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Today’s mathematicians would probably agree that the Riemann Hypothesis is the most significant open problem in all of math. It’s one of the seven Millennium Prize Problems, with a million dollar reward for its solution. It has implications deep into various branches of math, but it’s also simple enough that we can explain the basic idea right here.
There is a function, called the Riemann zeta function, written in the image above.
For each s, this function gives an infinite sum, which takes some basic calculus to approach for even the simplest values of s. For example, if s=2, then (s) is the well-known series 1 + 1/4 + 1/9 + 1/16 + …, which strangely adds up to exactly ²/6. When s is a complex number—one that looks like a+b, using the imaginary number —finding (s) gets tricky.
So tricky, in fact, that it’s become the ultimate math question. Specifically, the Riemann Hypothesis is about when (s)=0; the official statement is, “Every nontrivial zero of the Riemann zeta function has real part 1/2.” On the plane of complex numbers, this means the function has a certain behavior along a special vertical line. You can see this in the visualization of the function above—it’s along the boundary of the rainbow and the red. The hypothesis is that the behavior continues along that line infinitely.
The Hypothesis and the zeta function come from German mathematician Bernhard Riemann, who described them in 1859. Riemann developed them while studying prime numbers and their distribution. Our understanding of prime numbers has flourished in the 160 years since, and Riemann would never have imagined the power of supercomputers. But lacking a solution to the Riemann Hypothesis is a major setback.
If the Riemann Hypothesis were solved tomorrow, it would unlock an avalanche of further progress. It would be huge news throughout the subjects of Number Theory and Analysis. Until then, the Riemann Hypothesis remains one of the largest dams to the river of math research.
