How I proved the Riemann Hypothesis

The trouble with this modern age is that every few weeks someone goes and solves a problem that's been baffling Mathmos for centuries. Sometimes it's the Four Colour Problem, sometimes it's Fermat's Last Theorem, sometimes it's "Why are the Graph Theory books all miscatalogued?" You know how it is -- in households the length and breadth of the country, the following conversation takes place over breakfast:

"Well, I've been telling them it would happen for years, but they wouldn't believe me. 'It was claimed yesterday that four colours suffice to colour any map on the plane. Mrs Thatcher has promised to reduce this to three by 1995. In the House of Commons, Mr Dennis Skinner was suspended for saying "Poo-poo."'"

"Yes, dear. Did they explain how the theorem is proved?"

"Yes -- 'The intimate secrets of Appel and Haken revealed -- Sexy underwear in four colours to be won - see pages 6,7,8,9.' I think the Times has gone downhill a bit recently."

Time was running out and I had to decide quickly: if I wanted to make my name, should I prove Goldbach's conjecture, or the Riemann hypothesis? After some thought I decided: I'd make a serious attempt at cracking the Riemann hypothesis, and then, if it came out by lunchtime, I'd do Goldbach over tea.

The Riemann hypothesis was first formulated when Riemann wrote in the margin of a textbook he was reading: "All the nontrivial zeroes of the zeta function lie on the line Re s = 1/2. I have found a truly marvellous proof of this fact, but I'm certainly not going to write it in the margin -- I'll send it to the Cambridge Philosophical Society instead. Anyway, the book's due to go back to the library tomorrow." Riemann always claimed that his proof was lost in the post, and could never remember the details.

Of course there's not much money in unsolved problems -- after all I could have been earning three times as much if I had been bad at Maths, and done something to benefit mankind instead, such as buying shares and selling them at a profit -- but there's always the spin-offs: Riemann hypothesis tee-shirts, Zeta-function soap powder "Gets to the points that other brands cannot reach" Maybe, even, an appearance on Terry Wogan's chat show, though I might be able to avoid that. So I got out the pencil and paper, scratched my head, stared out of the window, and waited for inspiration.

At first things seemed to be going badly -- a good ten minutes passed, and I was beginning to think that the Goldbach conjecture looked a bit easier. I had even got to the stage of wondering whether there might be zeroes which didn't lie on the critical line, and had cautiously looked behind the filing cabinet in case there were any there.

Then there came to me a brilliantly simple idea, so ingenious that a child of ten could understand it, but so wide-reaching that the whole of mathematics would be instantly revolutionised.

(Part 2)

They say that all you have to do is prove the Riemann hypothesis and the world will beat a path to your door. In this cut-throat age, the world will sometimes beat such a path merely on the strength of a rumour that you were thinking of proving the Riemann hypothesis. So, as I stared at the loose tiles on my office ceiling, practised drawing zetas on the desk without spraining my fingers, and reflecting that if Riemann had called it the Z function, I would have finished much sooner, I heard a loud hammering at the door.

"Come in," I riposted wittily.

A large man bounded into the room, leaving a trail of trampled students behind him. "Hi, there! I have come to beat a path to your door. Be the envy of other major academics. Oh, sorry about the students -- they happened to be in my path when I start beating it."

"Oh, it's all right. They were merely queuing up for my autograph. We can always get some more. Now, how can you help me?"

"I represent a rich consortium and have come to offer YOU a large amount of money for doing very little work."

"Oh, like George ---- ?" I said, naming a prominent member of the department who never did any research.

"Well, not quite as little as that. We would expect you to solve the occasional Hilbert problem, or perhaps develop a totally new branch of Mathematics, once in a while. We pay well: 100,000 pounds per year, plus a bonus of 1,000 per lemma, 5,000 per theorem, and 2,000 per corollary. Free coffee, your own backgammon set, swimming pool and masseuse."

"Private Eye in the Common Room?"

"Sorry, we can only offer you the Computing Service Newsletter."

I decided to hold out for a better offer, and my visitor took his leave, beating a path to the office next door, from which a cry of "Eureka!" had just issued. Wondering whether writing 'Eureka' in Greek was any easier than 'Zeta', and whether I should have a bath in my office, I returned to my research. Would it perhaps be easier to start with the Goldbach conjecture? After all, even Euclid had decided to prove that there were infinitely many primes before studying the zeta function...

J.R. Partington, 1988