A trio of mathematicians have set the academic world abuzz with a recent paper that offers a new approach to what has been called "the greatest unsolved problem in mathematics," proving the so-called Riemann hypothesis.

For the mathematically uninitiated, this statement probably seems hopelessly obscure. But for those in the know, it means big dollar signs — $1 million, to be exact. That's because proving the Riemann hypothesis is one of seven Millennium Prize Problems — widely considered the most intractable problems in mathematics — each of which carry a $1 million award for anyone who can solve them.

While the Riemann hypothesis dates back to 1859, for the past 100 years or so mathematicians have been trying to find an operator function like the one discovered here, as it is considered a key step in the proof.

"To our knowledge, this is the first time that an explicit—and perhaps surprisingly relatively simple—operator has been identified whose eigenvalues ['solutions' in matrix terminology] correspond exactly to the nontrivial zeros of the Riemann zeta function,"

Dorje Brody, a mathematical physicist at Brunel University London and coauthor of the new study, told Phys.org.

What still remains to be proven is the second key step: that all of the eigenvalues are real numbers rather than imaginary ones. If future work can prove this, then it would finally prove the Riemann hypothesis.

Brody and his coauthors, mathematical physicists Carl Bender of Washington University in St. Louis and Markus Müller of the University of Western Ontario, have published their work in a recent issue of Physical Review Letters.

The Riemann hypothesis is named after the man who first proposed it, German mathematician Bernhard Riemann. The reason it's important is because it potentially offers a way to understand the distribution of prime numbers, which are notoriously difficult to wrangle; by all appearances, they seem to have a random distribution. If true, however, the Riemann hypothesis offers us a simple method to calculate how many primes there are below any given threshold, which would make the job of prime hunters extraordinarily easier.

## Spacing of primes

The Riemann hypothesis holds such a strong allure because it is deeply connected to number theory and, in particular, the prime numbers. In his 1859 paper, German mathematician Bernhard Riemann investigated the distribution of the prime numbers—or more precisely, the problem "given an integer N, how many prime numbers are there that are smaller than N?"

Riemann conjectured that the distribution of the prime numbers smaller than N is related to the nontrivial zeros of what's now called the Riemann zeta function, ζ(s). (The zeros are the solutions, or the values of s that make the function equal to zero. Although it was easy for mathematicians to see that there are zeros whenever s is a negative even number, these zeros are considered trivial zeros and are not the interesting part of the function.)

Riemann's hypothesis was that all of the nontrivial zeros lie along a single vertical line (½ + it) in the complex plane—meaning their real component is always ½, while their imaginary component i varies as you go up and down the line.

Over the past 150 years, mathematicians have found literally trillions of nontrivial zeros, and all of them have a real of component of ½, just as Riemann thought. It's widely believed that the Riemann hypothesis is true, and much work has been done based on this assumption. But despite intensive efforts, the Riemann hypothesis—that all of the infinitely many zeros lie on this single line—has not yet been proved.

## Identical solutions

## The operator

## Real solutions

## It's an exciting time to be a mathematician, a prime number enthusiast, or even just an admirer of the human intellect.

*You can learn more about the Riemann hypothesis with this incredibly informative video below:*

**Journal reference:** Physical Review Letters