Riemann Hypothesis

Among numbers, primes like 2, 3, 5, 7, and 11 are the "atoms of numbers" because they can't be broken into a product of other numbers. But you can never predict when the next prime will show up. They look like stars scattered at random across the night sky, and mathematicians have scratched their heads over them for thousands of years.

In 1859, the German mathematician Bernhard Riemann began studying a special formula that could capture how primes are distributed: the Riemann zeta function. It adds up the reciprocals of powers of every whole number, and he extended it into the world of complex numbers (numbers that combine real and imaginary parts).

Riemann found something astonishing. If you know every point where this function equals zero, you can build a formula that tells you exactly how many primes there are up to any given number. The zeta function was a magic map for unlocking the secrets of seemingly random primes.

The special points where the zeta function's result is zero are called "zeros." Leaving aside the trivial zeros, which are easy to find, the remaining important zeros are plotted on a special two-dimensional space called the complex plane.

In an age before computers, Riemann estimated the first few zeros by hand. To his surprise, the horizontal coordinates (the real parts) of those zeros all seemed to sit on a vertical line at one half.

The Riemann Hypothesis is exactly this claim: every important point where the zeta function equals zero lies on this one-half line, with no exceptions. If even a single point is found off the line, the hypothesis collapses at once.

To be more precise, the Riemann Hypothesis is more than a mathematical curiosity. More than a thousand theorems in modern mathematics have been proved under the assumption "if the Riemann Hypothesis is true."

If the hypothesis were proved true, we could pin down how small the error is when estimating how many primes there are up to a given number. That doesn't mean we could pull out the exact position of each prime from a formula. Primes are also an ingredient of modern encryption, but proving the hypothesis would not instantly break codes.

The spacing between the zeta function's zeros also looks statistically strikingly similar to the spacing between energy levels in heavy atomic nuclei. That has made the hypothesis a research topic linking mathematics and quantum physics.

So far, supercomputers have checked trillions of zeros, and every one sat on the one-half line. But in the world of infinite numbers, there must be not a single exception. That is why the Clay Mathematics Institute has put a $1 million prize on this problem and is waiting for a complete logical proof.