The post Bounded gaps between primes: the epic breakthroughs of the early 21st century first appeared on Fifteen Eighty Four | Cambridge University Press.

]]>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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]]>The post Behind the Scenes: Equivalents of the Riemann Hypothesis first appeared on Fifteen Eighty Four | Cambridge University Press.

]]>I had been interested in RH for some time, studying the zeta function through flows such as ds/dt=xi(s), which provided an equivalence. However this work, which had a topological basis, ‘’hit the wall” at the point where the structure of the flow near an essential singularity appeared to be important. The underlying theory was not available, and in the circumstances, I was not able to develop it.

A visit to the University of Waikato by Tim Trudgian stimulated work together on aspects of Robin’s inequality and its RH equivalence. In addition to his sterling detailed work on Volume One Chapter 7, his own published work improving Turing’s method for zeta zero analysis was of great value in many chapters.

I approached CUP at some stage near the completion of a draft of volume 1 and they showed interest. However the expert feedback they received was mixed – not only had volume one not covered some of the most valuable equivalences to RH, it did not cover GRH. This was considered to be much more useful than RH for applications, and the idea of two volumes took shape.

Regarding Cambridge, I had been impressed with their expertise and dedication to publishing good mathematics when I worked with them, supplying an appendix and software for Dorian Goldfeld’s book *Automorphic Forms and L-functions for the group GL(n,R) *(Cambridge, 2006). The new experience writing “Equivalents” showed that this was no exception.

The writing process did not always go smoothly. Some parts, including whole chapters in one case, were scraped. I decided that the details were either too technical or would be too taxing for the reader. My conceptualized target average reader was a graduate student considering potential research problems in pure mathematics and looking for accessible problems. I avoided using results which were at the pre-print stage at the time of writing. This meant sometimes leaving out some published results which depended on unpublished work.

For volume one, the seminal paper of Rosser and Schoenfeld of 1962, and other related papers, provided particular organizational challenges. For volume two, I spent a long time working with Zagier’s group representation equivalence. Eventually I decided to give up. It would be too difficult to give the average reader an adequate background in the specialized theory. In addition Zagier’s method was not able to be extended to number fields.

Given this graduate student target audience, I included quite a lot of background material. For example the chapter on numerical estimates for arithmetic functions, and work of Erdos and others on abundant numbers in volume one. In volume two, an extensive set of appendices provided proofs for the more specialized results referred to in the body of the text, which the reader might not necessarily meet in graduate courses.

I was often asked whether I had (by now) solved RH! Writing a tome of this size does not leave too much energy for such grandiosity, but I did twice believe I might have disproved RH. This was while writing volume two, once when considering integral equations, namely the method of Sekatskii, Beltraminelli ad Merlini in Volume Two Section 8.3, and once when developing examples for Weil’s explicit formula in Volume Two Section 9.5. In both cases the approach came to nothing.

After the volumes were published I created a website for Errata and notes, GRHpack and RHpack. I have had an excellent volume of feedback giving corrections and other comments which have been included or will be once time permits – this is especially welcome. Some folk even indicated they had been right through both volumes!

As expected, equivalents to RH and GRH continue to evolve. In late 2017, the University of Waikato had a visit from Ken Ono who gave a fascinating lecture related to the Jensen polynomial equivalence of Polya from 1927, namely that RH is equivalent to all of the Jensen polynomials of the Xi function being hyperbolic. He described a discovery Don Zagier, Michael Griffin, Larry Rolen and himself which, among other advances, shows that for each degree all but a finite number of the Jensen polynomials are hyperbolic. This work is being written up and will be referenced in the “Errata and notes” relating to Volume Two Section 4.4. when a pre-print appears on ArXiv.

And In February 2018 Brad Rogers and Terence Tao published on ArXiv an article entitled “The de Bruijn-Newman constant is non-negative”, giving the RH equivalence Lambda=0. A full report on this work would make a nice addition to Volume Two Chapter 5. Both of the works of Rogers/Tao and Griffen/Ono/Rolen/Zagier will be included in a second edition, should one be published.

Find out more about Kevin Broughan’s 2-volume work *Equivalents of the Riemann Hypothesis *here*.*

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]]>The post On Software for Automorphic forms and L-functions for the group GL(n,R) first appeared on Fifteen Eighty Four | Cambridge University Press.

]]>I was also able to attend a special seminar series at the Institute for Advance Study Princeton on automorphic forms and L-functions and take several courses with Dorian Goldfeld on a related topic, and attend the New York joint CUNY, NYU, Columbia number theory seminary and dinners. It was during one of these latter that Dorian asked if I would write a software package in Mathematica for the book he was writing on “GL(n,R)”, a group which for quite deep reasons is important in the field of automorphic forms and L-functions in number theory. His concept was that the intensive matrix manipulation, and other computations, such as finding the symbolic form of Casimir operators, would be greatly assisted by having a package of related functions.

Now I had had some experience with Mathematica programming having supervised a PhD candidate Rene Ferdinands (now at the University of Sydney) who used its symbolic matrix inversion and simplification facilities as part of solving a system of ODE’s with an application to that popular Australian sport cricket. I said yes to Dorian, not knowing the task ahead was not light.

As the chapters of Dorian’s book appeared as drafts I would go through them, doing some editing and thinking through which parts would lend themselves to computation. Along the way I learnt some of the tricks of the trade of mathematics expository book writing – such as repeating definitions near where they are used, lightening the burden on the reader’s memory. In the end the manual for the software came to over 70 pages, and Cambridge generously agreed to include it as an appendix.

This book I believe fills a large gap in our available texts, especially for beginning researchers. It is very concrete, focusing on the group SL(n,Z) acting on GL(n,R), but building up progressively through n=2, n=3 and then the general case. It includes finely worked full proofs. Dorian had the benefit of many conversations with another Columbia mathematician, Herve Jaquet, who was a founder of the theory of L-functions and automorphic forms in the wider setting of Adels, which Dorian covered in later works, also published by Cambridge.

**The software “GL(n)pack” is available here. **

**Find out more about Equivalents of the Riemann Hypothesis, available as a 2-volume hardback set and separate volumes 1 and 2. **

**Also check out Automorphic Forms and L-Functions for the Group GL(n,R) by Dorian Goldfeld**

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