Calculating values of the zeta function

In summary, you need a computer, some nifty maths, and a program like Maple, Matlab, or Matlab (??) in order to calculate zeta.
  • #1
c_d
6
0
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 :smile: ,
 
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  • #3
c_d said:
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:

[tex]\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^s}[/tex]

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).
 

1. What is the zeta function and why is it important in mathematics?

The zeta function, denoted as ζ(s), is a mathematical function that has important applications in number theory, particularly in the study of prime numbers. It is defined as the infinite sum of the reciprocal of natural numbers raised to a power s. The zeta function has connections to other mathematical concepts such as the Riemann hypothesis and the distribution of prime numbers.

2. How is the zeta function calculated?

The zeta function can be calculated using various methods, the most common being the Euler-Maclaurin formula and the functional equation. The Euler-Maclaurin formula is a series expansion that approximates the value of the zeta function at a given point. The functional equation relates the value of ζ(s) to ζ(1-s), allowing for the computation of values on the critical line s = 1/2.

3. What is the significance of the critical line in the zeta function?

The critical line, where Re(s) = 1/2, is a key area of interest in the study of the zeta function. It is related to the Riemann hypothesis, one of the most famous unsolved problems in mathematics. The Riemann hypothesis states that all non-trivial zeros of the zeta function lie on the critical line, and its proof would have far-reaching consequences in number theory.

4. Can the zeta function be evaluated for complex values of s?

Yes, the zeta function can be evaluated for any complex value of s, except for s = 1. In fact, the zeta function is an entire function, meaning it is defined for all complex numbers. This allows for the exploration of the behavior of the zeta function in the complex plane, which has led to many interesting discoveries and conjectures.

5. What are some real-world applications of the zeta function?

Although the zeta function is a purely mathematical concept, it has found practical applications in fields such as physics, engineering, and computer science. It has been used in the study of quantum mechanics, electrical circuits, and data compression algorithms. The zeta function also has connections to the distribution of energy levels in atomic spectra, making it a valuable tool in atomic physics.

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