Unexpected biases in the distribution of consecutive primes

Abstract: While the sequence of primes is very well distributed in the reduced residue classes (mod q), the distribution of pairs of consecutive primes among the permissible ϕ(q)2 pairs of reduced residue classes (mod q) is surprisingly erratic. This paper proposes a conjectural explanation for this phenomenon, based on the Hardy-Littlewood conjectures. The conjectures are then compared to numerical data, and the observed fit is very good. - https://arxiv.org/abs/1603.03720

paper: "Unexpected biases in the distribution of consecutive primes" http://arxiv.org/abs/1603.03720 explanation for non-experts: "The Last Digit of Prime Numbers - Numberphile" https://www.youtube.com/watch?v=YVvfY_lFUZ8

The article starts out a bit slow in a convoluted way which may be off-putting. There has been recently a major surprising discovery related to consecutive primes (prepub 11 Mar 2016 http://arxiv.org/abs/1603.03720). The value of the article here imho. compared with the referenced original paper (lots of math formulas) is that visualization may help exploring the topic. The author is showing some results of his exploration and is inviting us to join by sharing his code.

Well, whether it's obvious or not, I think it's in the paper. Immediately under equation (1.1) of http://arxiv.org/abs/1603.03720, the authors are discussing the second correction term to the distribution of primes mod q, and they state: "We can also show that c2(q; (a, b)) = c2(q; (−b,−a)) for any two reduced residue classes a and b (mod q)." I'm not 100% certain this is responsible for the phenomenon that we're seeing, but it seems exceedingly likely. I think I'd need to stare at their formula for c2 for a long time to understand where this relation comes from, though.

I don't know why you're nitpicking. The article's written for a more general audience that may be interested in the property, but not necessarily the nitty-gritty math behind it. The first paragraph of the article links to the paper, so people who want more detail can get it. http://arxiv.org/pdf/1603.03720v1.pdf For your second point, I don't think there's anything wrong with a paper announcing they found something interesting, even if they haven't completely analyzed every aspect of it. Getting the info out early lets a wider audience look at it, and opens their current research up to scrutiny.