Barry Mazur and William Stein, authors of Prime Number and the Riemann Hypothesis, discuss the Riemann Hypothesis and the writing of their book.
How can different readers adapt the book to their different needs and levels of proficiency?
Barry Mazur: The Riemann Hypothesis is one of the great unsolved problems in mathematics. It has been with us for over a century-and-a-half as a goal to be achieved, and has broad and important consequences. The more usual way of discussing it requires knowledge of—and some facility with—complex analysis. Our book goes about explaining it by taking quite a different route; at first, we take just a much simpler route. We do this in hopes that a neat understanding of this important mathematical goal and why it is important, is therefore made available to a much wider public.
Our book is in four parts. The first part requires very little mathematical experience, and yet conveys the essence of Riemann’s idea. Readers of Part I alone—which constitutes a ‘full story’ with ‘beginning, middle, and end’— will have experienced the force of this hypothesis, and of some of its consequences. We expect readers of Part II and III to have had some calculus; nothing more is required. At the end of Part III they will have seen a ‘full story’ at a more advanced level. Part IV requires some acquaintance with complex analysis, where the more traditional formulation of the Riemann Hypothesis is given, with a discussion of its ramifications.
William Stein: Some of our more computationally inclined readers have delighted in reproducing from scratch, using their favorite tools, some of the plots and computations in the book. This has even led in some cases to new observations.
One of the standout qualities of this book is that it does not only present the information to readers. Instead, they are also encouraged to put in the work in exploring the Riemann Hypothesis. How do you think you accomplish this?
William Stein: Not only do we explain abstract ideas, we also profusely illustrate everything we describe in ways that will strongly appeal to those (like me!) who love to compute examples. For example, we describe an explicit and very elementary construction of a plot, which any reader can reproduce from scratch on a computer, from which one can tease out the first few nontrivial zeros of the Riemann Zeta function. We also make complete source code available online to reproduce all plots in the book.
What prompted you to write Prime Numbers and the Riemann Hypothesis? What was the writing process like?
Barry Mazur: For me it was a wonderful way of being in extended conversation with William, experimenting with various ways of getting deeply familiar with the mysteries of number theory, in particular with what we call the ‘Riemann Spectrum’ (technically: the zeroes of the Riemann zeta-function) that underlies so much of the phenomena of number theory—in fact, of mathematics. We wrote and rewrote our book over a number of years, working intensely for one week each August. At the end of each of these weeks we posted our latest draft on the web (mistakes and all) to get feedback, and to therefore have the delight of conversations about it, and possibly contributions to it, from whomever read it.
William Stein: For me, I let my curiosity flow, asking questions about how to think about Riemann’s ideas that were motivated by my desire to be able to “compute everything”. I also wanted an approach to Riemann that would make sense to advanced high school students and undergraduates, who I often introduced this problem to in workshops and classes.
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