Michel Gondran
email: michel.gondran@polytechnique.org

Alexandre Gondran
email: alexandre.gondran@enac.fr

Replacing the Singlet Spinor of the EPR-B Experiment in the Configuration Space with two Single-particle Spinors in Physical Space

published or not. The documents may come     Replacing the Singlet Spinor of the EPR-B Experiment in the Configuration Space with two Single-particle Spinors in Physical Space

Introduction

A major difficulty of the wave function interpretation of N particles in quantum mechanics is its definition in a 3N-dimensional configuration space. Since the Solvay Conference in 1927, de Broglie and Schrödinger considered the wave function of N particles introduced by Schrödinger in the 3N-dimensional configuration space as fictitious and proposed to replace it by N single-particle wave functions in 3D-space:

"It appears to us certain that if one wants to physically represent the evolution of a system of N corpuscles, one must consider the propagation of N waves in space, each N propagation being determined by the action of the N-1 corpuscles connected to the other waves. Nevertheless, if one focusses one's attention only on the corpuscles, one can represent their states by a point in configuration space, and one can try to relate the motion of this representative point to the propagation of a fictitious wave Ψ in configuration space. It appears to us very probable that the wave Ψ = a(q 1 , q 2 , ..., q n )cos 2π h ϕ(t, q 1 , ...q n ), "a solution of the Schrödinger equation, is only a fictitious wave, which in the Newtonian approximation, plays for the representative point of the system in configuration space the same role of pilot wave and of probability wave that the wave Ψ plays in ordinary space in the case of a single material point." Louis de Broglie [START_REF] De Broglie | La Nouvelle Dynamique des quanta[END_REF], cited by Norsen [START_REF] Norsen | The theory of (Exclusively) local Beables[END_REF] "This use of the q-space [configuration space] is to be seen only as a mathematical tool, as it is often applied also in the old mechanics; ultimately... the process to be described is one in space and time."

Erwin Schrödinger [4], cited by Norsen et al. [START_REF] Noren | Can the wave function in configuration space be replaced by single-particle wave function in physical space[END_REF] p.26

However, this program to replace the wave function in a 3N-dimensional configuration space by N single-particle wave functions was prematurely abandoned. It was recently re-opened by Norsen, Marian and Oriols [START_REF] Norsen | The theory of (Exclusively) local Beables[END_REF][START_REF] Noren | Can the wave function in configuration space be replaced by single-particle wave function in physical space[END_REF]. For spinless non-relativistic particles, these authors show that it is possible in the de Broglie-Bohm pilot-wave theory to replace the wave function in the configuration space by N single-particle wave functions in physical space [START_REF] Noren | Can the wave function in configuration space be replaced by single-particle wave function in physical space[END_REF]. These N wave functions in 3D space are the N conditional wave functions of a subsystem introduced by Dürr, Goldstein and Zanghi [START_REF] Dürr | Quantum equilibrium and the Origin of Absolute Uncertainty[END_REF][START_REF] Dürr | Quantum equilibrium and the role of operators as observables in quantum theory[END_REF]. For a N-particle wave function Ψ(x 1 , x 2 , ..., x N , t), the N conditional wave functions are:

Ψ 1 (x, t) = Ψ(x, x 2 , ..., x N , t)| x2=X2(t);x N =X N (t) Ψ 2 (x, t) = Ψ(x 1 , x, ..., x N , t)| x1=X1(t);x N =X N (t) Ψ N (x, t) = Ψ(x 1 , ..., x N -1 , x, t)| x1=X1(t);x N -1 =X N -1 (t)
where X i (t) is the position of the particle i at time t in the Bohmian mechanics. The evolutions of these positions X(t) = {X 1 (t), X 2 (t), ..., X N (t)} are given by the guidance formula:

dX i (t) dt = m i Im ∇ i Ψ Ψ | x=X(t) ≡ m i Im ∇Ψ i Ψ i | x=Xi(t)
The aim of this paper is to show that this replacement of the wave function in the configuration space by singleparticle functions in the 3D space is also possible in the de Broglie-Bohm interpretation for particles with spin, in particular for the particles in the singlet state of the EPR-B experiment, the Bohm version of the Einstein-Podolsky-Rosen experiment.

To realize, in Bohmian mechanics, this decomposition, we use an explicit solution of the wave function of the EPR-B experiment. The first analytic expression of the wave function and of the probability density for the EPR-B experiment was done in 1987 by Dewdney, Holland and Kyprianidis [START_REF] Dewdney | A causal account of non-local Einstein-Podolsky-Rosen spin correlations[END_REF] via a complete integration of the two-body Pauli equation over time and space. They give also the first causal interpretation of the EPR-B experiment [START_REF] Dewdney | A causal account of non-local Einstein-Podolsky-Rosen spin correlations[END_REF][START_REF] Dewdney | What happens in a spin measurement?[END_REF]. However, this interpretation presents a flaw: the spin module of each particle varied during the experiment from 0 to

2

. The contribution of this paper is, first, to correct this flaw by considering a spin module always equal to 2 , and, second, to replace the singlet spinor of two entangled particles by two single-particle spinors.

The rest of the paper is organized as follows: section 2 recalls how Bohmian mechanics defines the spin of a quantum particle and how it interprets its measurement in a Stern-Gerlach apparatus. The explicit solution of the two-body Pauli equation over time and space for EPR-B experiment, is presented in section 3. A new causal interpretation of the EPR-B experiment is proposed in section 4, correcting the flaw of the previous studies and allowing to replace the singlet spinor by two single-particle spinors.

Spin and its measurement in Bohmian mechanics

In the Copenhagen interpretation, the state of a spin 1/2 particle is given by the wave function Ψ(x, t), called Pauli spinor, which has two complex components Ψ + (x, t) and Ψ -(x, t). The non-relativist evolution of the spinor Ψ(x, t) = Ψ + (x,t) Ψ -(x,t) , for a neutral spin-1/2 particle with a mass m and a magnetic moment µ in a magnetic field B, is given by the Pauli equation:

i ∂Ψ + (x,t) ∂t ∂Ψ -(x,t) ∂t = - 2 2m (∇) 2 Ψ + (x, t) Ψ -(x, t) + µBσ Ψ + (x, t) Ψ -(x, t) (1)
where σ = (σ 1 , σ 2 , σ 3 ) corresponds to the three Pauli matrices.

In the de Broglie-Bohm interpretation, the wave function does not completely represent the state of the quantum particle and it is necessary to add the particle position X(t). The evolution of the spinor is still given by the Pauli equation ( 1) and the evolution of the position is given by the guidance formula introduced by Takabayasi [8], Bohm et al. [START_REF] Bohm | A causal interpretation of the Pauli equation[END_REF]:

dX(t) dt = 2mρ Im(Ψ † ∇Ψ) (2)
where Ψ † = (Ψ + * , Ψ - * ) and ρ = Ψ † Ψ. A two-component spinor can be linked to the three Euler angles (θ,ϕ,χ) and we can write (cf. Bohm and Hiley [START_REF] Bohm | The Undivided Universe[END_REF] p.206):

Ψ(x, t) = Ψ + (x, t) Ψ -(x, t) = √ ρe i χ 2 cos θ 2 e i ϕ 2 isin θ 2 e -i ϕ 2 ,
where ρ, χ, θ and ϕ are functions of x and t. Bohm and al. [START_REF] Bohm | A causal interpretation of the Pauli equation[END_REF] define the spin vector as

s(x, t) = 2ρ Ψ † (x, t)σΨ(x, t) = 2 (sinθ sinϕ, sinθ cosϕ, cosθ). (3)
More properly, s(x, t) is a spin vector field: in each point of the 3D space, a vector is defined by the orientation of θ and ϕ. The spin vector of an individual particle defined by Ψ(x, 0) and X(0) is given by the Eq. ( 3) evaluated along its trajectory X(t):

s(t) = s (x, t)| x=X(t) = s(X(t), t). (4)
The spin vector therefore depends on the spinor and on the position of the particle. As Dürr, Goldstein and Zanghi remark [START_REF] Dürr | Quantum equilibrium and the role of operators as observables in quantum theory[END_REF]: "Unlike position, spin is not primitive, i.e., no actual discrete degrees of freedom, analogous to the actual positions of the particles, are added to the state description in order to deal with "particles with spin". Roughly speaking, spin is merely in the wave function".

