Eli5: How is the Riemann Hypothesis linked to our understanding of prime numbers?
In: Mathematics
The Riemann Zeta Function encodes information about primes and their distribution. Essentially, every prime contributes a factor to the value of the Riemann Zeta Function weighted by how big the prime is, and different inputs to the Riemann Zeta Function weight them differently. So an understanding of the Riemann Zeta Function as a whole can, theoretically reproduce information about primes. For instance, the fact that the Riemann Zeta Function is infinity at 1 implies that there are infinitely many primes and says a bit about how they are distributed along the number line.
So on one hand we can view the Riemann Zeta Function as wholly constructed from the prime numbers. On the other hand, the Riemann Zeta Function is what we call an “Entire Function” (well, meromorphic). One key thing about entire functions is that they, in some ways, behave like “infinite degree polynomials”. What this means is that, like polynomials, they are almost completely determined by where they are zero. This means that the Riemann Zeta Function is almost completely determined by where it equals zero. Some further analysis of the function constrains the (meaningful) zeros of the Riemann Zeta Function to all having to live in some vertical strip on the Complex Plane, called the Critical Strip.
So, the Riemann Zeta Function is completely determined by the primes and completely determined by its zeros. That must mean that we can use the zeros of the Riemann Zeta Function to get new information about the primes! And this is true, we have [formulas directly relating the primes to the zeros of the Riemann Zeta Function](https://people.reed.edu/~jerry/361/lectures/rvm.pdf).
For instance, the Prime Number Theorem was one of the biggest open questions in math during the 1800s. It conjectured that the number of primes less than a number x was “about” x/log(x) (where there is a technical condition for “about”). This was very hard to prove. But Riemann came along and proved all this stuff about the Riemann Zeta Function and showed that if we could prove that there were no zeros on the edges of the Critical Strip, then we would get the Prime Number Theorem. There was still a ways to go, to make Riemann’s analysis rigorous, but eventually the Prime Number Theorem was proved using this strategy.
Riemann noted that the closer to the center that the zeros were in the Critical Strip, the more “nicely” the primes would be distributed. So he conjectured that all the zeros would lie exactly in the center of the Critical Strip. This is the Riemann Hypothesis and is one of the most important open questions in math today.
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