It's a big game that remains at large -- a "Loch Ness monster." That's how Dr. Daniel N. Rockmore, professor of mathematics and computer science at the Dartmouth College, Hanover, New Hampshire, describes the biggest unsolved problem in his own subject. He's right, for, like the mythical beast rearing its head in a Scottish lake, the mathematical riddle is daring the wits of experts. No one has been able to crack it for nearly 150 years. And the game is at large despite a reward of $1,000,000 on its head for the last five years.
Like all his peers, Rockmore is intrigued by the imbroglio. No, he isn't in the fray to claim the prize. He earned his PhD from the Harvard University, and taught math for two years at the Columbia University before coming to the Dartmouth College. Besides contributing papers in scientific journals, he has written three technical books on designing algorithms for data analysis and applied his skill to, among other areas, climate modelling, mathematical biology and neuroscience. He is possibly the only radio math commentator in the US, regularly reading essays on the Vermont Public Radio, illustrating the subject's scope and range through real-life experiences. Some of the items have been included in the popular series "Sounds Like Science."
Maybe, it's this other hat that Rockmore wears which has goaded him to churn out a title, so that the uninitiated get a feel of the experts' 150-year-old futile chase after the math monster. The book - "Stalking The Riemann Hypothesis: The Quest to Find the Hidden Law of Prime Numbers" - will be published by Pantheon Books shortly.
"My overarching goal has always been to provide a window through which the public might get some sense of what mathematical research is like while showing the surprising breadth of math as a subject," says Rockmore in an e-mail interview. "I felt that the best way to achieve this was through a single story, for I believe that narrative is the key to holding a readerŐs interest. It seemed to me that the history of the Riemann Hypothesis - given its central place in modern mathematical history and research - would provide such an opportunity. The story of the search for its resolution would provide a structure off which I could hang a broader story of modern math."
The Riemann Hypothesis (RH) deals with prime numbers. They can be called the "elements of arithmetic." Chemists long ago told us not to be baffled by the jungle of objects all around. Everything that we see - a pebble of stone or a petal of rose - are made of elements. That's a huge step in making sense of the complexity of the world, an endeavour known as science. To recognise that an apple's fall and the moon's merry-go-round are one and the same act is as nice an example of perceiving the order amidst chaos as to know that everything is made of only a handful of elements.
Just as hydrogen and oxygen combine to form water, the numbers 2 and 3, when multiplied, make 6. Which is why numbers like 6 (2 x3), 15 (3 x5) or 91 (7 x13) are known as composites. And 2, 3, 5, 7, 11 or 13, which aren't multiples of other numbers - and so are divisible only by them or 1 - are called primes. A number is either a prime itself, or a product of other primes (composite), just as there are pure hydrogen or oxygen, and water.
The comparison, however, ends with that. There are only one hundred-or-so elements, whereas primes are innumerable. In fact, it's the infinitude of the primes that accords them a mystique.
How do we know that the series 2, 3, 5, 7, 11, 13, ... has no end to it, that is, there's no biggest prime? Greek mathematician Euclid, in the fourth century BC, offered a logically invincible proof of that - in a theorem that the British mathematician Godfrey Harold Hardy found not only "first-rate," but also "as fresh and significant [today] as when it was discovered."
Euclid's argument goes something like this: Let's assume that there's no bigger prime than 13. Then a number can be made by multiplying all the primes through 13, and then adding 1 to it. So we get 2 x3 x5 x7 x11 x13+1=30,031. What kind of a number is 30,031? A composite? Hardly so, because it's been made in such a way that, when divided by supposedly all the primes (2, 3, 5, 7, 11 or 13), it'll always leave a remainder 1. Nevertheless, 30,031 can still be a composite, and indeed so; 30,031 is the product of two primes bigger than 13 -- 59 and 509. This fact also refutes our assumption about 13 being the biggest prime. Conclusion: 13 is hardly the biggest prime. A number can attain that status, only till such time as a new one snatches the title. The biggest prime to date, discovered only last February, is 1 less than 2 multiplied 25,964,951 times. The number has 7,816,230 digits, and if printed in its entirety, will fill up 235 newspaper pages!
