Calculating values of the zeta function

[c_d]

Hi Group,

I appologise in advance. My maths knowledge is pretty bad, so some of what I say may not make sense.

I'm interested in the Riemann Zeta function, and more specificaly, the Riemann zero's. I'm not trying to prove it, I just want to calculate some of the values. And that's what I'm having trouble with. I'm okay with complex numbers, but I'm struggling with the series. For example, if I have:

zeta(1/2 + 10i) = sigma(1 / n^(1/2 + 10i)) for n=1 to infinity

I can calculate the values for specific values of n, but n goes all the way to infinity. So, to actually calculate the final value of zeta(1/2 + 10i) I guess I need to use somesort of convergence check. Am I right in thinking this? And, could using somesort of convergence check allow me to calculate a final value for zeta(1/2 + 10i)? Could someone show me how to calculate the final value or point me in the right direction?

Thanks ,

 

[matt grime]

you need a computer, and some nifty maths. in ths very forum there is a thread that explains how to change it to a problem involving real numbers.

https://www.physicsforums.com/showthread.php?t=68125&highlight=riemann+zeta

 

[shmoe]

So, to actually calculate the final value of zeta(1/2 + 10i) I guess I need to use somesort of convergence check. Am I right in thinking this?

Yes, you are right to think about convergence. Actually you shold have thought of convergence before you stuck 1/2+10i into the Dirichlet series (that's the 1/n^s sum thingie), but I'll forgive you. You've obviously seen zeta defined as:

$$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}$$

What you need to know is that this is only valid when the real part of s>1. Everywhere else this is a divergent sum. Remember the harmonic series (take s=1)? In fact you use the divergence at s=1 as another proof there are infinitely many primes.

To find values of Zeta elsewhere, you need some trickery. You've hopefully seen some mention of the analytic continuation of Zeta to the rest of the complex plane (the part where the above sum is invalid)? The functional equation? There's something called the approximate functional equation that will actually let you caclulate zeta in the critical strip using a truncated form of the above sum (meaning it's a sum over only a finite number of terms), plus some error terms that you will have to live with. Versions of this will be in any text on zeta, what kind of references do you have handy?

That's not very sophisticated though. There's something much better called the Riemann-Siegel formula that you might want to look into. In the thread that matt linked too, you can follow a link to Odlyzko's webpage. He's a champ in approximating Zeta numerically, so his webpage is (as always) a great place to go for stuff like this. It will be pretty technical though.

Finally, programs like Maple, mathematica, Matlab (??) will have built in routines for doing this as well. Oh I know there's some java applets on the web that will plot pieces of zeta. Don't have any links off hand, but they shouldn't be hard to find (they probably all use methods based on the Riemann-Siegel formula, so look for that).