1

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?

wkpk11235
  • 111
  • They make estimations. – Shaun Sep 26 '17 at 14:40
  • How do they make estimations @Shaun ? – wkpk11235 Sep 26 '17 at 14:43
  • They take only a finite number of summands in the summation. – Shaun Sep 26 '17 at 14:44
  • Then how do mathematicians know if the zeros of the Zeta function are really zeros or not if the result is something like 1/1000012312378907128074801720384780? – wkpk11235 Sep 26 '17 at 14:45
  • 2
    in real life we only need to know the answer with acceptable precision. Every numerical method has a criteria when to stop calculations. Sometime the algorithm has a certain number of iterations, sometimes calculated value is compared with the previous iteration and if the values are close enough, the process stops. – Vasili Sep 26 '17 at 14:46
  • Bernouilli number have closed-form because they are the derivatives of $\frac{z}{e^z-1}$ at $z=0$. To estimate $\zeta(s)$ see this series everywhere fast convergent – reuns Sep 26 '17 at 17:27

1 Answers1

2

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!)