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14

Apr

2021

Bounded gaps between primes: the epic breakthroughs of the early 21st century

Written by: Kevin Broughan

 
 

Why did I write this book? Certainly there are quite a few mathematicians who could write a better book on bounded gaps. I thought that the series of wonderful breakthroughs deserved to be celebrated with several accounts of the mathematical content of the breakthroughs, so why not! In addition, the style adopted for Equivalents of the Riemann Hypothesis, with mathematically enriched appendices and computational assistance with associated Mathematica software, seemed to suit quite a few aspects of the developments, and readers seemed to like the style.

The book sets out the mathematical content of the breakthroughs, with all of the details but not those of the work based on Deligne’s solution to the Weil conjectures. Those would be for a different book, maybe one on the Bombieri-Vinogradov theorem and its extensions and applications. For the expert, striving to improve the best bound 246, most of this material will be familiar. However the main target audience is beginning researchers, for example graduate students. I have vivid memories of my time at Columbia having to scrap with other grad students for important books held behind the library desk. One could have these only for one hour at a time, completely insufficient for understanding a major proof. To assist this group of potential readers the appendices contain proofs for supporting mathematics such as the spectral theorem for compact operators, Weil’s inequality for curves modulo primes, Bessel functions, the Shiu-Braun-Titchmarsh estimate etc. I have tried to simplify this material down to only what is essential for the work in the chapters, and these have been simplified down to only what is essential for the breakthroughs. But it’s certainly not simple!

Along the way there appeared to be many ways in which the results could be improved. However I did not tarry since after starting, the worst outcome would be for the work not to be completed. Having completed the work, others it is hoped will find paths to take it forward, with or without the text. For this writer, there are other pressing tasks and the Erdos life time limit is not so far off.

What the book is not: it’s not an account of the breakthroughs as a human endeavour. That would be a different book. There is the odd comment here and there which would qualify and some highly abbreviated biographical paragraphs. It is this author’s hope that such a book will be written, and soon before the individual and collective memory of events fades. To this end, on the book’s web page there is a link to the “backstory”, a web page containing an annotated series of time-lines and links to sources which might inspire someone to write up the human story with an absolute minimum of mathematical detail. Because what happened and especially the way it happened is unique, I would say in the entire history of mathematics, an account of the human side of the developments, in the hands of someone with suitable skills and experience, would be of interest I believe to a very wide audience.

As usual mathematical arguments are often difficult to follow and I needed help. This was generously provided especially by Pat Gallagher, Dan Goldston, Yoichi Motohashi and Terry Tao. I was not able to obtain a reply from Yitang Zhang, in spite of repeated requests, other to be sent his image. In the end I did not include more than a summary account of the proof of his extension to Bombieri-Vinogradov’s theorem – a full report of his proof, or better that of Polymath8a, would be part of the other potential book mentioned before. In any case, Maynard, Tao and Polymath8b went so much further than Zhang with their multidivisor/multidimensional method, an approach which seems both accessible and able to be improved.

Which brings me to my final remark: where to next in the bounded gaps saga? As hinted before, the structure of narrow admissible tuples related to the structure of multiple divisors of Maynard/Tao, and variations of the perturbation structure of Polymath8b, and of the polynomial basis used in the optimization step, could assist progress to the next target. Based on “jumping champions” results, this should be 210. But who knows!

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About the Author: Kevin Broughan

Kevin Broughan is a mathematics researcher, writer and teacher at the University of Waikato, Hamilton, New Zealand. His research is an anaytical and combinatorial number theory. Currently he is interested in equivalents of the Riemann hypothesis and the subject of bounded gaps between prime numbers....

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