Number of dimensions in string theory and possible link with number theory This question has led me to ask somewhat a more specific question. I have read somewhere about a coincidence. Numbers of the form $8k + 2$ appears to be relevant for string theory. For k = 0 one gets 2 dimensional string world sheet, For k = 1 one gets 10 spacetime dimensions and for k = 3 one gets 26 dimensions of bosonic string theory. For k = 2 we get 18. I don't know whether it has any relevance or not in ST. Also the number 24, which can be thought of as number of dimensions perpendicular to 2 dimensional string world sheet in bosonic ST, is the largest number for which the sum of squares up to 24 is itself a square. $(1^2 + 2^2 + ..+24^2 = 70^2)$
My question is, is it a mere coincidence or something deeper than that?
 A: There is definitely something deep going on, but there is not yet a deep understanding of what it is. In math the topology of the orthogonal group has a mod 8 periodicity called Bott periodicity. I think this is related to the dimensions in which one can have Majorana-Weyl spinors with Lorentzian signature which is indeed $8k+2$. So this is part of the connection and allows both the world-sheet and the spacetime for $d=2,10$ to have M-W spinors. The $26$ you get for $k=3$ doesn't have any obvious connection with spinors and supersymmetry, but there are some indirect connections related to the construction of a Vertex Operator Algebra with the Monster as its symmetry group. This involves a $Z_2$ orbifold of the bosonic string on the torus $R^{24}/\Lambda$ where
$\Lambda$ is the Leech lattice. A $Z_2$ orbifold of this theory involves a twist field of dimension $24/16=3/2$ which is the dimension needed for a superconformal generator. So the fact that there are $24$ transverse dimensions does get related to world-sheet superconformal invariance.
Finally, the fact you mentioned involving the sum of squares up to $24^2$ has been exploited in the math literature to give a very elegant construction of the Leech lattice starting from the Lorentzian lattice $\Pi^{25,1}$ by projecting along a null vector $(1,2, \cdots 24;70)$ which is null by the identity you quoted. I can't think of anything off the top of my head related to $k=2$ in string theory, but I'm sure there must be something.
A: The descent from 24 to 8 seems to happen when the straight sums are substituted by alternating sums. This is known from the theory of the Riemann zeta funtion, whose only pole in $s=1$ is cancelled via a multiplication that produces the Dirichlet eta function,     $$\eta(s) = \left(1-2^{1-s}\right) \zeta(s)$$.
This function has better analiticity than Zeta. But it is alternating,
$$    \eta(s) = \frac{1}{1^s} - \frac{1}{2^s} + \frac{1}{3^s} - \frac{1}{4^s} + \cdots$$
And thus it could be related with the simple expresion I have mentioned above in the comments. But, more important,
$$\eta(-1)=\left(1-2^{2}\right) \zeta(-1) = -3 \times {-1\over 12} = {1\over 4}$$
And you can suspect that the Eta function does for the superstring the same role that the Zeta does for the bosonic string. And indeed it appears in very similar situations. For instance, Michael B. Green, in his 1986 Trieste lectures "String and Supertring Theory", section 5.11, calculates the NS sector spectrum and then the normal ordering constant, that appears formally as a difference between the bosonic term and the fermionic term. Such difference can be manipulated to obtain the Eta function as above, times $(D-2)/2$
So if anyone can tell how the Zeta regulator is related to the integer sum up to 24, then we could probably guess how the Eta regulator would be related to the alternating sum up to 8. 
A: 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.  
A: I don't know about the particular example that you mention, but there are certainly some interconnections with special numbers in mathematics and in string theory/supersymmetry. 
One worked out example is the connection of possible dimensions of supersymmetry in dimensions 3, 4, 6 or 10 which is connected to the existence of normed division algebras in dimensions 1, 2, 4 and 8. For more details see


*

*John C. Baez, John Huerta: Division Algebras and Supersymmetry I  (arXiv)


and related work about higher gauge theory.