In the de Broglie-Bohm interpretation, the initial wave function of a quantum particle, prepared with a spin vector having (θ 0 ,ϕ 0 ) as Euler angles, is described by a spinor as the following:

Ψ 0 (x, z) = (2πσ 2 0 ) -1 2 e -x 2 +z 2 4σ 2 0 cos θ0 2 e i ϕ 0 2 sin θ0 2 e -i ϕ 0 2 (5) = (2πσ 2 0 ) -1 2 e -x 2 +z 2 4σ 2 0 cos θ 0 2 e i ϕ 0 2 |+ + sin θ 0 2 e -i ϕ 0 2 |- (6)
corresponding to a pure state. Quantum mechanics textbooks [START_REF] Feynman | The Feynman Lectures on Physics[END_REF][START_REF] Cohen-Tannoudji | Quantum Mechanics[END_REF][START_REF] Sakurai | Modern Quantum Mechanics[END_REF][START_REF] Bellac | Quantum Physics[END_REF] do not take into account the spatial extension of the spinor (5) and simply use the simplified spinor without spatial extension:

Ψ 0 = cos θ0 2 e i ϕ 0 2 sin θ0 2 e -i ϕ 0 2 . ( 7
)
This spatial extension enables, in following the precursory works of Takabayasi [START_REF] Takabayasi | The Vector Representation of Spinning Particle in Quantum Theory[END_REF], Bohm et al. [START_REF] Bohm | A causal interpretation of the Pauli equation[END_REF] and Dewdney and al. [START_REF] Dewdney | A causal account of non-local Einstein-Podolsky-Rosen spin correlations[END_REF], to taking into account the spin evolution during the measurement. Indeed, the difference in the evolution of the spatial extension between the two spinor components has a key role in the explanation of the measurement process with Bohmian mechanics (cf. equation (32) in Appendix).

In the Stern-Gerlach experiment, the spin of equations ( 5) or ( 7) is not directly measured; its measurement is obtained, after passing through an electromagnet during a time ∆t, from the impact of the particle on a screen located 20 cm after the Stern-Gerlach electromagnet. This distance corresponds to the time required to separate the initial wave packet into two disjoint packets; this is the decoherence time t D (cf. equation (35) in Appendix).

The measurement of spin (up or down along the z-axis) has no a pre-existing value before measurement. For the spinor (5), the initial spin vector s(X(0), 0) = 2 (sin θ 0 sin ϕ 0 , sin θ 0 cos ϕ 0 , cos θ 0 ) does not depend on the initial position X(0) = (x 0 , y 0 , z 0 ), but will evolve as s(X(t), t) differently during measurement depending on the initial position z 0 of the particle.

Figure 1 presents in the (Oyz) plane, 10 trajectories of silver atoms having the same initial spinor orientations (θ 0 = π 3 , ϕ 0 = 0) but having 10 different initial positions z 0 . Those initial positions z 0 have been randomly chosen from a Gaussian distribution with standard deviation σ 0 . The spin orientation θ(z(t), t) of each atom is represented by arrows. The final orientation, obtained after the decoherence time t D (equation (35) in Appendix), depends on the specific initial particle position z 0 in the spinor with a spatial extension and on the initial angle θ 0 of the spin with the z-axis. In previous works [START_REF] Dewdney | What happens in a spin measurement?[END_REF][START_REF] Gondran | A complete analysis of the Stern-Gerlach experiment using Pauli spinors[END_REF],

θ(t D ) = + π 2 is obtained if z 0 > z θ0 and θ(t D ) = -π 2 if z 0 < z θ0 with z θ0 = σ 0 Φ -1 sin 2 θ 0 2 ( 8
)
where Φ is the cumulative distribution of the standard normal distribution. If we ignore the position of the atom in its wave function, we lose the determinism given by equation ( 8).

In the de Broglie-Bohm interpretation, the "measured" value is not a preexisting value. It is contextual and conforms to the Kochen and Specker theorem [START_REF] Kochen | The problem of hidden variables in quantum mechanics[END_REF].

Finally, the Bohmian mechanics proposes a clear interpretation of the spin measurement in quantum mechanics. There is interaction with the measuring apparatus as Bohr said; and there is indeed a minimum time required to measure. However this measurement and this time do not have the meaning that is usually attributed to them. The result of the Stern-Gerlach experiment is not the measure of the spin projection along the z-axis, but the continuous orientation of the spin either in the direction of the magnetic field gradient, or in the opposite direction. It depends on the position of the particle in the wave function. We have therefore a simple explanation for the non-compatibility of spin measurements along different axes. The measurement duration (t t D ) is then the time necessary for the particle to point its spin in the final direction. The electromagnet A "measures" the A spin in the direction of the z-axis and the electromagnet B "measures" the B spin in the direction of the z -axis, which is obtained after a rotation of an angle δ around the y-axis.

In most papers on EPR-B experiment, the initial wave function of the quantum system composed of two entangled particles is the singlet spinor:

Ψ 0 = 1 √ 2 (|+ A |-B -|-A |+ B ) (9)
where |± A (resp. |± B ) are the eigenvectors of the spin operators S z A (resp. S z B ) in the z-direction pertaining to particle A (resp.B):

S z A |± A = ±( 2 )|± A (resp. S z B |± B = ±( 2 )|± B ).
More specifically, the initial singlet wave function has a spatial extension:

Ψ 0 (r A , r B ) = 1 √ 2 f (r A )f (r B )(|+ A |-B -|-A |+ B ) (10)
where r = (x, z) and

f (r) = (2πσ 2 0 ) -1 2 e -x 2 +z 2 4σ 2
0 . This spatial extension is essential to solve correctly the Pauli equation in space. Moreover, in Bohmian mechanics, the spatial extension is necessary to take into account particle position.

In the Copenhaguen interpretation, the result of the simultaneous measurement of two spins is obtained directly from the initial wave function [START_REF] Bohm | A causal interpretation of the Pauli equation[END_REF] written in the basis of the eigenvectors of the measuring operators, using for this the quantum-mechanics measurement postulates. For this, we make a coordinate change of the particle B, placing it in the plane x Oy obtained from Oz by rotation δ around Oy. So we have: In the new basis |± B of the eigenvectors of the operator σ z B , the spinors |± B are written:

x B = x B cos δ + z B sin δ and z B = -x B sin δ + z B cos δ.
|+ B = cos δ 2 |+ B + sin δ 2 |-B and |-B = -sin δ 2 |+ B + cos δ 2 |-B . (11)
The initial wave function ( 9) is then written:

Ψ 0 = 1 √ 2 (-sin δ 2 |+ A |+ B + cos δ 2 |+ A |-B -cos δ 2 |-A |+ B -sin δ 2 |-A |-B ). (12)
Since, in the EPR-B experiment, A spin is measured along the z-axis and that of B along the z -axis, the wave function [START_REF] Kochen | The problem of hidden variables in quantum mechanics[END_REF] predicts the following probabilities for the measuring torques:

P (+, +) = P (-, -) = 1 2 sin 2 δ 2 , P (+, -) = P (-, +) = 1 2 cos 2 δ 2 (13)
where P (+, -), for example, corresponds to the probability to find A with the spin up (+) and B with the spin down (-).

The same probabilities are obtained if we take the initial singlet [START_REF] Dewdney | A causal account of non-local Einstein-Podolsky-Rosen spin correlations[END_REF] with a spatial extension. Indeed as f (r B ) = f (r B ), we have:

Ψ 0 (r A , r B ) = 1 √ 2 f (r A )f (r B ) -sin δ 2 |+ A |+ B + cos δ 2 |+ A |-B -cos δ 2 |-A |+ B -sin δ 2 |-A |-B (14)
and the calculation of P (+, -) for example is made by integration:

P (+, -) = 1 2 cos 2 δ 2 |f (r A )| 2 |f (r B )| 2 dr A dr B = 1 2 cos 2 δ 2 .
In the de Broglie-Bohm interpretation, the postulates of quantum-mechanics measurement are not used and results of the measurement are obtained by calculating the evolution of the wave function in interaction with measuring apparatuses (see Appendix for numerical values).

