representation of complex systems. Recall that what is behind the will of the representation of the reality is almost always the desire to find a causal reason. Here, this must be carefully considered only as a linear will. It may be a non-relevant extension of standard paradigms in the case of complex environment. In spite of the non-linearity of power laws, the scientist can easily fall into the trap of an artificial linearization. It is precisely to avoid this trap that it is necessary to understand the link between non-integer dynamics and Riemann zeta function as a topologic, or more precisely, homotopic parametrization of fractional dynamics.

6. Riemann Zeta Function and Non-Integer Differentiation

We recall that the expressions of the Zeta functions and are arithmetically related. We do not enter below in the subtilities of this relationship insofar it is well known for the experts in number theory. We considered only the function assuming that generalizations in the frame TEISI model are obvious. We recall that this function is expressed in the set of integers by (i) a power law under a form of a co-product:

(s) 1 , (23)

n 1 ns

with and also (ii) as a product based on the prime numbers (Tenenbaum2011):

The Riemann function therefore carries the combination of sums and products merged together. This merging recalls all the problems mentioned above. As consequence, the relationship between zeta function and fractality seems rather natural and the author has developed this question in details (Le Méhauté, 2010). The theory of categories makes echoes to this proximity in relation with the punctuated torus. In addition we have already derived a physical demonstration of the Riemann hypothesis based on the TEISI model by using a transition phase geometrical model. The basic diagram of the proof is given below (Figure 5).

According to the parametrizations of the hyperbolic distance v / u 1/ 1/ d this diagram suggests to introduce and then for building a topological morphism between each terms of the sum ( i ) and each integer point upon Z (i ) . Each couple of points is obviously parametrized using integer frequency.

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Figure 5. Key representation of the Riemann Zeta Function hearth in the frame of the TEISI model (Le Méhauté, 2010). Also representation of macro KMS condition versus Fractional Differentiation Riemann Approach through exchange of axis.

More generally, the zeta function (respectively its inverse) is part of a larger set, taking into account the sign of the rotation in the complex plan especially and according to the new set (s), (1 s), (s), (1 s) . This set is obviously centered upon 1/ 2 . This set may be associated through a morphism to a new set of transfer functions which appear clearly in the

diagram above. Both sets authorize the construction of a new morphism upon the sets .... .

The phase characterizing the rotations, and coupling the components of the sets .... , it constrains the morphism between the sets, and this constraint allows a thorough understanding of Riemann as well as Goldbach‟s hypothesis. The trapping of a geometric phase for defining the thermodynamic states is the key factor of the fractal approach of Riemann‟s hypothesis (Le Méhauté, 2010).

The Riemann hypothesis establishes that there are zeros for the zeta function in the only event . This constraint establishes a direct link between the diffusive process analysed via fractional operators and Riemann zeros. The Goldbach hypothesis asserts in this case that any even number is a sum of two primes. Zeros being related to a static state, the first condition implicitly says that there is no assignable stable equilibrium from, except for .The Zeit objekt character is therefore irreducible in the general case. Fractal system is always dynamic. The second conjecture leads to more subtle conclusions that we examine elsewhere.

At this stage a crucial property of zeta functions amplifies the heuristic power of the derivation of non-integer order. This property is the functional relation:

The morphism between transfer functions and zeta functions leads one to say that the functional relationship on zeta gives credit to a homologous relation upon the impedances. In other respects, due to morphism in the couple of sets, another functional relation exists between them. A brief analysis of its properties shows that this relation relates from the disjoint sum of the dual transfer functions and in the complex plan. As shown schematically above (Figure 5), by coupling both impedances, this relationship resets the invariance of the

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energy and raises the time singularity at infinity. This statement requires a theoretical and mathematical work currently in progress (Riot, 2013). Restoring the notherian consistency by extension (Kan extension), the understanding of the Riemann hypothesis through the theory of categories solves the paradox of a dynamic responses for which the energy is not an invariant. The sum hidden behind Kan extension closes the fractional process not upon a transport process as assumed usually, but upon a deterministic process that constitutes the underlying (or horizon) of the process when it is performed by an experimentalist (respectively via the test function in distribution theory). This assertion does not mean that the diffusive process could not constitute an underlying; quite the contrary. We may not to forget that determinism and diffusive process are strongly linked together. The issue that arises here is the question of the functional content of this Kan‟s sum and its links with the un-causal component of the diffusion process, which is given through component even if it is not accessible just via a physical measurement. It is here that the prime numbers enter the lists for giving a statistical meaning to the Kan extension, and therefore for stating the concept of Kan entropy. Each integer number, being associated with point upon the set (seen as hyperbolic distance), is indeed the product of prime numbers. It is written as a product in a set of primes each of them having a position upon. Each point can be seen as both (i) the time constant of the process which is a freedom factor of the experimentalist and (ii) the reference distance as the hyperbolic metric upon.

