The Ultimate Hypothesis
Author, Prime Obsession
Columnist, National Review
As we may all remember from elementary mathematics, a prime number is a number that can only be divided by itself and the number one. This simple property of certain numbers, however, has been the subject of centuries of mathematical research. And a conjecture associated with it, called the Riemann Hypothesis, has been proclaimed as the greatest unsolved problem in mathematics. So much so that the person who solves it can claim a million dollar prize and of course everlasting fame and fortune.
Well, joining us today to discuss this million dollar mathematical dillema is Mr. John Derbyshire. Mr. Derbyshire is a mathematician and linguist by education, a systems analyst by profession, and a celebrated writer whose work has appeared in the National Review and the New Criterion. He is the author of the book, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. And he joins us today to discuss prime numbers and the Riemann Hypothesis.
Charles Lee (CL) interviews John Derbyshire (JD) about the Riemann Hypothesis.
CL: Mr. Derbyshire, thank you very much for joining us today.
JD: Thank you for having me on the show, Charles.
CL: Well, it’s certainly our pleasure, and you’ve certainly written a very fascinating book, Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. But, before we jump into this problem, I’m wondering if you can give us some background into this complicated problem and how it compares with some of the other well-known mathematical problems, like Fermat’s Last Theorem.
JD: Well, how does it compare with other problems? Not well, I think is the answer, mainly because the Riemann Hypothesis is not easy to grasp. It’s buried quite deep in some advanced mathematics, whereas something like Fermat’s Last Theorem, you can explain to an intelligent ten year old, or even the Four Color Theorem. But, with the Riemann Hypothesis, you really need a lot of math just to understand what it says, which is why I unfortunately had to put so much math in the book. I tried to just give the minimum math that you would need, but you do need a fair bit of math just to understand what the Riemann Hypothesis is.
CL: But, does that complexity translate into making it a more difficult problem to solve then?
JD: Actually, no. Not necessarily. Sometimes problems whose statement is very abstruse and hard to understand can be quite easy to solve, and vice versa. Again, Fermat’s Last Theorem is quite easy to understand. It’s just a statement in basic arithmetic, really. But, it took almost four hundred years to solve it. So, there’s really no correlation between how hard the statement of a problem looks, and how difficult it’s going to be to crack it. I don’t think there is any correlation at all.
CL: So, let’s see if we can attempt an explanataion of the statement of this problem.
JD: Well, the Riemann hypothesis concerns a certain function. A function is a way of transforming one set of numbers into another set, like the squaring function, which is probably the most elementary function that is at all interesting, where you transform 2 into 4, and 3 into 9, and 10 into 100, every number gets transformed into its square. Well, the Riemann Hypothesis concerns a function that transforms, or as mathematicians say, maps one set of numbers into a different set of numbers. It’s a much more complicated function than the simple squaring function. It’s called the Zeta Function.
Now, there’s a thing about functions in general that mathematicians like. The thing that mathematicians really like to know is: which numbers does this function send to zero? If you take the squaring function, the only number that the squarig function sends to zero is zero, because zero times zero is zero. With more complicated functions, you can get a lot of numbers that the function turns into zero. And the Riemann Hypothesis concerns all those numbers that the Zeta Function turns into zero, and it says something about all of those zeros. The Zeta Function is actually so complicated that it has an infinite number of these numbers that go into zero. And, the Riemann Hypothesis says a certain thing about all of those numbers that go into zero when you apply the Zeta Function to them. Quite a simple thing really, but nobody has been able to prove it.
JD: Yes. It turns out once you start digging into prime numbers. If you write out the first few dozen prime numbers or look at a list of them, you notice two things right away. One thing you notice is that they are kind of irregular. There’s a sort of quality of randomness to them. You never quite know when the next one is going to show up. You get long gaps where there are no prime numbers at all. From 89 to 97 for example, there are no prime numbers in that gap. And, then you get little clusters and clumps where there are prime numbers close together. So, there’s a sort of randomness and unevenness about them. You never quite know when the next one is going to show up. And the other thing you notice is that they thin out as you go along. Around about 100, there are more prime numbers in the neighborhood of one hundred then there are in the neighborhood of one thousand. And there are way more in the neighborhood of one thousand than there are in the neighborhood of one million. And so on. So, they thin out as you go along.
And when most mathematicians investigate the prime numbers, they are usually investigating one or the other of these issues, either the issue of how they gradually thin out as you get into more and more stupendous numbers, or this issue of this clumping and spreading quality that they have. That’s what makes them fascinating to mathematicians, because of course prime numbers are defined very simply. They are numbers that nothing goes in to. And yet, they have this deep and mysterious property. And the combination of those two things that I mentioned, the one that they thin out in a way that turns out to be remarkably regular and smooth, and the other this odd clumping and scattering which is very irregular, this combination of regularity with irregularity is just very fascinating. It’s very fascinating to those who love numbers. And, that’s where most of the inquiries take place. And once you start digging deep, deep down into those issues, you bump up against the Zeta Function and Riemann’s Hypothesis.
CL: So, what does the Zeta Function say about this clumping of prime numbers then?
JD: That’s an interesting question. Just to go back, as I said, a function maps one bunch of numbers into another bunch. And, in the case of the Zeta Function, there’s a whole bunch of numbers, an infinity of them, that it maps into the humble zero. And, the Riemann Hypothesis says that all of the interesting numbers that do that lie on a single straight line. If you walk along that line, and inspect all of the numbers that do this, again they have a certain random quality to them, rather like the prime numbers do. And these two randomnesses are related in deep and subtle ways. I mean it’s not as straightforward as if the zeros are clumping together than the prime numbers are clumping together. It’s nothing as simple as that. But, there are deep connections between the way these two things are distributed. And because the prime numbers are in the realm of arithmetic, which is rather difficult to get good generalizations about, but the Zeta Function is in another realm. It is in the realm of calculus, where we have some really powerful tools for investigating functions. So, we can go through the calculus with all of these powerful tools that were developed in the 19th and 20th centuries to handle functions. We can go through calculus to attack these issues in arithmetic. And again, that’s the fascination of it.