Let us consider the wave function of the two particles A and B of the EPR-B experiment in the configuration space. These are two identical particles:both are electrically neutral, with magnetic moments µ, and are respectively subject to magnetic fields B A and B B . This wave function Ψ(r A , r B , t) admits 4 components Ψ a,b (r A , r B , t) on the basis [|± A , |± B ] with a = ± and b = ±, and its evolution is given by the two-body Pauli equation (see Holland [START_REF] Holland | The quantum Theory of Motion[END_REF] p. 417 and Dürr et al. [START_REF] Dürr | Quantum equilibrium and the role of operators as observables in quantum theory[END_REF]). In Einstein notation this is expressed as:

i ∂Ψ a,b ∂t = - 2 2m (∇ A ) 2 - 2 2m (∇ B ) 2 Ψ a,b + µB A j (r A )(σ j ) a c Ψ c,b + µB B j (r B )(σ j ) b d Ψ a,d (15)
with j=1 to 3. The initial conditions are:

Ψ a,b (r A , r B , 0) = Ψ a,b 0 (r A , r B ) (16)
where the Ψ a,b 0 (r A , r B ) correspond to the singlet state [START_REF] Dewdney | A causal account of non-local Einstein-Podolsky-Rosen spin correlations[END_REF]. One of the difficulties of the interpretation of the EPR-B experiment is the existence of two simultaneous measurements. By doing these measurements one after the other as proposed in 1987 by Dewdney, Holland and Kyprianidis [START_REF] Dewdney | What happens in a spin measurement?[END_REF], it facilitates the interpretation and calculation of the experiment. That is the purpose of the two-step version of the EPR-B experiment studied below. The latter experiment is equivalent to the previous experiment and gives the usual correlations (13) of the initial EPR-B experiment.

Classic treatments of the EPR-B experiment within Bohmian mechanics [START_REF] Holland | The quantum Theory of Motion[END_REF][START_REF] Dewdney | A causal account of non-local Einstein-Podolsky-Rosen spin correlations[END_REF][START_REF] Dewdney | [END_REF] focus only on final calculations in order to ensure consistency with experimental results. In our view, intermediate formulas (those after the first step) are also very interesting to present. That is why we detail below the complete calculations and the conclusions after each step.

First step: Measurement of A spin

In the first step we make, on a pair of entangled particles A and B, a Stern-Gerlach "measurement" for atom A, then in the second step a Stern-Gerlach "measurement" for atom B.

Consider that at time t 0 the particle A arrives at the entrance of electromagnet A. t is the duration of crossing electromagnet A and t is the time after the A exit. At time t 0 + t + t, wave function [START_REF] Dewdney | A causal account of non-local Einstein-Podolsky-Rosen spin correlations[END_REF] becomes [START_REF] Gondran | A new causal interpretation of the EPR-B Experiment[END_REF]:

Ψ(r A , r B , t 0 + t + t) = 1 √ 2 f (r B ) f + (r A , t)|+ A |-B -f -(r A , t)|-A |+ B ( 17
) with f ± (r, t) f (x, z ∓ z ∓ ut)e i( ±muz +ϕ ± (t)) . ( 18
)
where z ∆ and u are defined in Appendix by equation (33). The atomic density ρ(z A , z B , t 0 + ∆t + t) is found by integrating Ψ † (r A , r B , t 0 + t + t) × Ψ(r A , r B , t 0 + t + t) on x A and x B :

ρ(z A , z B , t 0 + ∆t + t) = (2πσ 2 0 ) -1 2 e - (z B ) 2 2σ 2 0 (19) × (2πσ 2 0 ) -1 2 1 2 e - (z A -z ∆ -ut) 2 2σ 2 0 + e - (z A +z ∆ +ut) 2 2σ 2 0
.

Our previous studies [START_REF] Gondran | A new causal interpretation of the EPR-B Experiment[END_REF] on the EPR-B experiment are based on equations ( 17) and [START_REF] Gondran | Decoherence time and spin measurement in the Stern-Gerlach experiment[END_REF], which are not explicitly presented in the classic Bohmian framework. We deduce from [START_REF] Gondran | Decoherence time and spin measurement in the Stern-Gerlach experiment[END_REF] that the beam of particles A is divided into two parts, while the B beam of particles is not, and remains stable. Moreover, we note that the space quantization of particle A is identical to that of an free particle in a single Stern-Gerlach apparatus: the distance δz = 2(z ∆ + ut) between the two spots N + (spin +) and N -(spin -) of a set of particles A is the same as the distance between the two spots N + and N -of a set of particles in a single Stern-Gerlach experiment [START_REF] Gondran | A complete analysis of the Stern-Gerlach experiment using Pauli spinors[END_REF], cf. (34) in Appendix. We finally deduce from [START_REF] Gondran | Decoherence time and spin measurement in the Stern-Gerlach experiment[END_REF] that:

• the density of B is not affected by the "measurement" of A,

• the density of A is the same, whether particle A is entangled with B [START_REF] Gondran | Decoherence time and spin measurement in the Stern-Gerlach experiment[END_REF] or not.

These two results can be tested experimentally. We also conclude from ( 17) that the spins of A and B remain opposite throughout the experiment. This analysis of the first step was not made in the previous Bohmian approaches. It also provides the possibility of replacing the single spinor of two entangled particles with two independent single-particle spinors, plus an interaction-at-a-distance that maintains the two spin vectors in opposite directions.

3.2 Second step: "Measurement" of B spin.

After a first step of a Stern-Gerlach "measurement" on atom A, between t 0 and t 1 = t 0 + t + t D , the second step corresponds to a Stern-Gerlach "measurement" on atom B, with an electromagnet B forming an angle δ with A between t 1 and t 1 + t + t D .

At time t 1 , the wave function in configuration space is given by ( 17) with t = t D . In the new basis [|± A , |± B ], this wave function is written:

Ψ(r A , r B , t 1 ) = 1 √ 2 f (r B ) -sin δ 2 f + (r A , t D )|+ A |+ B + cos δ 2 f + (r A , t D )|+ A |-B -cos δ 2 f -(r A , t D )|-A |+ B -sin δ 2 f -(r A , t D )|-A |-B . (20)
After the measurement of B at its exit of magnetic field B, at time t 1 + t + t D , the wave function [START_REF] Gondran | A new causal interpretation of the EPR-B Experiment[END_REF] becomes:

Ψ(r A , r B , t 1 + t + t D ) = 1 √ 2 -sin δ 2 f + (r A , t D )f + (r B , t D )|+ A |+ B + cos δ 2 f + (r A , t D )f -(r B , t D )|+ A |-B -cos δ 2 f -(r A , t D )f + (r B , t D )|-A |+ B -sin δ 2 f -(r A , t D )f -(r B , t D )|-A |-B . (21)
Equation ( 21) predicts probabilities of [START_REF] Bohm | The Undivided Universe[END_REF]. The calculation of P (+, -), for example, is made by integration:

P (+, -) = 1 2 cos 2 δ 2 |f + (r A , t D )| 2 |f -(r B , t D )| 2 dr A dr B = 1 2 cos 2 δ 2 .
Similarly to Holland, we obtain, for spatial quantization and correlations of spins in this two-step version of the EPR-B experiment, the same results as in the EPR-B experiment. The EPR correlations are therefore obtained only by solving the two-body Pauli equation without using the quantum-measurement postulates.