Surjection

Z Z1

n 2

N

pi ki

pi

Figure 6. Simplified representation of Kan extension in the frame of the TEISI model.

This operation is a canonical surjection, that is to say an epimorphism. A part of information contained in is then lost within the product decomposition, whereas another data are supplied through. Clearly mixed upon the information coming from through the Kan extension requires a statistic entropic calculus. Only merging all data coming from and its complement provides the whole physical information. This was evident since frequency measurement assumes, notwithstanding the choice of the exchange interface, the choice of the reaction developed on the interface; therefore an a priori choice of a time constant (chosen within a discrete set of numbers). This freedom limited by the natural set of integer number must be taken into account. Even constrained experimentally the choice for observing the geometry using a dedicated process is an epistemological choice. The operation of surjection of the time constant upon forces the representation of the process to break his pure thermodynamic state, and to plunge into a disordered set of primes, that, due to the mixing,

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has a higher cardinal than the one of the primes; irreversible time appears at this step. From surjection both complex structure and its complement merges pure and partitioned states, multiplication and addition. The linear reference being always the pure states is solely given by a prime number. One can indeed observe that certain values n (respectively may themselves be primes. In this case the value is pinned upon the unit (respectively n). The surjection becomes a simple application. Then, by inversion, the application is an injection. This shows that if is an ordinate distribution of frequencies, the correlations are only controlled via. may then be associated with a prime numbers distribution, rightly ordinated according to its own structure, but mixed with respect the distribution of the frequencies upon

. This function of mixing via the surjection leads the negentropic/entropic balance. It is not only the origin of the closure of the fractional processes upon the Noether axiomes but the source of the auto organization and of the emergence. These new phenomena is required to close experimentally „non causal components‟ and to regularize the singularity at infinity. Due to the use of energy to do that, it must be related to a competition between hyperbolic distances upon and. Let us notice that the second term, strongly related to zeta function, provides the pure stable states used as references after Kan extension. This observation requires more comments.

7. Auto Organization and Emergence

The origin of emergence in Complex Systems (Holland, 1998) is an open question that always baffles the scientists. Like phase transition, the phenomenon of emergence is indeed a highly nonlinear process led by generally unknown criterions. Nevertheless we know that emergence, characterized by holistic properties, must be sourced within local chaotic behaviour. The Riemann zeta function remarkably illustrates this feature. The local chaos is associated with the stochastic occurrence of the prime numbers whereas, the global behaviour is ordered according to the mathematical well known distribution

Similarly (i) the ideal gas rules PV=NRT that comes from the molecular chaos, (ii) the WLF law of renormalization for polymers comes from the molecular properties of the monomers, (iii) the Peukert law (Peukert, 1897) comes from the surface disorder of the electrodes of batteries, (iv) the processes of aggregation with or without fields raise dendritic fractal structures, etc. From the considerations given above let us seek to understand why macroscopic rules emerge so naturally, from chaotic local process. Especially, from above analysis, we would like to know why the competition between internal and external referential plays a so crucial role in the emergence of auto organization. As noticed previously the relation

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means that the fractional process represented by is less dissipative than the deterministic process that serves as the underlying. Nevertheless it is more dissipative than the diffusive process which bounds via d=2 the geometrical embedding. However, since the energy is not the invariant of the fractional process, it seems impossible at this general stage to evoke a variable that we could call entropy. For giving a meaning to his concept the analysis requires a precise analysis of the Kan extension, that is to say the embedment of the fractal structure of dissipation into a larger infinite set. The surjection using the set of prime number then gives a statistical status of the concept of entropy. Moreover, the properties of scaling of allows to define two distinct types of statistical entropy namely which has a thermodynamic and statistical meaning of the ball of measurement, and which has only a statistical signification evoked in the frame of the interpretation of the Kan extension. Within the standard statistical meaning, the relation ordering the impedances leads the inequality. Because it is empirically based on the performing of energy, the fractional process involves only local dissipative properties (for all integer point of the transfer function) but it involves also global correlations that can be pointed via by means of a distribution of prime adequate time constants. The categorical analysis of Riemann‟s conjecture (Riot, 2013) then suggests designing a set of infinite orthogonal of primes, in which natural numbers can find their locations. Such a referential is natural for fractional process because any scaling is simply a change of unit. It follows that, the deterministic process Z1 may use as Euclidean dynamic external referential

to source the fractional processes Z , whereas diffusive process Z1/ 2 defines in the same referential an hyperbolic distance able to minimize the geometrical entropy. Along the browsing upon process Z from high to low frequencies, the partition function associated with the distribution of prime decreases dually.

Figure 7. Schematic representation of the problematic of „entropy‟ production on self-similar structure, within integer environment (Fractal merged with a co Fractal). In order to balance the energy between deterministic and diffusive process, the irreversible process leads the creation of holistic emerging properties.

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