Computers and infinity

Question

How do computers calculate the Riemann Zeta Function? I also heard that the computers could calculate Bernoulli numbers. How is this possible? The computers cannot calculate up to infinity, but how do they do it?

Answer 1

The same way humans do — they're just capable of more brute force arithmetic.

For studying Riemann zeroes specifically, I think the usual approach is to find the zeroes of the Riemann Xi function, which are at the same places on the critical line. However, $\xi(\frac{1}{2} + i t)$ is purely real, for real $t$.

If you want to point someplace on the line and say "there's a zero here", then a very simple way to do that is just find a lower and upper bounds $a,b$ on the location of the zero, and show that $\xi(\frac{1}{2} + i a)$ and $\xi(\frac{1}{2} + i b)$ have opposite signs. Since $\xi$ is continuous, that guarantees there is a zero between them.

Furthermore, if you can show the derivative $\frac{d}{dt} \xi(\frac{1}{2} + it)$ is strictly positive or strictly negative on the interval, you're guaranteed there is only one zero on that interval.

You learn the basics of this sort of thing in calculus — pay attention to the error terms or remainder terms that come with your theorems (e.g. the remainder term to the Taylor series). This subject is further expanded upon in numerical analysis.

For a more elementary introduction to these ideas, you might start with the simpler topic of finding zeroes of polynomials; look both at methods for finding numerical estimates (e.g. Newton's method) and understand how Sturm's theorem works to prove where zeroes are located.

Finding properties (like zeroes) of special functions like $\zeta$ is a harder topic; people have spent lots of time studying the different special functions looking for ways to study them fruitfully. (fortunately, for many special functions people have already done this, so we just have to learn the individual methods rather than try to discover them!)