The infinitude of primes is a counter-intuitive concept. They ought to thin out as we keep counting along the number line. For example, there are four primes between 1 and 10 (2, 3, 5, 7), and another four (11, 13, 17, 19) between 11 and 20, but only two (23, 29) between 21 and 30. Even numbers can't be primes, as they are all multiples of 2. Only some of the odd numbers - which aren't multiples of odd primes - are themselves primes. Which is why 21, 25 and 27 (multiples of 7, 5 and 3 respectively) aren't primes. Such multiples - more numerous the farther we go along the number line - are bound to make primes more of a rarity the higher the range we consider. However, since there is an infinity of numbers, and also since Euclid has shown how easy it is to concoct bigger and bigger primes from the smaller ones, they don't dry up.
Humanity's yearning to decipher order in chaos has met its match in the prime numbers. Their distribution in the boundless ocean of numbers defies any rule. Just to give an example, numbers like 31; 331; 3,331; 33,331; 333,331; 3,333,331 and 33,333,331 are all primes. But that doesn't mean 333,333,331 will also be one. In fact, 333,333,331=17 x19,607,843. Is the list of the 'twin primes' - 3, 5; 17, 19; 29, 31 (pairs only 2 apart) - finite or infinite? There are scores of such unanswered questions. For example, why are there nine primes between 9,999,900 and 10,000,000, as against two between 10,000,000 and 10,000,100? The RH, says Rockmore, was conceived to decipher "a very precise law governing the way in which primes are distributed along the number line."
According to Prof. Harold M. Edwards, whose 1974 book on the RH (Riemann's Zeta Function) Rockmore praises in his bibliography as a 'bible', "no mathematical work is more clearly a classic than Bernhard RiemannŐs 8-page paper Ueber die Anzahl der Primzahlen unter einer gegebenen Grosse." It was presented by the 32-year-old German mathematician at the Berlin Academy when he was made a member of the institution in August 1859. Translated, the title of the paper reads On the Number of Primes Less Than a Given Magnitude. Riemann's predecessors, Frenchman Adrien-Marie Legendre and German Carl Freidrich Gauss, had come up with good guesses about how many primes there would be less than one thousand, one million, one billion or more.To their surprise, they noticed that the formula they had devised for this purpose worked better the higher the number they considered. Still, those were approximations, not exact counts. Is there a way to predict the exact number of primes below a given number? Riemann tried to answer that fascinating question.
And for that he exploited a mathematical expression introduced in 1740 by the Swiss mathematician Leonhard Euler. Called the 'zeta function' (simply because it's symbolically represented using the Greek letter zeta), the expression is the sum of the series -
The dots at the end of the line indicate that there are an infinity of terms in the series. (Don't think that this makes the calculation of the value of the series an impossible task, experts can tackle the problem). Also note that the 's' in the expression stands for any numerical value - a whole number or a fraction.
Riemann's genius was to extend the value of that 's' to a new range - numbers of the form a+ib, where 'a' and 'b' can have any value other than zero, and i=sqrt(-1). Numbers of this class are known as 'complex numbers', because it's not easy to visualise what sqrt(-1) is. sqrt(+1)=1, but sqrt(-1)=? Don't be put off by this weird sqrt(-1); it's a calculational tool that proves handy in many branches of science. The new series that Riemann thought of, now known as 'Riemann zeta function', is -
Riemann calculated that the new zeta function amounted to zero for some typical values of a and b. Which values? Well, the Riemann zeta function could be equal to zero, if 'a' were even negative numbers like -2, -4, -6, etc, and 'b' were 0. Riemann wasn't interested in these values of 'a' and 'b', because in such cases the zeta function did not signify anything. On the other hand, the zeta function amounted to zero if 'b', instead of being 0, took some typical values. And what would be 'a' in such cases? To his utter surprise, Riemann noticed that then a had to be equal to 1/2. To put it mathematically, the Riemann zeta function, if it takes the form
That 'a'=1/2 is a precondition for the Riemann zeta function to be zero is known as the RH. The math wizard only hazarded a guess in his 1859 paper that 'a' had to be 1/2 for the zeta function to be zero, but he couldn't devise a proof showing why 'a' couldn't be anything else. That proof is still eluding experts, even five years after the Clay Mathematics Institute (CMI) in Cambridge, Massachusetts, announced a reward of $1,000,000 for it.
