Mathematician 1859 Hypothesis Statement
The first million-dollar maths puzzle is called the Riemann Hypothesis. First proposed by Bernhard Riemann in 1859 it offers valuable insights into prime numbers but it is based on an unexplored mathematical landscape. If you can show that its mathematical path will always lie true, $1m (£600,000) is all yours.
Mathematicians are obsessed with primes because they are the foundation of all other numbers. Prime numbers in mathematics are like atoms in chemistry, bricks in the construction industry and ludicrous pay cheques in professional football. Everything is built up from these fundamental units and you can investigate the integrity of something by taking a close look at the units from which it is made. To investigate how a number behaves you look at its prime factors, for example 63 is 3 x 3 x 7. Primes do not have factors: they are as simple as numbers get.
They are simple in this one respect – but are otherwise extremely enigmatic and slip away just when you think you have a grip on them. Part of the problem is that, by definition, they have no factors, which is normally the first foothold in investigating a number problem. This is also the key to their usefulness. It is their difficulty to grasp that makes primes the basis for our modern information security. Whenever you use a cash machine or visit a secure website, it is huge prime numbers that encrypt your information and make it extremely difficult for anyone else to pry it back out of the electronic cipher.
Prime numbers also have the annoying habit of not following any pattern. 3,137 is a prime and the next one after that is not until 3,163, but then 3,167 and 3,169 suddenly appear in quick succession, followed by another gap until 3,187. If you find one prime number, there is no way to tell where the next one is without checking all the numbers as you go. One possible way to get a handle on how primes are spaced is to calculate, for any number, how many primes there are smaller than it. This is exactly what Riemann did in 1859: he found a formula that would calculate how many primes there are below any given threshold.
Riemann's formula is based on what are called the "Zeta Function zeroes". The Zeta Function is a function that starts with any two coordinates and performs a set calculation on them to return a value. If you imagine the two initial coordinates to be values for latitude and longitude, for example, then the Zeta Function returns the altitude for every point, forming a kind of mathematical landscape full of hills and valleys. Riemann was exploring this landscape when he noticed that all of the locations that have zero altitude (points at "sea level" in our example) lie along an straight line with a "longitude" of 0.5 – which was completely unexpected. It's as if all the places in England that are at sea level (ignoring the coast) are on a dead straight line that runs directly north along the 0.5 longitude line.
Riemann used these zeroes as part of his prime distribution formula, but the problem is that no one knows for sure that all of the zeroes are on that same straight line. Sure, mathematicians have checked that the first ten trillion zeroes all fall on that line, but that's no guarantee that the ten trillionth and one zero might be somewhere else, throwing the whole prime distribution formula out the proverbial window, along with vast amounts of related number theory. Which is why there is a $1m prize for anyone who can show that all of the Zeta Function zeroes line up on the "0.5 line" without resorting to the impossible task of walking along this infinite line to check.
I've given you the Zeta Function to get you started and if you dust off a bit of "complex variable" maths, you will be well on your way to exploring the Riemann landscape. However – if that's a bit much – here is an easier starting problem: All prime numbers (greater than five) squared are one more than a multiple of 24. Check it for a few – it works. You can even prove that it works for all of the infinite number of primes.
Now if you can just do that for the Zeta zeroes, you can stop kicking a football around in the cold in hope of a big pay day.
Matt Parker is based in the mathematics department at Queen Mary, University of London, and can be found online at www.standupmaths.com His favourite prime is 31
WHEN Andrew Wiles, a British mathematician working at Princeton University, announced a decade ago that he had solved Fermat's last theorem, his discovery was reported on front pages around the world. The Frenchman's mathematical conundrum, which had taken more than 350 years to unravel, went on to inspire a television documentary, a bestselling book, and even “Fermat's Last Tango”—a musical that boasts such lines as: “Elliptical curves, modular forms, Shimura-Taniyama. It's all made up, it doesn't exist, algebraic melodrama.”
Three new books grapple with what is arguably an even tougher problem: the Riemann hypothesis, a puzzle that has perplexed mathematicians for the last century and a half.
