Riemann Hypothesis | Research & Encyclopedia Articles

Riemann Hypothesis

The Riemann hypothesis is a question in the field of number theory that is perhaps the most famous unsolved problem in mathematics. First posed by the great mathematician Bernhard Riemann in 1859, it has captured the imagination of generations of mathematicians. David Hilbert, one of the most important mathematicians of the early twentieth century, once said that if he could look 500 years into the future, his first question would be, "Has someone solved the Riemann hypothesis?" In 1900, at a meeting of the International Congress of Mathematicians in Paris, Hilbert placed the Riemann hypothesis on his famous list of the most important mathematics problems for the next century, a list that has set the course of twentieth-century mathematics. And in April 2000, the Clay Mathematics Institute (in Paris, as a tribute to Hilbert) announced that the Riemann hypothesis is to be one of its seven Millennium Prize Problems, whose solutions will each garner a $1 million prize.

The Riemann hypothesis concerns the Riemann zeta function, a function that transforms each number s into the new number 1/1^s+1/2^s+1/3^s+1/4^s+.... So for example, when s is equal to 2, the Riemann zeta function gives out 1/1[sup2 ]+1/2[sup2 ]+1/3[sup2 ]+..., which, surprisingly, Leonhard Euler proved in 1734 is equal to [sup2 ]/6. More than a century after Euler, Riemann examined the zeta function from the standpoint of the emerging field of complex analysis: he studied the behavior of the zeta function when the input s is allowed to be a complex number. When s is complex, something unexpected can happen: even though the numbers 1, 2, 3, ... are all positive, the sum 1/1^s+1/2^s+1/3^s+... may come out to zero. Riemann calculated the first few of these zero-producing exponents, called zeros for short, and they all lay on the same line in the complex plane: a vertical line whose x-coordinate (the real coordinate) is equal to 1/2. Riemann proceeded to make his famous hypothesis: except for certain well-understood exceptions, all the zeros of the zeta function lie on that 1/2-line.

One of the reasons that the Riemann hypothesis has fascinated mathematicians is that the location of the Riemann zeros is exactly what controls the distribution of prime numbers, those numbers that are only divisible by 1 and themselves. As early as the 4th century B.C., Euclid proved that the number of primes is infinite. But even though there are infinitely many of them, they are they exception, not the rule, and the task of finding them is complicated by the fact that they crop up randomly, seemingly unpredictably. For that reason, searching for large prime numbers is a difficult task, and the discovery of a new prime number is a cause for celebration. But although on the local scale the distribution of prime numbers is haphazard, on the large scale the primes display a surprising order. In 1896, Jacques Hadamard and Charles-Jean-Gustave de la Vallée Poussin proved a remarkable theorem: the probability that a given large number x is prime is roughly 1/log(x). Thus, the primes maintain a delicate balance between order and randomness.

The prime number theorem of Hadamard and de la Vallée Poussin gives the probability that a number is prime, but it does not say how far the actual distribution of primes strays from this rough probability law. It was Riemann who realized that the location of the Riemann zeros holds the key to just how closely the primes follow the 1/log(x) distribution. A century earlier, Euler had discovered a connection between the zeta function and the prime numbers, in the famous formula 1/1^s+1/2^s+1/3^s+...=(2^s3^s5^s7^s11^s...)/(2^s-1)(3^s-1)(5^s-1)(7^s-1)(11^s-1).... Using this formula, Riemann was able to show that if the Riemann zeros are located on the 1/2-line, as he hypothesized, then the prime numbers stay close to the 1/log(x) distribution, with about as much digression from the distribution as you would expect if you were flipping a coin that was weighted to have a 1/log(x) probability. If the zeros do not all lie on the 1/2-line, then the primes are much more unpredictable than has hitherto been imagined.

The first 1,500,000,000 zeros have been calculated by computer, and they do all lie on the 1/2-line, evidence that substantially bolsters Riemann's claim. But since the Riemann hypothesis is a statement about an infinite collection of numbers, it cannot be proved simply by calculating more and more zeros; there will always be more zeros to check.

Although the Riemann hypothesis appears to deal with objects that lie firmly within the realm of pure mathematics, recently researchers have discovered a surprising connection with an emerging branch of physics: quantum chaos. A huge amount of numerical evidence points to the likelihood that the Riemann zeros correspond to the energy levels of a quantum mechanical system whose classical counterpart is chaotic. This new connection is important for both physicists and mathematicians. Energy levels of quantum chaotic systems are generally difficult to calculate, while Riemann zeros are comparatively easy to calculate; thus the link between the two gives physicists a way to generate a wealth of useful date. And for mathematicians, the connection to physics gives a potential way finally to prove the Riemann hypothesis: if mathematicians could find a quantum system whose energy levels correspond to the Riemann zeros, then the symmetries of the system would prove the Riemann hypothesis instantly.