Some Comments on the Physical Interpretation of Riemann Zeta in TGD

Abstract

The Riemann zeta function ζ and its generalizations are very interesting from the point of view of the TGD inspired physics. M8 - H duality assumes that rational polynomials define cognitive representations as unique discretizations of space-time regions interpreted in terms of a finite measurement resolution. One implication is that virtual momenta for fermions are algebraic integers in an extension of rationals defined by a rational polynomial P and by Galois confinement integers for the physical states. In principle, also real analytic functions, with possibly rational coefficients, make sense. The notion of conformal confinement with zeros of ζ interpreted as mass squared values and conformal weights, makes ζ and L-functions as its generalizations physically unique real analytic functions. If the conjecture stating that the roots of ζ are algebraic numbers is true, the virtual momenta of fermions could be algebraic integers for virtual fermions and integers for the physical states also for ζ. This makes sense if the notions of Galois group and Galois confinement are sensible notions for ζ . In this essay, the properties of ζ and its symmetric variant ξ and their multi-valued inverses are studied. In particular, the question whether ξ might have no finite critical points is raised.