# References on Erdos conjecture on arithmetic progressions

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.

## 1 Answer

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).