About the generalization of the holography = holomorphy ansatz to general analytic functions

The general ansatz works also for analytic functions with poles since (f₁=0,f₂=0) implies that the poles do not belong to the space-time surface. What is required is that the roots are not essential singularities. For rational functions R_i=P_i/Q_i the vanishing conditions reduce to those for the polynomials P_i.

The generalization Rieman zeta to polyzeta S_n(s₁,...,s_n) is s function of n complex variables (see this) and satisfies identities analogous to those satisfied by Riemann zeta. This generalization is extremely interesting from the point of view of physics of chaotic and quantum critical systems. Polyzeta S₄ with four complex arguments would define as its roots a 6-D analog of the twistor space of the space-time surface expected to have an infinite number of 6-D roots having interpretation as a generalization of zeros of Riemann zeta.

One could have f₁=S₄ so that its roots would correspond to 6-D zeros of polyzeta S₄(s₁,...,s₄) defining the counterparts of twistor surfaces! f₂=0 could define a map from the M⁴ twistor sphere S²₁ to CP₂ twistor sphere S²₂ characterized by a winding number or vice versa.

A further extremely nice feature is that the space-time surfaces form a number field in the sense that one can sum, multiply and divide the members of f_i and g_i of (f₁,f₂) and (g₁,g₂) elementwise. Also functional composition is possible. One could say that the space-time surface is a number. One can also consider polynomials and polynomials with prime order behave like multiplicative primes. It is also possible to identify prime polynomials with respect to functional composition (see this).

See the article TGD as it is towards end of 2024: part I or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.