Two single-particle spinors in the EPR-B experiment

In this section, we present the principal contribution of the paper: how to replace the singlet spinor of EPR-B experiment with two single-particle spinors plus an interaction-at-a-distance that maintains the two spin vectors in opposite directions. When the entangled pair of particles A and B is created, we assume that each particle has the initial wave function: 6):

Ψ A 0 (r A , θ A 0 , ϕ A 0 ) and Ψ B 0 (r B , θ B 0 , ϕ B 0 ) like in equation (
Ψ A 0 (r A , θ A 0 , ϕ A 0 ) = f (r A ) cos θ A 0 2 |+ A + sin θ A 0 2 e iϕ A 0 |-A (22)
and

Ψ B 0 (r B , θ B 0 , ϕ B 0 ) = f (r B ) cos θ B 0 2 |+ B + sin θ B 0 2 e iϕ B 0 |-B . (23)
Moreover those spinors have opposite spins:

θ B 0 = π -θ A 0 , ϕ B 0 = ϕ A 0 -π.
We treat the dependence on y classically: speed -v 0 for A and v 0 for B. Then the Pauli principle tells us that the two-body wave function must be antisymmetric; it is written:

Ψ 0 (r A , θ A , ϕ A , r B , θ B , ϕ B ) = Ψ 0 A (r A , θ A , ϕ A )Ψ 0 B (r B , θ B , ϕ B ) -Ψ 0 A (r B , θ B , ϕ B )Ψ 0 B (r A , θ A , ϕ A ) i.e. Ψ 0 (r A , θ A , ϕ A , r B , θ B , ϕ B ) = f (r A )f (r B )[(cos θ A 2 |+ A +sin θ A 2 e iϕ A |-A )(cos θ B 2 |+ B +sin θ B 2 e iϕ B |-B )-(cos θ B 2 |+ A + sin θ B 2 e iϕ B |-A )(cos θ A 2 |+ B + sin θ A 2 e iϕ A |-B )],
and after calculation we obtain the same singlet state as [START_REF] Dewdney | A causal account of non-local Einstein-Podolsky-Rosen spin correlations[END_REF], factorwise:

Ψ 0 (r A , θ A , ϕ A , r B , θ B , ϕ B ) = -e iϕ A f (r A )f (r B )(|+ A |-B -|-A |+ B )
The assumption of the existence of initial wave functions Ψ A 0 (r A , θ A 0 , ϕ A 0 ) and Ψ B 0 (r B , θ B 0 , ϕ B 0 ) (equations (22-23)) is consistent with singlet state [START_REF] Dewdney | A causal account of non-local Einstein-Podolsky-Rosen spin correlations[END_REF] and new in Bohmian interpretations. It is important to note that each entangled pair of atoms has a different value of (θ A 0 , ϕ A 0 ) and thus (θ

B 0 = π -θ A 0 , ϕ B 0 = ϕ A 0 -π).
At each emission of one EPR-B pair, the initial spin directions are unknown: θ A 0 have a uniform distribution over [0,π] and ϕ A 0 have a uniform distribution over [0,2π], according to the invariance in all rotations in 3D space.

This interpretation differs from the classic Bohmian interpretation. Indeed, in Dewdney and al. [START_REF] Dewdney | A causal account of non-local Einstein-Podolsky-Rosen spin correlations[END_REF][START_REF] Dewdney | [END_REF], Bohm and Hiley [START_REF] Bohm | The Undivided Universe[END_REF] (p. 226) and Holland [START_REF] Holland | The quantum Theory of Motion[END_REF] (p. 417 and 467), the spin vectors are defined by a generalization of the definition (3) applied to the singlet wave function, and no to a wave function in the 3D space. These authors find for each particle an initial spin that is strictly zero and a variation of the spin module during the experiment from 0 to 2 . This solution gives mathematically a causal interpretation of the EPR-B experiment, but variability of spin module causes it to lose its physical sense.

With our assumptions, we consider two initial spin vectors s A and s B with a module 2 as in the one-body case. It is the total spin of the singlet that is equal to zero. We therefore assume that, at the initial time, we know the wave functions ( 22) and ( 23) of the particles A and B. In the de Broglie-Bohm interpretation, we assume also that the intial position of particle A is known (x A 0 , y A 0 = 0, z A 0 ) as well as of the particle B (x B 0 , y B 0 = 0, z B 0 ). It remains to determine the evolution of these wave functions and the trajectories of particles A and B. Let's start with particle A. Equation [START_REF] Gondran | Decoherence time and spin measurement in the Stern-Gerlach experiment[END_REF] shows that the density of A is independent of that of B: it is equal to the density of a family of free particles in a Stern-Gerlach apparatus, whose initial spin orientation has been randomly chosen (it is exactly the density given by equation (34) in the Appendix). Since the particle A can be described by the initial wave function [START_REF] Holland | The quantum Theory of Motion[END_REF], we can assume that its evolution is that to a free particle in a Stern-Gerlach apparatus, i.e.:

Ψ A (r A , t 0 + t + t) = cos θ A 0 2 f + (r A , t)|+ A + sin θ A 0 2 e iϕ A 0 f -(r A , t)|-A (24)
For an initial polarization (θ A 0 , ϕ A 0 ) and an initial position (z A 0 ), we obtain, in the de Broglie-Bohm interpretation [START_REF] Bohm | The Undivided Universe[END_REF], an evolution of the position (z A (t)) and of the spin orientation of A (θ A (z A (t), t)) [START_REF] Gondran | A complete analysis of the Stern-Gerlach experiment using Pauli spinors[END_REF]. In the interval [t 0 , t 0 + ∆t] during passage through the electromagnet, we obtain:

dz A dt = µ 0 B 0 t m cosθ(z A , t) with tan θ(z A (t), t) 2 = tan θ 0 2 e - µ 0 B 0 t 2 z A 2mσ 2 0 ( 25
)
with the initial condition z A (t 0 ) = z A 0 ; and in the interval [t 0 + ∆t, t 0 + ∆t + t] (t ≥ 0) after passing through the electromagnet:

dz A dt = u tanh( (z∆+ut)z A σ 2 0 ) + cos θ 0 1 + tanh( (z∆+ut)z A σ 2 0 ) cos θ 0 and tan θ(z A (t), t) 2 = tan θ 0 2 e - (z ∆ +ut)z A σ 2 0 . ( 26
)
It is this evolution of the polarization which is shown in Figure 1 for the initial polarization (θ 0 = π 3

). The behavior of particle A is independent of B, whether the particle is entangled or not.

Let us now study particle B. Equation [START_REF] Gondran | Decoherence time and spin measurement in the Stern-Gerlach experiment[END_REF] shows that the density of B is independent of time and of the density of B: it is equal to the density of a family of free particles, which is constant in x and z. Therefore we can assume that the particle B is immobile in x and z: z B (t) = z B 0 and x B (t) = x B 0 . Moreover, B follows a rectilinear classical trajectory in y with y B (t) = v 0 t.

Equation [START_REF] Bellac | Quantum Physics[END_REF] shows that spins of A and B remain opposite throughout step 1. The spin of a particle A is oriented gradually following the position of the particle. The spin of particle B follows that of A, while remaining opposite. Therefore, we can assume that the orientation of B spin is driven by the orientation of A spin, like an interaction-at-a-distance:

θ B (t) = π -θ(z A (t), t) and ϕ B (t) = ϕ(z A (t), t) -π. ( 27
)
Since the particle B can be described by the initial wave function [START_REF] Albert | Quantum mechanics and experience[END_REF], we can then associate to the particle B the wave function:

Ψ B (r B , t 0 + t + t) = f (r B ) cos θ B (t) 2 |+ B + sin θ B (t) 2 e iϕ B (t) |-B . ( 28
)
This wave funtion is specific, because it depends upon initial conditions of A (positions and spins). The orientation of B spin is driven by that of particle A through the singlet wave function. Thus, the singlet wave function is the non-local hidden variable.

Finally, during the first step, the singlet spinor in configuration space ( 17) can be replaced by the two single-particle spinors given by equations ( 24) and (28).

After the "measurement" of A at time

t 1 = t 0 + ∆t + t D , if the A measurement is + (respectively -), i.e. θ(z A (t), t) = + π 2 (resp. - π 2
), we can deduce from equations ( 27) and (28) that:

Ψ B (r B , t 1 ) = f (r B )e iϕ B (t1) |-B (resp. f (r B )|+ B ) (29)
We have also x B (t 1 ) = x B 0 and x B (t 1 ) = z B 0 . After this first "measurement", the second step of the EPR-B experiment is exactly the case of a single particle in a Stern-Gerlach magnet B which is at an angle δ (resp. π -δ) in relation to A.

Figure 3 represents the evolution of trajectories and the spin orientations of three pairs of entangled A-B atoms in the first step of the EPR-B experiment. The three pairs of A-B particles are represented in black, gray and white on the figure. There are created at the initial moment in y = 0 (y A 0 = y B 0 = 0). The respective positions of A and B in relation to z, z A 0 and z B 0 , at the initial moment have been randomly chosen. The orientation of the spin of A at the initial moment, θ A 0 , is randomly chosen, in opposition to B, i.e.:

θ B 0 = π -θ A 0 (and ϕ B 0 = ϕ A 0 -π).

