In school, we all learned about prime numbers. They're numbers that can only be divided by themselves or one. Some examples are 2, 3, 5, 7, and so on. Mathematicians go nuts over them. Specifically, they spend a lot of time trying to predict exactly where the prime numbers occur on the number line. They seem random, but equations like the Riemann zeta function might predict where we can find them.
We aren't going to go too much into the nitty-gritty on the equation, but it has a really weird property. If you graph the function, all the places where the line hits zero can be connected by another line, one related to complex numbers. So what, you might say. That doesn't seem very exciting.
Well, this property, called the Riemann conjecture, influences nearly everything. Researchers see it pop up randomly in quantum mechanics, number theory, and most importantly, figuring out where prime numbers will appear on the number line. And yet the equation remains unproven. Sure, researchers can find proofs for specific solutions (there have already been 10,000,000,000,000 discovered so far), but nobody can find a general proof that works every time. It's such a big problem that mathematicians can win $1 million to solve it.
Before we break out our calculators, it's worth noting that this is one of the most complicated problems of all time. Remember in the movie A Beautiful Mind how Josh Nash (played by Russell Crowe) went crazy? This was the equation that led the real-life Josh Nash into a descent into madness. So enter at your own risk.