Manuel de León, Director of ICMAT, reflects on how we should measure the social importance of scientific problems, in a world so fixated on immediate economic impacts. He examines one of the great unsolved mathematical problems, the Riemann hypothesis, which deals with the relationship between a function of complex numbers and the distribution of prime numbers. Despite its apparent abstract nature, if this problem is resolved it could change the way that data is transmitted over the Internet.
In 1900 Paris hosted an event that had an enormous impact on the future of mathematics, the second International Congress of Mathematics (ICM in its English acronym). At the event, the renown German mathematician David Hilbert listed 23 unresolved problems that he believed would shape the work of mathematicians over the following century. Problem number eight was the so-called Riemann hypothesis, which had been formulated by the mathematician of the same name a decade previously.
Bernhard Riemann was a German mathematician and professor at the University of Gotinga, the center of mathematical excellence at the time, where great names from the history of science worked, such as Gauss, Riemann, Klein, Hilbert, Minkowski, Heisenberg, Born, Jordan, Wigner, Teller, Von Neuman and many others. In 1859, when studying the distribution of prime numbers, Riemann observed a close relationship with a defined function on complex numbers, the so-called Riemann zeta function. He sensed that there was a correlation between the zeroes of the function (i.e. the point at which it is cancelled out) and prime numbers.
Remember that complex numbers are used to solve the problem of negative numbers, and that the imaginary unit i is Ö-1; in other words, complex numbers are identified as points on a plane, allowing them to be added or multiplied according to a certain rule. Furthermore, all complex numbers have a real part and an imaginary part. Riemann conjectured that the real part of all (nontrivial) zeroes is ½. .
Proving the Riemann hypothesis or finding a counterexample would unlock huge amounts of information on prime numbers. Prime numbers are those that can only be divided by themselves and one: 2, 3, 5, 7, 11, 13… They are important because, metaphorically speaking, they are the bricks used to build all numbers. This is seen in the high school exercises that ask students to break down a number by primes (sometimes with powers, such as 72 = 2^3 x 3^2).
As Euclid proved with an ingenious argument, based on a reduction to the absurd, prime numbers are infinite. But their distribution remains a mystery. Their structure within natural numbers is still unknown. As you go down the list of numbers they seem to become increasingly scarce, appearing less frequently.
As well as being of mathematical interest, prime numbers are also very important to digital communications. The RSA cryptographic algorithm, developed by Rivest, Shamir and Adleman in 1977, is based precisely on the problem of factorization of whole numbers into prime numbers. Messages are represented in numbers, with the functioning based on the known product of two large prime numbers selected at random and kept secret.
Like all public key systems, each user has two encryption keys: one public and another private. When sending a message, the sender enters the recipient’s public key, encrypts the message using that key, and when the encrypted message reaches the recipient, said person decrypts it using their private key. This is based on a theorem from another great mathematician, Leonard Euler. To crack the key requires finding the prime factors, but breaking a number into its factors, when these are around 100 digits long, is a colossal task.
So prime numbers are important for business. In fact they are vital! Online business transactions depend on them. And understanding their distribution, which we would achieve by resolving the Riemann hypothesis, is also crucial.
However, despite the best efforts of the world’s greatest mathematicians, no-one has yet resolved the problem. Even the great Alan Turing tried to solve it by building a machine for that very purpose, but he too failed. The conjecture has become one of the most fascinating problems in mathematics. As well as being listed by Hibert, it deserves its place among the seven problems of the millennium, offering a million dollar prize money to the person able to unravel the mystery. Hilbert himself, when asked what he would do if he woke up after sleeping for 500 years, said his first question would be whether the Riemann hypothesis had been proved.
But we live at a time when social relevance is measured by the so-called markets. This is also reflected in the funding provided for scientific research. In the Horizon 2020 program, which reflects the national plans of European countries, the central theme is “from idea to market”. But, do not ideas have value simply because they are ideas, even if there are no plans to take them to the market? The Riemann hypothesis and general number theory are examples of ideas that should not just be financed based on their potential to transform markets. Godfrey H. Hardy, in his Apology of a mathematician, called himself a pure mathematician, priding himself on never having done anything useful. But we have seen that number theory is always behind the markets…
So when we talk of social challenges, I once again call for the Riemann hypothesis to be included among them, even if official journals continue to ignore it.
Manuel de León
(CSIC, Royal Academy of Sciences, Academy of Sciences of the Canary Islands) Research Professor at the CSIC and Member of the Executive Committee of the IMU.