Conclusion

We first recalled the definition of the spin vector in Bohmian mechanics (dependent on both wave function and position) and its evolution during the phenomenon of measurement. In Bohmian mechanics, the "measured" value is not a preexisting value. It is the value obtained after a continuous orientation of the spin, either in the direction of the magnetic field gradient, or in the opposite direction.

Next, we have shown that, for the two entangled particles of the two-step version of the EPR-B experiment, it is possible to replace the singlet spinor in configuration space ( 17) by two single-particle spinors in physical space, given by equations ( 24) and (28).

We have demonstrated that the "first-measured" particle A behaves in a Stern-Gerlach apparatus as if it were not entangled. During the measurement of A, the particle density of B evolves as if it were not entangled. These two properties could be tested experimentally when the EPR-B experiment can be carried out easily. This result was obtained in the de Broglie-Bohm interpretation using an integration of the two-body Pauli equation over time and space from an initial singlet with a spatial extension [START_REF] Dewdney | A causal account of non-local Einstein-Podolsky-Rosen spin correlations[END_REF].

As de Broglie and Schrödinger stated at the Solvay Conference in 1927, the wave function in configuration space may only be a mathematical tool that can be replaced by more physical wave functions. In our model, the A wave function is the same as that of a free particle in a Stern-Gerlach apparatus, the B wave function is the same as that of a free particle whose spin orientation vector is driven by the orientation of the A spin. Thus, we obtain a possible physical understanding of the EPR-B experience and the entanglement.

Our interpretation goes beyond the limit of the one put forward by Dewdney, Holland and Kyprianidis [START_REF] Dewdney | A causal account of non-local Einstein-Podolsky-Rosen spin correlations[END_REF],

where EPR-B spins evolved from 0 to 2 from creation to measurement, which fits badly with spin quantification. We thus make the de Broglie-Bohm interpretation more credible.

These results also reopen the discussion about the completeness of quantum mechanics and the existence of hidden variables. Firstly, it clearly shows that Bohmian mechanics, which only uses resolution of the Pauli equation, gives the same statistical results as the Copenhagen interpretation for the Stern-Gerlach and EPR-B experiments. It is the two-body Pauli equation that couples spin and spatial degrees of freedom in equations ( 17) and [START_REF] Dewdney | [END_REF]. Moreover, the measurement postulates and the postulate of wave packet reduction are not used in Bohmian mechanics and we show that they can be demonstrated (cf. Appendix).

The wave function of the singlet state alone introduces non locality: when we replace a singlet wave function in the configuration space with two wave functions in the 3D physical space, we must introduce interaction-at-a-distance (equation ( 27)) between the A spin orientation and that of B.

Thus, the non-local influence in the EPR-B experiment only concerns the spin orientation, not the motion of the particles themselves. This is a key point in the search for a physical understanding of this non-local influence.

Appendix: Spin "measurement " in the Stern-Gerlach experiment

The measurement of spin of a silver atom is carried out by a Stern-Gerlach apparatus: an electromagnet A, where there is a strongly inhomogeneous magnetic field, followed by a screen P (Fig. 4). In the Stern-Gerlach experiment, silver atoms contained in the oven E are heated to a high temperature and escape through a narrow opening. A second aperture, T, selects those atoms whose velocity, v 0 , is parallel to the y-axis. The atomic beam passes through the gap of the electromagnet A, before condensing on the screen P on two spots of equal intensity N + and N -. The magnetic moment of each silver atom before crossing the electromagnet is oriented randomly (isotropically).

In the beam, we represent the atoms by their wave function; one can assume that at the entrance to the electromagnet A (at the initial time t = 0), each atom can be approximatively described by a Gaussian spinor in x and z:

Ψ 0 (x, z) = (2πσ 2 0 ) -1 2 e -x 2 +z 2 4σ 2 0 cos θ0 2 e i ϕ 0 2 sin θ0 2 e -i ϕ 0 2 (30)
corresponding to a pure state. The variable y is treated in a classical way with y = v 0 t. In (30), θ 0 and ϕ 0 characterize the initial orientation of the spin. This initial orientation being randomized, on may suppose that θ 0 is drawn in a uniform law from [0, π] and that ϕ 0 is drawn in a uniform law from [0, 2π]; In this way, we obtain a beam of atoms in which each atom has a different spinor: this is a model of a mixture of pure states.

In the Copenhagen interpretation, it is not necessary to resolve the Pauli equation. Just applying the postulates of quantum-mechanics measurement is sufficient. For measurement of the spin along the z-axis, the postulate of quantification states that the measurement corresponds to an eigenvalue of the spin operator S z = 2 σ z , and the spectral decomposition postulate states that equation [START_REF] Dürr | Quantum equilibrium and the role of operators as observables in quantum theory[END_REF] gives probability cos 2 θ 2 (resp. sin 2 θ 2 ) to measure the particle in the spin state + 2 (resp.-2 ).

In the de Broglie-Bohm interpretation, the postulates of quantum mechanics measurement are not used, but demonstrated (see below). The results of the measurement are obtained, first by calculating the evolution of the wave function in interaction with the measuring apparatus with the Pauli equation (equation( 1)), secondly by using the calculation of the wave function in space and time to pilot the particle (equation ( 2)).

Let us consider the evolution of the initial wave function ( 5) in the Stern-Gerlach apparatus. To obtain an explicit solution to the Stern-Gerlach experiment, we take the numerical values used in the Cohen-Tannoudji textbook [START_REF] Cohen-Tannoudji | Quantum Mechanics[END_REF]. For a silver atom, we have m = 1, 8×10 -25 kg, v 0 = 500 m/s , σ 0 =10 -4 m. For the electromagnetic field B, B x = B 0 x; B y = 0 and B z = B 0 -B 0 z with B 0 = 5 T esla, B 0 = ∂B ∂z = -∂B ∂x = 10 3 T esla/m over a length ∆l = 1 cm. The variable y will be treated classically with y = v 0 t. The particle stays within the magnetic field for a time ∆t = ∆l v0 = 2 × 10 -5 s. On exiting the magnetic field, the particle is free until it reaches screen P placed at a D = 20 cm distance.

During this time [0, ∆t], the spinor is calculated (Dürr and al. [START_REF] Dürr | Quantum equilibrium and the role of operators as observables in quantum theory[END_REF], Gondran [START_REF] Gondran | A complete analysis of the Stern-Gerlach experiment using Pauli spinors[END_REF]) with the Pauli equation [START_REF] Norsen | The theory of (Exclusively) local Beables[END_REF], where µ = e 2me is the Bohr magneton:

Ψ(x, z, t)     cos θ0 2 (2πσ 2 0 ) -1 2 e - (z- µ B B 0 2m t 2 ) 2 +x 2 4σ 2 0 e i µ B B 0 tz-µ 2 0 B 2 0 6m t 3 +µ B B 0 t+ ϕ 0 2 i sin θ0 2 (2πσ 2 0 ) -1 2 e - (z+ µ B B 0 2m t 2 ) 2 +x 2 4σ 2 0 e i -µ B B 0 tz-µ 2 0 B 2 0 6m t 3 -µ B B 0 t- ϕ 0 2     (31)
After the magnetic field, at time t + ∆t (t ≥ 0), in the free space, the spinor becomes [START_REF] Gondran | A complete analysis of the Stern-Gerlach experiment using Pauli spinors[END_REF] Ψ(x, z, t + ∆t)

   cos θ0 2 (2πσ 2 0 ) -1 2 e - (z-z ∆ -ut) 2 +x 2 2σ 2 0 e i muz+ ϕ + i sin θ0 2 (2πσ 2 0 ) -1 2 e - (z+z ∆ +ut) 2 +x 2 2σ 2 0 e i -muz+ ϕ -    (32)
where

z ∆ = µ B B 0 (∆t) 2 2m = 10 -5 m, u = µ B B 0 (∆t) m = 1m/s. (33)
Equation (32) takes into account the spatial extension of the spinor and we note that the two-spinor components have very different values.