Riemann's hypothesis is just 15 words: “The non-trivial zeros of the Riemann zeta function have real part equal to 1/2”. But explaining it so that non-mathematicians can understand it is more complicated. The distribution of prime numbers, such as 5, 7 and 11, does not follow any regular pattern but they become less common as they grow bigger. In an 1859 paper, Bernhard Riemann, a 33-year-old German mathematician, observed that the frequency of prime numbers is very closely related to the behaviour of an elaborate function, “ξ (s)”, which came to be named the Riemann zeta function.
The Riemann hypothesis asserts that all non-trivial solutions of the equation ξ (s) = 0 lie on a straight line. This has been checked for the first 100 billion zeros, but no proof appears to be in sight. The real significance of the hypothesis lies in its consequences, and particularly in what it says about prime numbers. A proof that held for every zero would shed new light on many of the mysteries surrounding the distribution of primes.
Mathematicians have been fascinated with prime numbers ever since the third century BC, when Euclid proved that there are an infinite number of them. Early in the 19th century, they began to look more closely at how often primes occur among the integers. Although Riemann, who died at the age of only 39, also laid the foundations in geometry for Einstein's general theory of relativity, his paper on prime sattracted little attention in the decades after it was first published.
A major breakthrough came in 1896, when two mathematicians independently proved the Prime Number theorem, which showed not just that prime numbers become rarer as they grow larger, but that their reoccurrence follows a specific formula (for those who care, the average gap between consecutive prime numbers is roughly proportional to the logarithm). The Prime Number theorem also encouraged mathematicians to begin working on Riemann's hypothesis, which gives a far more detailed picture of how the primes are distributed and which has consequences, not just for prime numbers, but also for many other areas of mathematics, including the analytic properties of L-functions and representations of non-singular cubic forms.
The fame of the Riemann hypothesis has grown steadily over the past 150 years. In 1900, David Hilbert, a German mathematician and contemporary of Einstein's, in a famous speech to the International Congress of Mathematicians, presented a list of what he considered to be the 23 most important problems for the new century. Hilbert's problems came from a variety of mathematical fields, and were to have a significant influence on 20th-century mathematics. One hundred years later, the Clay Mathematics Institute, which was founded by Landon T. Clay, a Boston businessman, published its list of what it called the seven millennium-prize problems, and promised a $1m reward for solving any one of them. The Riemann hypothesis appears on both lists.
Riemann's riddle has become a fixture on the mathematical landscape. Over the years, hundreds of results have been published that assume the truth of the hypothesis, but a proof of the conjecture would have immense consequences. Equally, a disproof would be the mathematical equivalent of an earthquake, destroying decades of work at a stroke.
All three of these books offer fascinating accounts of the story surrounding the Riemann hypothesis. Karl Sabbagh approaches the subject in the manner of an anthropologist, recording dozens of conversations with mathematicians working on various aspects of the problem. As a non-mathematician, he appreciates the difficulty of explaining a recondite mathematical problem to a general audience, and gives a vivid account both of the passion that mathematicians have for their subject and of the fascination for this particular problem.
John Derbyshire's book is more of a historical adventure. Chapters alternate between broad-scale historical accounts and detailed mathematical presentation. Some readers may find the mathematical chapters heavy going, but Mr Derbyshire makes a valiant attempt at explaining the mathematical ideas around the problem. His historical chapters link mathematical developments to the lives and personalities of the mathematicians involved and are full of interesting stories.
The most detailed account is given by Marcus du Sautoy. A professor of mathematics at Oxford University, Mr du Sautoy provides an engaging and accessible history of work on prime numbers and the Riemann hypothesis. He also has an eye for modern applications, and offers detailed discussion of the relevance of the Riemann hypothesis to cryptographic security as well as an interesting account of its possible links with quantum physics.
What, then, are the prospects that the hypothesis will ever be proved? In the 18th century, Leonhard Euler suggested that the distribution of prime numbers could be “a mystery into which the human mind will never penetrate”. But, to judge from Mr Sabbagh's book, most mathematicians would incline more towards Hilbert's view that a solution will one day be found for even the most difficult mathematical problem. Hilbert's words, expressed in a speech he made in 1930, were carved on his tombstone after his death: Wir müssen wissen. Wir werden wissen. “We must know. We will know.”
This article appeared in the Books and arts section of the print edition