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I was reading some questions about prime numbers posted in latest days and a question came to my mind:

What is the state of art of the research into prime numbers distribution?

I read then several other questions (this, this or this for example), all interesting but no one really fitting: is there any recent new from researchers in this field? Any milestone or breakthrough discovery related to it in the latest few years?

Is approach to this open problem changing in time? Thanks to somebody in particular?

On one side I do not check regularly such high topics, on the other I seriously risk not to understand properly some of what I ask. This was meant to be a sort of soft question, hope it's on topic. Any links to (soft or not) dissertations over the recent evolution of this study is welcome.

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closed as too broad by 6005, Eric Stucky, colormegone, vadim123, Claude Leibovici May 11 '14 at 4:05

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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Essentially you are asking for a survey of the modern developments in understanding prime numbers, and I believe it would be best to start by looking at some expository papers.

Here is a paper by Granville title Analytic Number Theory which appears in the Princeton companion to mathematics. It starts off very slow, and I think it is a great place to start. (Alternatively, here is another expository paper by Granville title A good new millennium for prime)

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For a partial answer see:

http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes

Everything there is certainly state of the art.

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