# Writing the book "Prime Numbers and the Riemann Hypothesis" 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. This talk is meant to provide a glimpse into the writing process and also content of the book. It's about making research math a little more accessible, math education, and technology. ## Content of the book It's here: http://wstein.org/rh/ Download a copy before we have to remove it from the web! 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. 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 Ph.D. thesis). The expert analytic number theorist Andrew Granville helped us out of certain confusing thickets. ## Our Formulations of RH In the book we present X equivalent formulations of RH: [state them precisely here] [illustrate with some computations] ## Clay Lecture The book started in May 2005 when the Clay Math Institute asked Barry Mazur [link] to give a large lecture to a popular audience in Cambridge, MA, and he chose to talk about RH, with me helping with preparations: "Are there still unsolved problems about the numbers 1, 2, 3, 4, ... ?" -- see http://www.claymath.org/library/public_lectures/mazur_riemann_hypothesis.pdf 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 (and I finally switched from Linux to OS X when Ubuntu made a huge mistake pushing out a standard update that hosed X11 for everybody!). 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. We met again many times, in Boston and Berkeley, for further similar working sessions. These resulted in side projects, like our work-in-progress on "(not) random walks" associated to modular symbols, which may be useful in understanding certain behavior of L-functions, and a project we have on making the explicit formula for elliptic curve L-functions more explicit. ## SIMUW [SIMUW](http://www.math.washington.edu/~simuw/) is the "SUMMER INSTITUTE FOR MATHEMATICS AT THE UNIVERSITY OF WASHINGTON". You should definitely get involved at some point, if you have a chance. I taught a SIMUW course in Summer 20xx on the book. Rather, I spent one very intense week on this book, and another on the Birch and Swinnerton-Dyer conjecture (really: elliptic curves). The first non-calculus part of our presentation of RH, combined with interactive use of Sage, was perfect 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. [for example...] ## Reader feedback 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 another 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. 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 came up with a cover design that combined the main elements of the book: the title, Riemann's face, and the zeta function: [cover in our pdf] Then designers at Cambridge University Press with actual experience made our rough design more attractive. Of course, as non-mathematician designers, they initially made it look very pretty by messing with the Riemann Zeta function, e.g., by adding symmetries it doesn't have, etc. There was a lot of back and forth, which finally resulted in this compromise: [insert actual book cover cambridge made] ## Publishing with Cambridge University Press [insert CUP logo] We talked with people from AMS, Springer and Princeton Univ Press about publishing the book. I met Kaitlin [?] at the Joint Mathematics Meetings in [?], since the Cambridge University Press (CUP) booth was directly opposite the SageMath booth, which I was running. We decided, due to their great enthusiasm -- which lasted more than a few minutes while talking to them -- some similarly positioned books they had published, and general frustration with other publishers, to publish with CUP. Much of the feedback mentioned above from readers was at the same time as (or even after) the feedback from the official CUP copy editors, and combining this together was quite challenging. 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. I will have a better sense when I see how the book itself is handled when it actually exists, and how their planned publicity campaign for our book unfolds. 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 very slow to get their heads around, which motivated us publishing with CUP. For example, CUP was extremely diligent putting huge effort into tracking down permissions for every one of the XXX 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). On the other hand, our marketing contact at CUP vanished for a long time; evidently, they had left to another job, and CUP was recruiting somebody else to take over. 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 in the end. Email from this morning (!): > "Dear Professor Mazur and Professor Stein, > I hope you are both well. > My name is Chris Burrows and I'm a publicist in the Academic Books division at Cambridge University Press. > As you know, your new book, Prime Numbers and the Riemann Hypothesis, is soont to be published so I'm producing an infographic to be used online and in our email marketing campaigns. Exciting!