Swagger Again
Recall that one of the permutohedra series begins with the relation
$n(n+1)(2n + 1) = 6(1^2 + 2^2 + 3^2 + \cdots + n^2)$
With the surreals, this relation could extend all the way to $\omega$, resulting in the nonsensical equality
$\frac{1}{6} \omega (\omega + 1)(2 \omega + 1) = \zeta (-2)$
in contrast to the usual definition, where $\zeta (-2)$ is zero. Moreover, it needs to be zero to cancel the pole of $\Gamma (-2)$ in the functional relation defining $\zeta (3)$. This suggests that, paradoxically, $\Gamma (-2)$ should be expressed in terms of the infinitesimal $\omega^{-1} = \varepsilon$. Perhaps we got $\zeta$ and $\Gamma$ mixed up!
Now consider the interesting number $\zeta (2) = \frac{\pi^{2}}{6}$, defined using $\zeta (-1) = \frac{1}{2} \omega (\omega + 1)$. The $\Gamma$ function is conveniently infinite again, usually in order to balance a zero from the sin factor in the functional relation. It seems necessary to balance an awful lot of zeros and infinities just to define the $\zeta$ function. A surreal zeta function may distinguish different zeros with polynomials in $\varepsilon$. Wouldn't that be fun?
$n(n+1)(2n + 1) = 6(1^2 + 2^2 + 3^2 + \cdots + n^2)$
With the surreals, this relation could extend all the way to $\omega$, resulting in the nonsensical equality
$\frac{1}{6} \omega (\omega + 1)(2 \omega + 1) = \zeta (-2)$
in contrast to the usual definition, where $\zeta (-2)$ is zero. Moreover, it needs to be zero to cancel the pole of $\Gamma (-2)$ in the functional relation defining $\zeta (3)$. This suggests that, paradoxically, $\Gamma (-2)$ should be expressed in terms of the infinitesimal $\omega^{-1} = \varepsilon$. Perhaps we got $\zeta$ and $\Gamma$ mixed up!
Now consider the interesting number $\zeta (2) = \frac{\pi^{2}}{6}$, defined using $\zeta (-1) = \frac{1}{2} \omega (\omega + 1)$. The $\Gamma$ function is conveniently infinite again, usually in order to balance a zero from the sin factor in the functional relation. It seems necessary to balance an awful lot of zeros and infinities just to define the $\zeta$ function. A surreal zeta function may distinguish different zeros with polynomials in $\varepsilon$. Wouldn't that be fun?
posted by Kea | 6:09 PM
7 Comments:
I would love some more explanation on all of this, please.
Hi Lieven. Yeah, I'm sorry about the way my brain works! Basically, the QG angle strongly suggests (a) a non set theoretic axiomatic approach to RH and (b) NOT to follow the linear operator approach, but to think of (non linear) gravitational spectra instead. Now (b) is really interesting to think about: maybe in the end we could actually construct a Hermitean operator with the zeroes as spectrum, but it would require understanding complex geometry as an omega-categorical limit (in the sense of 'infinite prime') of some as yet undefined n-cat structure.
Now Kapranov's little Langlands' paper suggests looking at only 2-categorical structures, but QG (in particular, Machian holography) says that we need a duality between '2' and 'infinity' (as well as a whole lot more n-alities) and I just think we'll never understand mass spectra until we formalise this properly.
Now the obscene scale of such a project suggests that a more concrete approach to RH would (at least) have to re-define analytic continuation, and that might start out by defining a different zeta function (using the surreals, and categorified sums) since it no longer matters if we break all the set theoretic (eg. Russell type paradox) rules.
The fact that I keep getting zeroes and infinities mixed up is actually physically suggestive. Stringy dualities say that we should understand how these values are related to each other, or rather, they say that physical observables defined in terms of one are just as good as those defined in terms of the other. But String theory completely misses the importance of Triality to mass generation (not to mention higher levels of the heirarchy) so the question for the zeta function is: what are all the 'higher functional relations' that we're missing?!
Anyway, sorry for blathering. I am trying to sort out some ideas for a proper DARPA proposal, so I'll let you know if I write something up.
Oh, I should probably mention here again that the relation between associahedra and permutohedra (and other operad polytopes) might be relevant to a new notion of analytic continuation, albeit a crazy physics one. The real argument zeta values seem to obey an algebra related to the associahedra (this is discussed briefly in my linked thesis - see the references) and a natural (in Batanin's sense) 2D analogue would involve 2-ordinal polytopes. The integral zeta values are expressed as motivic integrals for the real points of Riemann moduli spaces (a la Devadoss). The surreals seem to be one nice way to generalise the zeta arguments to other reals, ie. as a binary tree instead of a linear string 1,2,3,... but then there's no reason to stop at the reals.
Dear Kea,
surreal infinities do not have number theoretical anatomy and I would not use them in generalization of permutohedra series.
This in mind I would look your identity for zeta(-2) from p-adic view point. Take any p-adic integer which is infinite as real integer (virtually all p-adic numbers are such). In this case the right hand side would have finite p-adic norm but its value would be ill-defined since there is infinite number of integers in the sum having p-adic norm equal to 1 for given p.
This allows the possibility that the terms sum up to zero: or that zeta(-2) vanishes by definition. This would mean that p-adic zetas would vanish at z=-2. If p-adic zeta exists in some sense it would be indeed natural to assume that it vanishes at same points as real zeta and therefore also at z=-2.
Could this identity have some geometric interpretation in terms of infinite (p-adic?)permutohedra?
Dear Kea,
immediately after clicking "Publish your comment" I realized that n(n+1)(2n+1) is completely well defined and non-vanishing for any p-adic integer which is usually infinite as real integer! Hence it cannot vanish, and the identification with the right hand side does not work in p-adic context. Hence also the identification as zeta(-2)=0 fails p-adically.
Note that product formula would give infinite value for zeta(n) for both positive and negative integers.
thanks kea for your attempt to explain things.
i think i have to catch up with a lot of things before i can begin to understand even your clarification.
ive worked through some kapranov papers and know a few things about surreal numbers, so at least i now have 'a way in'...
next week ill take a break, so maybe then ill (re)educate myself...
Thanks, Matti. Hi Lieven. You know, I really haven't done much with the idea of using the surreals. It might not be the right thing to think about.
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