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.



References on Erdos conjecture on arithmetic progressions
MR3203599 Gowers, W. Timothy, Erdős and arithmetic progressions. Erdős Centennial, 265–287, Bolyai Soc. Math. Stud., 25, János Bolyai Math. Soc., Budapest, 2013. The review says the author gives a survey of progress on the conjecture (and on another conjecture of Erdős). 


