Register to reply

Euler sum of positive integers = -1/12

by gabeeisenstei
Tags: 1 or 12, euler, integers, positive
Share this thread:
Mar3-13, 11:17 AM
P: 31
My question arises in the context of bosonic string theory calculating the number of dimensions, consistent with Lorentz invariance, one finds a factor that is an infinite sum of mode numbers, i.e. positive integers but it really goes back to Euler, and his argument that the sum of all positive integers is -1/12. (One gets the same result by evaluating the zeta function at -1, using the formula with Bernoulli numbers; but I don't find that very illuminating.)

This argument depends in turn on a previous result that the sum of 1-2+3-4+ is 1/4. Given that result, I can follow the manipulation into the all-positive result. But I have a problem with the alternating series. I am looking at the argument found here:

I can follow the shifting out of the 1 and subsequent matching of cancelling pairs amongst the four copies of the series. But my problem is this: after the four pairs are cancelled, it seems that the remaining four copies are no longer the same. The 4th copy has two terms fewer than the 1st, and one fewer than the 2nd and 3rd. Why doesn't that matter?
Phys.Org News Partner Physics news on
An interesting glimpse into how future state-of-the-art electronics might work
What is Nothing?
How computing is transforming materials science research
Mar3-13, 01:36 PM
Sci Advisor
P: 5,437
The best way to deal with these expressions IS zeta function regularization. Doing that one can be sure that one uses the same trick (regulator) for all expressions. I mean, all these expressions ae ill-defined and there is no a priori reason why different tricks should be compatible.
Mar4-13, 02:29 PM
PF Gold
arivero's Avatar
P: 2,907
Quote Quote by gabeeisenstei View Post
This argument depends in turn on a previous result that the sum of 1-2+3-4+ is 1/4.
In fact, the argument for alternating series is the one you use for superstrings (alternating signs are terms for angular momentum of bosons and fermions, up to a factor 1/2)

Mar4-13, 06:09 PM
Sci Advisor
P: 8,555
Euler sum of positive integers = -1/12

Tong gives the zeta function regularization argument in section 2.2.2 and one that seems cleaner in section 4.4.1.

Register to reply

Related Discussions
Let a, b be positive coprime integers. Show that if two positive integers x, y sat... Calculus & Beyond Homework 0
For what positive integers n does 15|M Calculus & Beyond Homework 0
If a,m and n are positive integers with m<n Linear & Abstract Algebra 1
One point compactification of the positive integers Calculus & Beyond Homework 4
Combinatroics 4-permuations of positive integers Calculus & Beyond Homework 1