1. ## zeta function problem

Time for me to eat some crow.

I am looking at $\displaystyle \zeta(-1) = \sum_{n = 1}^{\infty}n = -\frac{1}{12}$

Several weeks ago we had a problem from a member that posted a summation that seemed impossible. As I recall the consensus was that the problem was an example of how not to use the theorems about summing infinite series. (If someone knows the link please let me know. I can't find it.) Well, I've run into it myself now. I am quoting from "String Theory and M-Theory" by Becker, Becker, and Schwarz, page 50.

... (sum on the right hand side) is divergent and needs to be regularized. This can be achieved using $\displaystyle \zeta$-function regularization. First, one considers the general sum
$\displaystyle \zeta (s) = \sum_{n = 1}^{\infty} n^{-s}$

which is defined for any complex number s. For Re(s) > 1, this sum converges to the Riemann zeta function $\displaystyle \zeta(s)$. This zeta function has a unique analytic continuation to s = -1, where it takes the value $\displaystyle \zeta(-1) = -1/12$....
I've never been what you would call "good" at analytic continuation. How the heck can you do this?

Thanks!

-Dan

2. ## Re: zeta function problem

number theory: infinite series 1+2+3+4=-112.

3. ## Re: zeta function problem

Originally Posted by BobP
number theory: infinite series 1+2+3+4=-112.
I'm sorry, I'm not understanding this. What do you mean?

-Dan

4. ## Re: zeta function problem

Originally Posted by topsquark
I'm sorry, I'm not understanding this. What do you mean?

-Dan
I think it may have been a joke, poking fun at Laurence Krauss, remember? Infinite series 1+2+3+4... = -112?

5. ## Re: zeta function problem

Hi Dan

I thought that this was the thread you were looking for. Within it there is a link to a Wikipedia article and that then refers to the zeta function.

Bob

6. ## Re: zeta function problem

Originally Posted by BobP
Hi Dan

I thought that this was the thread you were looking for. Within it there is a link to a Wikipedia article and that then refers to the zeta function.

Bob
I looked at the wikipedia link and also found a link at the bottom of the page (1 + 2 + 3 + 4....) and found a link at the bottom of that page which addressed the "bosonic string" which is where this summation came from. But it didn't elaborate how the result was gotten.

-Dan

7. ## Re: zeta function problem

The plot thickens. I don't know why it took me so long, but Wolfram|Alpha gives the following:
Here
Here
and Here
to name a few.

Can anyone shed any light on how these are calculated?

Bribe: I'll give you a "thanks" in the post.

-Dan