A $ and $ 1_{A}(n)=0 $ otherwise.