Major breakthroughs in mathematics and science are usually the result of many people working over many years. In 2000, seven mathematical problems were awarded $1 million each, and only one has been solved to date.
Mathematicians are bracing for a possible breakthrough in a very old, very difficult problem in number theory. The Riemann hypothesis — concerning the distribution of prime numbers along a number line — is more than 160 years old. Although a new paper does not claim to solve the problem, it could be a significant step toward solving it. It could allow other number theorists to continue making strides toward solving it and, perhaps more importantly, win a $1 million prize.
The Millennium Problems are seven infamously unsolvable mathematical problems posed in 2000 by the prestigious Clay Institute, each with a $1 million reward for solving them. They span all areas of mathematics, as the Clay Institute was founded in 1998 to advance the field through funding for researchers and important breakthroughs.
But the only Millennium Problem solved to date, the Poincaré conjecture, illustrates one of the amusing pitfalls inherent in offering a large monetary prize for mathematics. The winner, Grigori Perelman, declined the Clay Prize, as well as the prestigious Fields Medal. He retired from mathematics and public life in 2006, and as late as 2010 he still insisted that his contribution was equal to that of the mathematician whose work laid the foundation on which he built his proof, Richard Hamilton.
Mathematics, all science, and perhaps all human exploration are filled with pairs or groups circling the same discovery at the same time until someone officially makes a breakthrough. Think of Sir Isaac Newton and Gottfried Leibniz, whose arguments over calculus led to the combined version of the field we still study today. Rosalind Franklin is now mentioned in the same breath as her fellow DNA discoverers James Watson and Francis Crick. Even the Bechdel test for women is sometimes called the Bechdel-Wallace test in the media, because the people almost always collaborate.
The Riemann hypothesis is one of the seven problems of the millennium, along with the Poincaré hypothesis proved by Grigori Perelman and the Yang-Mills theory. It is formulated as follows. Let us take a function – at each point s it is equal to the sum of the series:
This series converges for s greater than one. Using special mathematical techniques, this function can be extended to the entire complex plane — the result is the Riemann zeta function. Moreover, at some points of the complex plane, the values of this function will be equal to zero, for example, at negative even points. These real zeros are called trivial. But besides them, there are other zeros, complex ones — for example, s = 0.5 ± 21.022040i. The Riemann hypothesis states that all non-trivial zeros of the zeta function lie on the line Re=0.5 of the complex plane.
Riemann showed that knowing the non-trivial zeros of the zeta function, one can construct a prime distribution function that shows how many primes there are that do not exceed a given number. The validity of the Riemann hypothesis will allow one to prove statements not related to prime numbers, such as those concerning the computational complexity of various algorithms.
