Thursday, November 19, 2015

"Prime Numbers and the Riemann Hypothesis", Cambridge University Press, and SageMathCloud


Barry Mazur and I spent over a decade writing a popular math book "Prime Numbers and the Riemann Hypothesis", which will be published by Cambridge Univeristy Press in 2016.  The book involves a large number of illustrations created using SageMath, and was mostly written using the LaTeX editor in SageMathCloud.

This post is meant to provide a glimpse into the writing process and also content of the book.

This is about making research math a little more accessible, about math education, and about technology.

Intended Audience: Research mathematicians! Though there is no mathematics at all in this post.

The book is here:
Download a copy before we have to remove it from the web!

Goal: The goal of our book is simply to explain what the Riemann Hypothesis is really about. It is a book about mathematics by two mathematicians. The mathematics is front and center; we barely touch on people, history, or culture, since there are already numerous books that address the non-mathematical aspects of RH.  Our target audience is math-loving high school students, retired electrical engineers, and you.

Clay Mathematics Institute Lectures: 2005

The book started in May 2005 when the Clay Math Institute asked Barry Mazur to give a large lecture to a popular audience at MIT and he chose to talk about RH, with me helping with preparations. His talk was entitled "Are there still unsolved problems about the numbers 1, 2, 3, 4, ... ?"


Barry Mazur receiving a prize:

Barry's talk went well, and we decided to try to expand on it in the form of a book. We had a long summer working session in a vacation house near an Atlantic beach, in which we greatly refined our presentation. (I remember that I also finally switched from Linux to OS X on my laptop when Ubuntu made a huge mistake pushing out a standard update that hosed X11 for everybody in the world.)

Classical Fourier Transform

Going beyond the original Clay Lecture, I kept pushing Barry to see if he could describe RH as much as possible in terms of the classical Fourier transform applied to a function that could be derived via a very simple process from the prime counting function pi(x). Of course, he could. This led to more questions than it answered, and interesting numerical observations that are more precise than analytic number theorists typically consider.

Our approach to writing the book was to try to reverse engineer how Riemann might have been inspired to come up with RH in the first place, given how Fourier analysis of periodic functions was in the air. This led us to some surprisingly subtle mathematical questions, some of which we plan to investigate in research papers. They also indirectly play a role in Simon Spicer's recent UW Ph.D. thesis. (The expert analytic number theorist Andrew Granville helped us out of many confusing thickets.)

In order to use Fourier series we naturally have to rely heavily on Dirac/Schwartz distributions.


University of Washington has a great program called SIMUW: "Summer Institute for Mathematics at Univ of Washington.'' It's for high school; admission is free and based on student merit, not rich parents, thanks to an anonymous wealthy donor!  I taught a SIMUW course one summer from the RH book.  I spent one very intense week on the RH book, and another on the Birch and Swinnerton-Dyer conjecture.

The first part of our book worked well for high school students. For example, we interactively worked with prime races, multiplicative parity, prime counting, etc., using Sage interacts. The students could also prove facts in number theory. They also looked at misleading data and tried to come up with conjectures. In algebraic number theory, usually the first few examples are a pretty good indication of what is true. In analytic number theory, in contrast, looking at the first few million examples is usually deeply misleading.

Reader feedback: "I dare you to find a typo!"

In early 2015, we posted drafts on Google+ daring anybody to find typos. We got massive feedback. I couldn't believe the typos people found. One person would find a subtle issue with half of a bibliography reference in German, and somebody else would find a different subtle mistake in the same reference. Best of all, highly critical and careful non-mathematicians read straight through the book and found a large number of typos and minor issues that were just plain confusing to them, but could be easily clarified.

Now the book is hopefully not riddled with errors. Thanks entirely to the amazingly generous feedback of these readers, when you flip to a random page of our book (go ahead and try), you are now unlikely to see a typo or, what's worse, some corrupted mathematics, e.g., a formula with an undefined symbol.

Designing the cover

Barry and Gretchen Mazur, Will Hearst, and I designed a cover that combined the main elements of the book: title, Riemann, zeta:

Then designers at CUP made our rough design more attractive according their tastes. As non-mathematician designers, they made it look prettier by messing with the Riemann Zeta function...

Publishing with Cambridge University Press

Over years, we talked with people from AMS, Springer-Verlag and Princeton Univ Press about publishing our book. I met CUP editor Kaitlin Leach at the Joint Mathematics Meetings in Baltimore, since the Cambridge University Press (CUP) booth was directly opposite the SageMath booth, which I was running. We decided, due to their enthusiasm, which lasted more than for the few minutes while talking to them (!), past good experience, and general frustration with other publishers, to publish with CUP.

What is was like for us working with CUP

The actual process with CUP has had its ups and downs, and the production process has been frustrating at times, being in some ways not quite professional enough and in other ways extremely professional. Traditional book publication is currently in a state of rapid change. Working with CUP has been unlike my experiences with other publishers.

For example, CUP was extremely diligent putting huge effort into tracking down permissions for every one of the images in our book. And they weren't satisfy with a statement on Wikipedia that "this image is public domain", if the link didn't work. They tracked down alternatives for all images for which they could get permissions (or in some cases have us partly pay for them). This is in sharp contrast to my experience with Springer-Verlag, which spent about one second on images, just making sure I signed a statement that all possible copyright infringement was my fault (not their's).

The CUP copyediting and typesetting appeared to all be outsourced to India, organized by people who seemed far more comfortable with Word than LaTeX. Communication with people that were being contracted out about our book's copyediting was surprisingly difficult, a problem that I haven't experienced before with Springer and AMS. That said, everything seems to have worked out fine so far.

On the other hand, our marketing contact at CUP mysteriously vanished for a long time; evidently, they had left to another job, and CUP was recruiting somebody else to take over. However, now there are new people and they seem extremely passionate!

The Future

I'm particularly excited to see if we can produce an electronic (Kindle) version of the book later in 2016, and eventually a fully interactive complete for-pay SageMathCloud version of the book, which could be a foundation for something much broader with publishers, which addresses the shortcoming of the Kindle format for interactive computational books. Things like electronic versions of books are the sort of things that AMS is frustratingly slow to get their heads around...


  1. Publishing a high quality book is a long and involved process.
  2. Working with CUP has been frustrating at times; however, they have recruited a very strong team this year that addresses most issues.
  3. I hope mathematicians will put more effort into making mathematics accessible to non-mathematicians.
  4. Hopefully, this talk will give provide a more glimpse into the book writing process and encourage others (and also suggest things to think about when choosing a publisher and before signing a book contract!)