Yes, this should be the case; I currently have a graduate student checking the details of this. (One slight issue here is that whereas the original argument requires distribution results only for squarefree moduli, the extension to ak+b requires distribution results for moduli which are the product of a and a squarefree number, and so one needs a very slight generalisation of the existing distribution results. But this appears to only be a minor technical difficulty.)

]]>I meant at a distance Na.

]]>Can the proof be generalized to arithmetic progressions ak+b, gcd(a,b)=1 ? That is, are there infinitely many pairs of primes at a distance at most Nb in such progression? ]]>

Thank you Eytan. In passing, I’m also wondering whether the current Deligne-avoiding k0 of 1788 can be nudged down, though of course this is of more marginal interest to most observers than the headline 632 value.

]]>I will also participate of course. I’ll also use my own subversion setup independently of the Dropbox, so this might also help limit any possible difficulties with editing conflicts.

I will probably sketch the conductor bound in a blog post next week. Interestingly, it really turns out to be better to proceed with the algebraic analogue of the companion sum argument, although the latter certainly gives the right idea of what is going on.

]]>That’s a good idea, and thanks for offering to chip in! It may take a while before we get to the point where we can start writing seriously the Deligne stuff (it depends to some extent on how your writeup with Emmanuel and Etienne on the conductor bounds for pushforward sheaves turns out) but this is certainly something to discuss in the next post. I’ll try to set something up in a day or two (I’m hopping on a plane in a few hours).

Regarding alternatives to dropbox, I am familiar with Subversion, and with a lesser extent with Git, but the learning curve (particularly for the latter) is steeper with that for Dropbox, and for the fairly limited task we have at hand (writing a single paper) I think these more advanced version control platforms may be overkill. Of course, one pays a price for this, which is that Dropbox has more difficulty dealing with conflicts in which two people are trying to edit the same file. But I think we can coordinate this through the blog; I have split the paper up into one file for each section, and if people announce what section they are working on, one should be able to avoid serious conflicts in practice (particularly if we try to only modify the Dropbox files while connected to the internet). For instance, I am going to be working more or less exclusively on Section 2 (“Key subclaims”, or subtheorems.tex) for the next day or two to try to organise the overall logic, which of course we should also discuss in the next post.

]]>Maybe this is a good time to start a fresh new post devoted to the effective writing of the paper (or to decide of an different way to proceed) to share views and ideas on how redaction should be done.

Needless say I am in for helping on the writing Deligne/exponential sum section(s) and probably for other parts as well.

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