I am separating this answer from the other because it is overly speculative; mainly I wanted to list a few hints about the sequence I named in the comments.

The OP names a square sum that happens to be related to the critical dimension of space time of the bosonic string, D-2=24. It seems natural to ask if there is some similar sequence for the critical dimension of the superstring, D-2=8. On other hand, as I said in the other answer, for the open superstring the Zeta regularization is naturally substituted, in some cases, by the Dirichlet Eta, that happens to be an alternating sum. So it is natural for a numerologist to try the alternating square sum and, as said in the comments to the OP, it works:
$$1^2-2^2+3^2-4^2+5^2-6^2+7^2-8^2= -36 = -6^2$$

The main difference, number-wise, with the non alternating sum, is that here the solution is not unique. Still, it is the smallest non trivial one, and all the others can be generated iteratively: the (absolute value of the) sums are the triangular square numbers, and it was observed by Colin Dickson (alt.math.recreational March 7th 2004) that such numbers obey a recurrence law $$a_{n+1}={(a_n -1)^2 \over a_{n-1}}$$ with the first two terms being the trivial $a_1=1$ and the above $a_2=36$. For more info, see the OEIS sequences A001110 and A001108. Note that the sign in the actual solution depends of the number of terms in the sequence, alternating itself, so that actually the sum is $\sigma_n= (-1)^{n-1} a_n$

A way to produce the alternating solutions is to solve Pell equation, and then the sums are also produced from Pell numbers, via $a_n= P_n (P_n+P_{n-1})$ . This could be interesting because the root of the non alternating series, $70$, is a Pell number itself, the 6th, and the next Pell number, $70+(70+29)=169$, is the only Pell number that is an exact square (and the only one that is an exact power).

In A001108, Mohamed Bouhamida mentions some periodicities and some mod 8 relationships for the series. Also the page http://www.cut-the-knot.org/do_you_know/triSquare.shtml gives some hints on some eight factors appearing in a particular *subsequence* of square triangular numbers: "Eight triangles increased by unity produce a square". If these factors can be related to the 8-periodicities of Bott theory or theta functions, I can not tell.

EDIT: of course the use of triangular numbers can be telling us that all the business of alternating series is just a decoy: pairing the terms, we can reduce $ (m+1)^2 - m^2 = 2 m + 1 = (m+1) + m $ the alternating squared series to the non alternating non squared series and then simply $$1+2+3+4+5+6+7+8= 36 = 6^2$$ But the goal is to keep at least a formal likeliness with the OP series.

This post imported from StackExchange Physics at 2014-08-30 04:15 (UCT), posted by SE-user arivero