Matti Pitkanen now has a post about Farey sequences and the Riemann hypothesis in TGD. The idea that the hypothesis is not provable within standard mathematics appears to be gaining a foothold within physical constructions.

On the other hand, it is possible that the physical axioms could guide a concrete proof within a convenient model, such as the Jordan algebra M Theory, in which U duality is algebraically manifest. But the zeta function itself only enters here with the (operad) algebras associated to moduli integrals. So it is difficult to avoid the higher categorical framework in studying exact (eg. MHV) amplitudes, and this lands us back in the world of post ZF axioms.

After inhabiting this world for some time, it becomes difficult to look at zeta functions any other way. One simply can’t help looking at the Selberg axioms and thinking of closure under products, or factorisation, as topos-like axioms, even though these are radically different things. Recall that the interplay of + and x here is thought of as a higher distributive law for monads. This suggests that the Euler relation for zeta functions is about equating invariants based on monads, or rather that the distributivity $+ \times \rightarrow \times +$ is an identity. That is, that the distributivity of complex arithmetic is somehow more responsible for Euler’s product relation than the notion of primeness, which is used through the application of the fundamental theorem of arithmetic only after the product has been expanded.

This suggests that the higher dimensional versions of the Riemann zeta function should be thought of as non-commutative, non-associative and even non-distributive L-functions. Ah! So that’s why Goncharov likes Shimura varieties. Note that such considerations are necessary for understanding even the values of the Riemann function, since its arguments extend throughout the heirarchy.

Update: Khalkhali has a new post on Determinants and Traces in which he notes: “… Bost and Connes in their paper Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory, Selecta Math. (N.S.) 1 (1995), no. 3, 411–457, right in the beginning show that the above formula (5) gives the Euler product formula for the zeta function … In fact their paper starts by quantizing the set of prime numbers … Another interesting issue with regard to the boson-fermion duality formula (6) is its relation with Koszul duality.”

Formula (6) is $Tr_{s}(\Lambda A) Tr(SA) = 1$, a relation between trace and supertrace. Hmm. I would like to understand Koszul duality better because it applies to operads, and more generally PROPS.