Nonuniqueness of -1/12

Carson Chow Mathematics, Physics

I’ve been asked to give an example of how the sum of the natural numbers could lead to another value in the comments to my previous post so I thought it may be of general interest to more people. Consider again S=1+2+3+4\dots to be the sum of the natural numbers.  The video in the previous slide gives a simple proof by combining divergent sums. In essence, the manipulation is doing renormalization by subtracting away infinities and the left over of this renormalization is -1/12. There is another video that gives the proof through analytic continuation of the Riemann zeta function

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

The zeta function is only strictly convergent when the real part of s is greater than 1. However, you can use analytic continuation to extract values of the zeta function to values where the sum is divergent. What this means is that the zeta function is no longer the “same sum” per se, but a version of the sum taken to a domain where it was not originally defined but smoothly (analytically) connected to the sum. Hence, the sum of the natural numbers is given by \zeta(-1) and \zeta(0)=\sum_{n=1}^\infty 1, (infinite sum over ones). By analytic continuation, we obtain the values \zeta(-1)=-1/12 and \zeta(0)=-1/2.

Now notice that if I subtract the sum over ones from the sum over the natural numbers I still get the sum over the natural numbers, e.g.

1+2+3+4\dots - (1+1+1+1\dots)=0+1+2+3+4\dots.

Now, let me define a new function \xi(s)=\zeta(s)-\zeta(s+1) so \xi(-1) is the sum over the natural numbers and by analytic continuation \xi(-1)=-1/12+1/2=5/12 and thus the sum over the natural numbers is now 5/12. Again, if you try to do arithmetic with infinity, you can get almost anything. A fun exercise is to create some other examples.

3 thoughts on “Nonuniqueness of -1/12”

  1. thats a nice one, but way above my pay grade. (I did graduate 6th—pete seeger played for us–and i saw someone who had my same music teacher on u st last nite (asking about the police—parked in front of our building)

    i think there are also examples on http://www.math. exeter.ac.uk/~mwatkins/zeta/physics.htm

    (number theory and physics archive)

    you can also look at ‘ultrafinitism’ on wikipedia (also mentioned on ‘math under the microscope’ blog) or check out edward nelson (princeton—peano arithmatic is inconsistant (FOM archives—he retracted it; he also had to retract his approach to quantum theory. born rules.. ). or http://www.math.rutgers.edu/~zeilberg/Opinion68.html comments are funny..
    snow coming up etc.should be fun. (plus then i’m going to visit family in hawaii).

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  2. A calc student not knowing analytic continuation might say that you cannot assume \sum(a_n+b_n)=\sum(a_n)+\sum(b_n) unless the LHS converges. But I guess one is assuming ‘convergence’ of sum of natural numbers. I always thought it was neat how a conditionally convergent series can be re-arranged to sum to any real number; I wonder if this is used anywhere in physics?

    In the previous post, if setting the sum of (-1)^n to 1/2 is based on some uniform prob. of stopping, then could one assume another prob. of stopping and make that sum equal to anything between 0 and 1? I might be out of my realm here.

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  3. The rearranging of sums is how renormalization works in quantum field theory. Yes, when you operate on nonconvergent series you can get all sorts of answers. Yes, the sum of (-1)^n is not well defined. However, if you make particular choices you can get assignments of divergent sums to finite numbers. This is just a mapping though and one of an infinite number of possibilities.

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