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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.

Sylvain JULIEN
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1 Answers1

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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).

Gerry Myerson
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  • The arxiv version is at https://arxiv.org/abs/1509.03421. A related question is https://mathoverflow.net/questions/198387/most-dense-subset-of-numbers-that-avoids-arbitrarily-long-arithmetic-progression, also https://math.stackexchange.com/questions/232023/has-erdős-conjecture-on-arithmetic-progressions-been-proved – Gerry Myerson Apr 04 '18 at 06:14