Some comments of the physical interpretation of Riemann zeta in TGD

Abstract

The Riemann zeta function  $\zeta$ and its generalizations   are very interesting from the point of view of the  TGD inspired physics.  $M^8-H$ 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 $\zeta$ interpreted as mass squared values and conformal weights, makes $\zeta$ and  L-functions as its generalizations physically unique real analytic functions.

If the  conjecture stating that the roots of $\zeta$ 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 $\zeta$. This makes sense if the notions of Galois group and Galois confinement are sensible notions  for $\zeta$.

In this article, the  properties of $\zeta$ and its symmetric variant $\xi$ and their multi-valued inverses are studied. In particular, the question whether $\xi$ might have no finite critical points  is raised.