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Math Help - zeta function problem

  1. #1
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    zeta function problem

    Time for me to eat some crow.

    I am looking at \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 \zeta-function regularization. First, one considers the general sum
    \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 \zeta(s). This zeta function has a unique analytic continuation to s = -1, where it takes the value \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
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    Re: zeta function problem

    number theory: infinite series 1+2+3+4=-112.
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    Re: zeta function problem

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

    -Dan
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    Re: zeta function problem

    Quote Originally Posted by topsquark View Post
    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?
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    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
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    Forum Admin topsquark's Avatar
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    Re: zeta function problem

    Quote Originally Posted by BobP View Post
    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
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    Forum Admin topsquark's Avatar
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    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
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