Since we have a mixture of pure states, the atomic density ρ(z, t + ∆t) is found by integrating ρ(x, z, t + ∆t) on x and on (θ 0 , ϕ 0 ): ρ(z, t + ∆t) = (2πσ 2 0 ) - The decoherence time t D , where the beam is separated into the two spots N + and N -( when z ∆ + ut D 3σ 0 ), is then given by the equation:

t D 3σ 0 -z ∆ u = 3 × 10 -4 s. ( 35
)
We then obtain atoms with spins oriented only along the z-axis (positively or negatively). Experimentally, we do not measure the spin directly, but position ( x, z) of the particle impact on P . If z ∈ N + , the term ψ -of (32) is numerically equal to zero, and the spinor Ψ is proportional to 1 0 , one of the eigenvectors of σ z :

Ψ(z, t + ∆t) (2πσ e i muz+ ϕ + 1 0 .

If z ∈ N -, the term ψ + of (32) is numerically equal to zero and the spinor Ψ is proportional to 0 1 , the other eigenvector of σ z : Ψ(z, t + ∆t) (2πσ e i -muz+ ϕ - 0 1 .

Therefore, the measurement of the spin corresponds to an eigenvalue of the spin operator S z = 2 σ z . It is a proof of the postulate of quantization in Bohmian mechanics. Equation (32) gives the probability cos 2 θ0 2 (resp.sin 2 θ0 2 ) of measuring the particle in the spin state + 2 (resp.-2 ). It is a proof of the spatial decomposition postulate in Bohmian mechanics.

Fig. 5 presents in x0y a set of 10 silver-atom trajectories of which initial characteristics (θ 0 ,ϕ 0 ,z 0 ) have been randomly chosen: θ 0 and ϕ 0 , which define on one hand the wave function, have uniform distributions, and z A 0 , which on the other hand defines the particle position in the wave function, has a normal distribution N (0, σ 0 ). This double representation of quantum particles allows to take into account a mixture of pure states which satisfies the density of (34). Spin orientation θ(z(t), t) is represented by arrows.

We can see that the final orientation, obtained after the decoherence time t D , will depend on the initial particle position z A 0 in the wave packet and on the initial angle θ A 0 of the atom magnetic moment with the z axis. Finally, we can also give a clear explanation of the Albert's example on contextuality [START_REF] Albert | Quantum mechanics and experience[END_REF]

Figure 1 :Figure 2

 12 Figure 1: Ten silver atom trajectories with the same initial spinor orientation (θ 0 = π 3 ) and with 10 different initial positions z 0 ; arrows represent the spin orientation θ(z(t), t) along the trajectories.

Figure 2 :

 2 Figure 2: Schematic configuration of the EPR-B experiment.

  Particle A goes to the right and cross a Stern-Gerlach device. Particle B goes to the left without crossing any device. The spin orientation (θ) is indicated by an arrow: the arrows on the right (respectively on the left) represent the evolution of the spin orientation of A (respectively of B).

Figure 3 :

 3 Figure 3: Evolution of the trajectories and the spin orientations (arrow) of three pairs of entangled A-B atoms (represented in black, gray and white) in the first step of the EPR-B experiment. There are created at the initial moment in y = 0 (y A 0 = y B 0 = 0). z A 0 and z B 0 have been randomly chosen. θ A 0 is also randomly chosen, θ B 0 is in opposition to θ A 0 , i.e.: θ B 0 = π -θ A 0 . Particle A goes to the right and cross a Stern-Gerlach device. Particle B goes to the left without crossing any device. The spin orientation along a trajectory (θ) is indicated by an arrow.

Figure 4 :

 4 Figure 4: Schematic configuration of a Stern-Gerlach apparatus.

Figure 5 :

 5 Figure 5: 10 silver atom trajectories after the electro-magnet where the initial characteristics (θ 0 ,ϕ 0 ,z 0 ) have been randomly chosen; Arrows represent the spin orientation θ(z(t), t).

  (p. 153-155). He considers a Stern-Gerlach experiment where the initial wave function is a pure state with symmetric spin orientation(θ 0 = π 4and so z θ0 = 0). He changes, in a second experiment, the orientation of the magnetic field inside the Stern-Gerlach apparatus (B into -B). For the same initial position of atoms ( for example z 0 > 0), the first experiment gives a spin + and, in the second, a spin -. Bohmian mechanics explains this result: u and z ∆ are proportional to B 0 = ∂B z ∂z (equation (33)), and therefore change their signs like B. By solving the Pauli equation, Bohmian mechanics is naturally contextual (involves the measuring device).

Acknowledgement

Authors gratefully acknowledge the two anonymous reviewers for their useful comments and constructive suggestions which helped to improve the paper.


Federico Pellarin
email: federico.pellarin@univ-st-etienne.fr

A NOTE ON MULTIPLE ZETA VALUES IN TATE ALGEBRAS

Keywords: Mathematics Subject Classification. 11M38 (primary). Key words and phrases. Multiple zeta values, Carlitz module, A-harmonic sums

In this note, we shall discuss a generalization of Thakur's multiple zeta values and allied objects, in the framework of function fields of positive characteristic and more precisely, of periods in Tate algebras.

Introduction

Let A = F q [θ] be the ring of polynomials in an indeterminate θ with coefficients in F q the finite field with q elements and characteristic p, let K be the fraction field of A and K ∞ the completion of K at the infinity place ∞. For d ≥ 0 an integer, we denote by A + (d) the set of monic polynomials of A of degree d. Carlitz studied, in [START_REF] Carlitz | On certain functions connected with polynomials in a Galois field[END_REF], the so-called Carlitz zeta values:

ζ A (n) := a∈A + a -n ∈ K ∞ , n ≥ 1.
It is likely that the formal analogy of these objects with the classical zeta values

ζ(n) := i≥1 i -n ∈ R
with n integer (convergence occurs only if n ≥ 2) was the main motivation for his study (so that, in some way, "ζ = ζ Z "). In a more modern approach, we can say that Carlitz suggested, with his first pioneering papers, to develop an arithmetic theory of periods over the ring F q [θ] in parallel with the study of the arithmetic theory of periods over Z.

In all the following, if R is a ring, R × denotes the group of the multiplicative invertible elements of R. It was proved by Carlitz in [START_REF] Carlitz | On certain functions connected with polynomials in a Galois field[END_REF] that, if n ≡ 0 (mod q -1), [START_REF] Akiyama | On analytic continuation of multiple L-functions and related zeta functions[END_REF] ζ

A (n) ∈ K × π n ,
where π is the value in C ∞ = K ac ∞ ( 1 ) of a convergent infinite product

(2)

π := -(-θ) q q-1 ∞ i=1 (1 -θ 1-q i ) -1 ∈ (-θ) 1 q-1 K ∞ ,
uniquely defined up to the multiplication by an element of F × q = F q \ {0} (corresponding to the choice of a root (-θ) 1 q-1 ). It has been proved in a variety of ways (see [START_REF] Pellarin | Aspects de l'indépendance algébrique en caractéristique non nulle[END_REF] to see the most relevant ones) that π is moreover transcendental over K.

The element π is a fundamental period of the Carlitz exponential exp C (Goss,[START_REF] Goss | Basic Structures of Function Field Arithmetic[END_REF]§3.2]), that is, the unique surjective, entire, F q -linear function

exp C : C ∞ → C ∞ of kernel πF q [θ
] such that its first derivative satisfies exp ′ C = 1 (note that, since we are in a characteristic p > 0 environment, a function with constant derivative is not necessarily C ∞ -linear).

In his book [26, §5.10], Thakur also consider several variants of classical multiple zeta values in the context of the Carlitzian arithmetic over the ring A. We mention here what we think is the most relevant. For n 1 , . . . , n r ∈ Z ≥1 , Thakur defines, as one of the analogues of the classical multiple zeta values in the Carlitzian setting: [START_REF] Anglès | Arithmetic of "units[END_REF] ζ A (n 1 , . . . , n r ) =

a i ∈A + |a 1 |>•••>|ar | 1 a n1 1 • • • a nr r ∈ K ∞ .
Here, for x ∈ C × ∞ , we write |x| = q -v∞(x) where v ∞ is the valuation of C ∞ (so that v ∞ (θ) = -1) and we define |0| := 0. If r = 0 we further set the corresponding Thakur multiple zeta value ζ A (∅) to be equal to 1.

