## Sunday, June 10, 2018

### How the Villarino-Gasarch-Regan paper came about

(This post overlaps a prior one here. The paper I am blogging about was also blogged about by Lipton here. The paper itself is on arxiv here. My slides for a talk I will be giving on this material are here)'

Todays post is on how the paper came about. A later post will be about why someone else didn't do it earlier.

Many years ago Bill noticed that while several books on Ramsey theory (see my prior post for quotes) state that the HCL was the first Ramseyian theorem. I think one source mentioned in passing that Hilbert used it to prove the Hilbert Irreducibility theorem (HIT). Bill could not find a modern English exposition of the proof.  So he asked Ken Regan (who not only knows German but can recite The Lewis Carol Poem Jabberwocky in German!) to translate it, and then Bill would put it in modern language, and there would be an exposition. Bill got bogged down in some of the math, and they both got bogged down with other things (For Ken catching chess-cheaters, for Bill mentoring HS students, for both of them, blogging.) Many years passed.

Sometime before 2015 Larry Washington showed me a nice proof that (ignoring mult constants)

∑ 1/p ≤ ln(ln(n)) + O(1) (the sum is over all primes p ≤n )

Read that carefully. There are many proofs in the web that the sum isat least  ≥ ln(lg n) but I could not find any that the sum was ≤ ln(ln n).  Larry Washington told me that the result and
the proof were not new. I told him that, even so, it doesn't seem to be out there. So we agreed to write and and post to arXiv but not publish in a journal. It's here.

This arXiv paper caught the attention of Mark since he had an exposition of Merten's proof  see here that that sum diverges. Mertens proof had explicit bounds which are missing from modern proofs.

I got into an email discussion with Mark about Math and History and I casually mentioned that Ken and I had worked on-and-off on HRL and HIT.  A few weeks later he emailed me a translation of the paper.  WOW! We worked together on polishing it, combining it with what Ken had done, and later brought Ken back into the project. I say without embarasment that we NEEDED Mark to resolve some of the issues we had and push the paper out the door. A Win-Win-Win.

And a lesson here--- Larry Washington was reluctant to publish on arXiv a paper on stuff that was already known. I should have told him

But Larry, if we do that I might find someone to help me finish the Hilbert paper

In a word: Serendipity.