CL: So, it provides a roundabout way of getting at some of the issues of the prime number distribution.
JD: Exactly. This very often happens in math. It happens in algebra. People spent hundreds of years trying to figure out how to solve equations. Solving the quadratic equation was easy. The cubic equation gave us a bit more trouble, but they cracked that in the late renaissance. And, the quartic equation came soon after. And, they really got stuck on the fifth degree equation. They just couldn’t solve that. And then this wonderful detour turned up. Instead of trying to crack these things head on, you start looking at the underlying symmetries of the roots of equations. And through that, they got the solution. So, sometimes you sort of go around the back door and investigate things. But, of course, finding the back door is very difficult, because it’s pitch dark.
CL: So, why are mathematicians so fascinated in this particular problem?
JD: That’s a good question. I guess mostly it’s the challenge of the difficult. It’s the challenge of difficulty. Here’s a problem that people have struggled with, the great minds of mathematics. You know, if you have a mathematical education and training, you have these towering figures from the past whose names you learn and have to study. It’s like if you go to Julliard, and learn to play the cello. You’ve got these marbled busts of Mozart, Beethoven, and Bach glowering down at you. People who have a mathematical training also have these great names from mathematics looking down at them. And you find yourself thinking, you know these great figures from mathematics from the past have taken a crack at this problem and they couldn’t do it. Wouldn’t it be wonderful if I could do it? And, the person who’s confident of having a high level of mathematical powers is just drawn to have a try at it. I don’t have that level of mathematical powers myself, but I’ve met some people who have. And, I think that is what drives them, the lure of the immensely difficult. It’s like climbing Mount Everest. It’s there.
CL: Well, I recall you mentioning in the book that in this field of research, everyone can find there way back to some towering figure, like Gauss for instance.
JD: Yes. Pretty much any branch of mathematics that you dig into, sooner or later you bump up against Gauss. He was really a tremendous giant of the field. Gauss and an earlier Swiss mathematician, Leonard Euler, you could say between them laid the foundation of all the mathematics that followed them. They were really tremendous figures. I just stand in awe of minds like that. And they weren’t freakish people. They were normal people who had family lives. Euler was a great family man, loved kids, traveled a lot, and attended at court. They were human beings with full lives, and yet they churned out this amazing, high-powered intellectual work. It’s just incredible. It leaves you in awe of what the human mind can accomplish.
CL: I’m curious what are some of the current figures or in the history of the Riemann Hypothesis?
JD: Well, the thing that attracted me to it was actually the figure of Riemann himself. I say somewhere in the book, in the preface, that it’s much easier to write a book if you can peg it on a human personality. Human beings are social animals, and the thing we are mostly interested in is each other. And, if you can peg a story on a human personality, it’s much easier to write, and it’s much easier to focus your attention. And, it was really the personality of Bernhard Riemann that got my attention. And, in particular, it was the contrast between the inner Bernhard Riemann and the outer one. Outwardly, he was rather a pathetic character. He was in ill health most of his life. He was chronically poor. Most of the people he loved died off before his eyes. He was very shy, not very social, and very difficult to draw out of his shell, rather a pathetic character. But inwardly, he had this tremendous power of imagination, not only mathematical, but also a physical imagination. In a way he was a great physicist too. His work laid the foundations for Einstein’s general theory of relativity sixty years later. So, that was the thing that got me. The contrast between this rather shabby, unhealthy, pathetic outward man, and this figure of tremendous vigor and imagination inside caught my attention, and that’s really where it started.
Other great names have been involved. Well, it’s been such a big problem that pretty much everyone from the late 19th century on has had something to do with it. I think, David Hilbert, the great Prussian mathematician who flourished in the late 19th and early 20th century, deserves a special mention, not so much for any work that he did on it, but more for the inspiration that he gave to others. And of course, the greats of the past who laid the foundation on which Riemann built, Gauss and Euler. Riemann actually studied under Gauss who was teaching at Gottingen University when Riemann was there. And in modern times, Alan Turing, the 20th century British mathematician about whom there was a rather good play a few years ago deserves a mention. And some mathematicians still alive and still working that I mentioned, like Hugh Montgomerie and Andrew Lesko get a lot of play in my book. You can almost open a catalogue of mathematicians to any page and find someone who has worked on the problem.
CL: But, I’m curious what do you think are the prospects for having this problem solved any time soon?
JD: Well, as I say in the book, that’s a bit of a mug’s game. I relate a famous story about David Hilbert in 1925, and who was asked to rate three great outstanding problems and the prospects for their solution. And, he got it totally wrong, and he was a great mathematician. So, what are my odds about being able to guess this? My gut feeling, having spent several months talking to mathematicians about it, my gut feeling is that we are not even close. But, if you pick up your New York Times tomorrow and see that the Riemann Hypothesis has been cracked, don’t be too surprised, because these things can take you by surprise. My gut feeling though is that we are not even close. I feel we are a bit lost at the moment.
CL: Well, I suppose we’ll have to wait and see who will solve it.
CL: Mr. Derbyshire, I want to thank you very much for joining us today to discuss your book, Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics.
JD: It was a pleasure. Thank you.