Classically, one of the reasons we could get interested in multiple zeta values is the need of "enveloping" zeta values in the "simplest" Q-algebra possible. From Euler, it is well known that the zeta values ζ(2), ζ(4), . . . all belong to the Q-algebra Q[ζ [START_REF] Anderson | Tensor powers of the Carlitz module and zeta values[END_REF]]. In general, the other zeta values are not expected to belong to this algebra. However, they belong to the Q-algebra Z R ⊂ R generated by the multiple zeta values. It is known that the product of two multiple zeta values is a Q-linear combination of multiple zeta values, and this algebra also has a more natural structure.

We expect that Z R is isomorphic to the algebra Q f 3 , f 5 , . . .

X ⊗ Q Q[ζ(2)],
where Q f 3 , f 5 , . . . X is the Q-algebra generated by the non-commutative words in the alphabet with letters f 3 , f 5 . . . with, as a product, the shuffle product X (see Brown's [START_REF] Brown | Mixed Tate motives over Z[END_REF]). A folklore conjecture comes in support of this guess; the number π and the zeta values ζ(3), ζ(5), . . . are expected to be algebraically independent over Q. Multiple zeta values are thus expected to provide a natural basis of this Qalgebra. See also [START_REF] Kaneko | Finite multiple zeta values[END_REF] for the definition of a Q-algebra of finite multi-zeta values which could offer a nice realization of the algebra Q f 3 , f 5 , . . . X .

Similarly, in the Carlitzian setting we note that, after [START_REF] Akiyama | On analytic continuation of multiple L-functions and related zeta functions[END_REF], the values ζ A (n) with n > 0 divisible by q -1 are all contained in the K-algebra K[ζ A (q -1)], which is isomorphic to K[X] for an indeterminate X. However, the remaining Carlitz zeta values ζ A (1), . . . do not belong to this algebra (if q > 2). Indeed, Chang and Yu proved in [START_REF] Chang | Determination of algebraic relations among special zeta values in positive characteristic[END_REF] that π and the Carlitz zeta values ζ A (n) with n ≥ 1, q -1 ∤ n and p ∤ n with p the prime number dividing q are algebraically independent (we recall that these authors, in ibid., use the powerful algebraic independence methods introduced by Papanikolas in [START_REF] Papanikolas | Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms[END_REF]); see also [START_REF] Chang | Linear independence of monomials of multizeta values in positive characteristic[END_REF][START_REF] Chang | Linear relations among double zeta values in positive characteristic[END_REF][START_REF] Chang | An effective criterion for Eulerian multizeta values in positive characteristic[END_REF].

Just as for the algebra Z R , Thakur proved in [START_REF] Thakur | Shuffle Relations for Function Field Multizeta Values[END_REF] that the product of two multiple zeta values as in ( 3) is a linear combination (this time with coefficients in F p ) of such multiple zeta values. Thakur also mentioned to the author of the present note that G. Todd's numerical computations have led to a good understanding of (conjectural) relations among Thakur's multiple zeta values; the relations are universal in a sense that is described in ibid. Compared to the classical setting, the difficulty here is to handle the product of the so-called power sums (see later). We denote by Z K∞ the K-sub-algebra of K ∞ generated by the multiple zeta values (3); even conjecturally, in spite of the striking results of algebraic independence mentioned above, we know very little about the structure of this algebra. In particular, we presently do not know what could be the analogue structure which could play the role of the algebra Q f 3 , f 5 , . . . X in this setting.

In this note, we shall discuss of a generalization of the Thakur multiple zeta values which, so far, has no counterpart in the classical setting. For this purpose, we note that A is an algebra over F q (Z is not an algebra over a field). Therefore, a series of advantages occurs in the Carlitzian framework, notably the possibility to use the tensor product over F q . We consider variables t 1 , . . . , t s over K and we write t s for the family of variables (t 1 , . . . , t s ). We denote by F s the field F q (t s ), so that

F 0 = F q .
In all the following, if R is a ring, we denote by R * the underlying multiplicative monoid (inclusive of the element 0). Note that A + is a multiplicative sub-monoid of A * . We denote by F ac q the algebraic closure of F q in C ∞ . Definition 1. A monoid homomorphism σ : A + → F ac q (t) * is called a semi-character. The trivial semi-character is the map 1 : A + → {1}. Let σ be a semi-character. We say that it is of Dirichlet type if there exist F q -algebra homomorphisms

ρ i : A → F ac q (t), i = 1, . . . , s, such that σ(a) = ρ 1 (a) • • • ρ s (a)
for all a ∈ A + . The integer s is called the length. By convention, the semi-character 1 is the unique semi-character of Dirichlet type of length 0.

For example, setting t = t 1 , the map χ t :

A + → F q [t] * ⊂ F ac q (t) * defined by χ t (a) = a(t) ( 2 ) is a semi-character of Dirichlet type. Let ζ be an element of F ac q . The map a ∈ A + → χ ζ (a) = a(ζ) ∈ F ac
q is also a semi-character of Dirichlet type, and the same can be said if we now pick elements ζ 1 , . . . , ζ s ∈ F ac q and consider the map a) is a semi-character, but it can be proved that it is not of Dirichlet type. Definition 2. Let σ : A + → F ac q (t) * be a semi-character. The associated twisted power sum of order k and degree d is the sum:

χ ζ : a → χ ζ1 (a) • • • χ ζs (a) (this is more commonly called a "Dirichlet character"). The map a ∈ A + → F q [t] * which sends a to t deg θ (
S d (k; σ) = a∈A + (d) a -k σ(a) ∈ F ac q (θ)(t s ).
More generally, let σ 1 , . . . , σ r be semi-characters, let n 1 , . . . , n r be integers, and d a non-negative integer. The associated multiple twisted power sum of degree d is the sum

S d σ 1 σ 2 • • • σ r n 1 n 2 • • • n r = S d (n 1 ; σ 1 ) d>i2>•••>ir ≥0 S i2 (n 2 ; σ 2 ) • • • S ir (n r ; σ r ) ∈ F ac q (θ)(t s ).
The integer i n i is called the weight and the integer r is called its depth.

We can write in both ways S d (n; σ) = S d σ n . Observe also that

S d 1 1 • • • 1 n 1 n 2 • • • n r = S d (n 1 , n 2 , . . . , n r ) ∈ K,
in the notations of Thakur, [28, §1.2]. We hope that all these slightly different notations will not bother the reader.

Definition 3. With n 1 , . . . , n r ≥ 1 and semi-characters σ 1 , . . . , σ r as above, we introduce the associated multiple zeta value

ζ A σ 1 σ 2 • • • σ r n 1 n 2 • • • n r := d≥0 S d σ 1 σ 2 • • • σ r n 1 n 2 • • • n r ∈ F ac q (t)((θ -1 )).
The sum thus converges with respect to the unique valuation extending the ∞-adic valuation of K ∞ and inducing the trivial valuation over F ac q (t). Explicitly, we have:

ζ A σ 1 σ 2 • • • σ r n 1 n 2 • • • n r = d≥0 a 1 ,...,ar ∈A + d=deg θ (a 1 )>•••>deg θ (ar )≥0 σ 1 (a 1 ) • • • σ r (a r ) a n1 1 • • • a nr r .
We will say that this is the multiple zeta value associated to the composition array

σ 1 σ 2 • • • σ r n 1 n 2 • • • n r .
The integer i n i is called the weight of the above multiple zeta value and the integer r is called its depth. If all the semi-characters σ 1 , . . . , σ r are of Dirichlet type, then, for all

1 ≤ i ≤ r, σ i = ρ i,1 • • • ρ i,ni
for ring homomorphisms ρ i,j . Then, we say that the multiple zeta value associated to the above matrix data is of Dirichlet type, and the cardinality of the set {ρ i,j ; i, j} is called its length.

The referee has pointed out that these multiple zeta values can be seen as some kind of analogues, for functional analysis in Tate algebras, of the Dirichlet multiple L-functions as in the paper of Akiyama and Ishikawa [START_REF] Akiyama | On analytic continuation of multiple L-functions and related zeta functions[END_REF].