Why on earth is a proof of the RH so important? What's it got to do with prime? As Riemann showed in 1859, the secrets of the primes are locked inside the zeta function. If 'a'=1/2 is a must for it to be zero, then although you can't predict where in the number line the next prime will occur, you find that the overall pattern of the distribution of primes is extremely regular. Put simply, you can predict the exact count of primes below a given number. On the other hand, if 'a' won't have to be 1/2 for the zeta function to be zero, then the primes are really unruly. As Prof. Enrico Bomberi, of the Institute for Advanced Study in Princeton, New Jersey, says, "The failure of the RH would create havoc in the distribution of prime numbers." And math wizards all over the world know the havoc will spread further. Hundreds of results in theorems dealing with numbers begin, "If the RH is true, then ..."
Perhaps there is no greater indication of the difficulty and importance of RH than the roll call of distinguished mathematicians who've tried - and failed - to prove it. And the list includes such stalwarts as Jacques Hadamard, Charles de la Valle-Pousin, Edmund Landau, Harald Bohr, G.H. Hardy, John Littlewood, George Polya, Atle Selberg, Carl Ludwig Siegel, Alan Turing, Paul Erdos, Alain Connes, Peter Sarnak, Andrew Odlyzko and Hugh Montgomery. There have been false starts and dead ends in the long chase. For example, our very own Srinivasa Ramanujan, in his famous introductory letter to Hardy, boasted of a formula for predicting the count of primes less than a given number - a claim that did not pan out. John Forbes Nash, Nobel laureate in economics and hero of the movie A Beautiful Mind, also claimed to have a proof of the RH in possession just before schizophrenia gripped him.
Interestingly, tireless labour, though not enough for reaching the summit, has nonetheless helped wizards scale some peaks towards it. "As is so often the case in science, and in life, the more you know, the more you realise what remains to be learned," writes Rockmore in Stalking the Riemann Hypothesis. "The story of science is, in part, an infinite tape loop of the tale of Icarus and Daedalus, one generation's achievement serving as the means and motivation for the next generation's aspirations. Grandparents trudge to the corner stone, parents learn to fly over the oceans, and children dream of reaching the moon."
Rockmore's book isn't the only one trying to evoke public awareness on the RH. Ever since the announcement of the million-dollar prize, titles on it have come out. What distinguishes Stalking the Riemann Hypothesis from others is its least dependence on mathematical calculations. "My goal was to try and reach as large an audience as possible," says Rockmore who has even made a documentary film on math. "The fact is that many people freeze at the sight of symbols and equations, even though they can understand the underlying mathematical ideas. My experience is that often when people say they aren't good in math, it's really the case that there was a bad teacher in their school. Nevertheless, if you try to reach people on their own terms, often they can understand mathematical concepts - assuming that they want to."
Will the million-dollar prize ensure a speedy resolution of math's most famous riddle? Rockmore isn't sure. "It's certainly possible that the publicity of the prize will heighten interest in math and thereby attract a larger number of experts who might not have been aware of the RH," says he. "I think for most professionals the existence of a cash reward is probably not a spur. You're pretty sure if you solve the problem, you'll be forever famous, and rewards will come your way. I know of several well-known experts who only work on problems for which a solution will make them famous." In his book Rockmore discusses the new vista that has opened up since the serendipitous discovery of a similarity between the zero values of the zeta function and a queer property shown by atoms - a connection that has engendered one of the most vigorous attacks on the RH till date.
Who among those stalking the RH does he think most likely to nab it? "Too hard to say," says Rockmore. "There are plenty of experts working on RH and problems related to it. I won't be surprised if the winner turns out to be a relatively unknown wizard. This happens all the time."