Talk:Elliott–Halberstam conjecture

Question

on Dan Goldston, it says "they can also show that primes within 16 of each other occur infinitely often"

this article says "shows (assuming this conjecture) that there are infinitely many pairs of primes which differ by at most 20. '20' instead of '16'.

What gives? GangofOne 04:09, 29 September 2005 (UTC)

I checked the source material. Looks like there was an earlier result of 20 and a few months later it was improved to 16. I've updated the page accordingly. Terry 04:22, 29 September 2005 (UTC)

According to http://arxiv.org/abs/1311.4600 it appears this number should be revised to 12. However I'm not sure if that paper has been peer reviewed yet. Can someone with more knowledge on the subject check whether this edit should be made? Alsee (talk) 14:25, 20 November 2013 (UTC)

The Goldston, Pintz, Yildirim reference is pretty sketchy

Just a untitled mystery pointer into the arxiv. I believe there are better surveys on the web; perhaps a real mathematician could choose one. Weren't there (subsequently fixed) problems with the early G-P-Y papers? Hopefully not the referenced one! Also, "recent result" has a way of dating. Thanks to whoever eventually makes the effort to fix this. Maybe it will even be me.

Original conjecture

There are a number of statements in the literature and in the MR which are referred to as the Elliott-Halberstam conjecture, and which are all very similar to the version described in the page. I thought the original EH conjecture should at least be mentioned (which we know of course is false, but can be reformulated by replacing the condition $X<x(\log x)^{-A-1}$ by $X<x^{\theta }$ for any $\theta <1$). These versions are probably all more or less equivalent, at least in the weak sense that their assumptions can be indifferently used to prove the same conditional results. Sapphorain (talk) 14:45, 20 September 2024 (UTC)