Again note that if

σ 1 = • • • = σ r = 1, then, we can write ζ A σ 1 σ 2 • • • σ r n 1 n 2 • • • n r = ζ A (n 1 , . . . , n r ) ∈ K ∞
with ζ A (n 1 , . . . , n r ) as in (3) (of Dirichlet type, depth r and length 0). Further, let us assume that r = 1, n = n 1 > 0 and that σ = χ t1 • • • χ ts . Then, we have that

ζ A (n; σ) = ζ A σ n = a∈A + a(t 1 ) • • • a(t s ) a n = P 1 - P (t 1 ) • • • P (t s ) P n -1 ∈ T × s .
These series have been introduced in [START_REF] Pellarin | Values of certain L-series in positive characteristic[END_REF] and extensively studied in e.g. [START_REF] Anglès | Functional identities for L-series values in positive characteristic[END_REF][START_REF] Anglès | Universal Gauss-Thakur sums and L-series[END_REF][START_REF] Anglès | Arithmetic of positive characteristic L-series values in Tate algebras Preprint[END_REF]. The product runs over the irreducible polynomials P of A + and the convergence holds in the standard s-dimensional Tate algebra, which can be identified with the C ∞ -algebra of the rigid analytic functions B(0, 1) s → C ∞ , where B(0, 1) = {z ∈ C ∞ ; |z| ≤ 1}. In fact, these functions extend to entire functions [START_REF] Anglès | Functional identities for L-series values in positive characteristic[END_REF]Corollary 8]). More generally, we assume that for all i = 1, . . . , r, there exists a subset I i ⊂ {1, . . . , s} such that

C s ∞ → C ∞ (see
σ i = j∈Ii χ tj .
In particular, if I i = ∅, we set σ i = 1. In this setting, the multiple zeta value

(4) ζ A σ 1 σ 2 • • • σ r n 1 n 2 • • • n r belongs to T s .
Proposition 4. With the above assumption over the semi-characters σ 1 , . . . , σ r , the multiple zeta value (4), hence of Dirichlet type and of length ≤ s, extends to an entire function C s ∞ → C ∞ . Proof. Note that there exists a non-negative integer κ, depending of σ 1 , . . . , σ r , such that if we set

Σ d := S d σ 1 σ 2 • • • σ r n 1 n 2 • • • n r ∈ K[t 1 , . . . , t s ],
then the degree in t i of Σ d is ≤ κd for all i = 1, . . . , s.

If s > 0 and σ = χ t1 • • • χ ts then, by [START_REF] Anglès | Functional identities for L-series values in positive characteristic[END_REF]Lemma 7], we have the following estimate for the Gauss norm S d (n; σ) of S d (n; σ) associated with our valuation of F ac q (t)((θ -1 )) such that θ = q:

S d (n; σ) ≤ q -dn q -q [d-s q-1 ]-1 , d > s -1 q -1 ,
where [•] denotes the lower integer part (replace x = θ n and y = -n ∈ Z p with p | q in the statement of that Lemma). This implies that, if σ 1 is not the trivial semi-character,

Σ d ≤ S d (n; σ 1 ) ≤ q -dn1 q -q [d-µ q-1 ]-1 , d ≫ 0,
for some constant µ, which immediately implies the proposition. If σ 1 is the trivial semi-character we cannot directly apply the statement of the above lemma but the problem is superficial, as we can more easily conclude by using [START_REF] Goss | Basic Structures of Function Field Arithmetic[END_REF]Proposition 8.8.2] as, in this case:

Σ d ≤ S d (n 1 ; 1) ≤ q -(q-1) d(d+1) 2 .
Remark 5. A more general result of this type can now be found in [START_REF] Anglès | Twisted characteristic p zeta functions[END_REF].

We set F s := F q (t s ). It is presently a work in progress of the author to show that the product of two multiple zeta values as in Definition 3 is a linear combination, with coefficients in the field K ⊗ Fq F ac q ⊗ Fq F s , of such multiple zeta values (with the various matrices of associated data not including, necessarily, the same semi-characters). We hope this will allow us to exhibit new multiple zeta values algebras Z F ac q ((θ -1 ))⊗ Fq F s containing the algebra Z K∞ and collecting the algebraic relations of Z K∞ in families by specialization.

Content of the present note

Waiting for more general results, in this note we will accomplish a more modest objective, as we will only give a few explicit examples of shuffle products of such multiple zeta values in the following case: s = 2, weight ≤ 2, and the semi-characters 1, σ, ψ and σψ of Dirichlet type, where

σ : a → a(t 1 ) ∈ F q [t 1 , t 2 ], ψ : a → a(t 2 ) ∈ F q [t 1 , t 2 ], (σψ)(a) = σ(a)ψ(a) = a(t 1 )a(t 2 ).
As an advantage of our explicit and restrictive viewpoint, we will see beautiful formulas dropping out from this new theory that we will apply to some new properties of the so-called "Bernoulli-Goss" polynomials.

The matrix data we are going to handle are:

Four in weight 1. 1 1 , σ 1 , ψ 1 , σψ 1 .
Four in weight 2 depth 1.

1 2 , σ 2 , ψ 2 , σψ 2 .
Nine in weight 2 depth 2.

1 1 1 1 , σ 1 1 1 , 1 σ 1 1 , ψ 1 1 1 , 1 ψ 1 1 , σψ 1 1 1 , σ ψ 1 1 , ψ σ 1 1 , 1 σψ 1 1 .
We shall show the following Theorem, which provides, taking into account the above tables, a complete picture of all the products of two weight one multiple zeta values in the restrictive context we have prefixed (in two variables t 1 , t 2 , and with the semi-characters 1, σ, ψ and σψ), unveiling partly an extremely complex and mysterious algebra structure. From now on, we suppose that q > 2. All the arguments presented below under this restriction can be also developed in the case q = 2 with appropriate modifications, but we refrain from giving full details here. Theorem 6. The following formulas hold.

(1

) ζ A (1) 2 = ζ A (2) + 2ζ A (1, 1), (2) ζ A (1; σ)ζ A (1) = ζ A (2; σ) + ζ A σ 1 1 1 , (3) ζ A (1; ψ)ζ A (1) = ζ A (2; ψ) + ζ A ψ 1 1 1 , (4) ζ A (1; σ)ζ A (1; ψ) = ζ A (2; σψ), (5) ζ A (1; σψ)ζ A (1) = ζ A (2; σψ) -ζ A σ ψ 1 1 -ζ A ψ σ 1 1 + ζ A 1 σψ 1 1 + ζ A σψ 1 1 1 .
We observe that the formula (1) can be found in Thakur's [START_REF] Thakur | Function Field Arithmetic[END_REF]Theorem 5.10.13]. Further, due to the symmetry of the roles of t 1 and t 2 , the formulas (2) and (3) are equivalent. Moreover, the formula (4) is in fact well known (see Perkins' [START_REF] Perkins | Explicit formulae for L-values in positive characteristic[END_REF] for more general formulas of this type). However, we will give a proof of this in the spirit of multiple zeta values. The formulas (2) is, on the other side, new, as far as we can see.

Twisted power sums

We need a few tools in order to obtain our formulas; more precisely, we have to improve our skill in computing twisted powers sums. For this purpose, we are going to use the tools introduced in the recent preprint of Perkins and the author [START_REF] Pellarin | On twisted A-harmonic sums and Carlitz finite zeta values[END_REF]; we are going to use for a while the notations of this reference. Let s be an integer ≥ 0. We set, for an integer d ≥ 0:

S d (n; s) = a∈A + (d) a(t 1 ) • • • a(t s ) a n ∈ K[t s ].
We are thus considering a special case of Definition 2. We also set, for d ≥ 0, F 0 (n; s) = 0 and

F d (n; s) = d-1 i=0 S d (n; s) ∈ K[t s ],
so that

lim d→∞ F d (n; s) = ζ A (n; s) := P 1 - P (t 1 ) • • • P (t s ) P n -1 ∈ T × s ,
where the product runs over the irreducible polynomials of A + (in general, all along this note, empty sums are by convention equal to zero). We are using the notation of [START_REF] Pellarin | On twisted A-harmonic sums and Carlitz finite zeta values[END_REF]. In particular, if σ is the semi-character χ t1 • • • χ ts (of Dirichlet type), then, the comparison between the notations of ibid. and those of the present note are:

S d (n; s) = S d (n; σ), ζ A (n; s) = ζ A (n; σ). It is easy to show that, if 0 ≤ s ′ < s, S d (n; s ′ ) is the coefficient of (t s ′ +1 • • • t s ) d in F d+1 (n; s).
We define inductively l 0 = 1 and l i = (θθ q i )l i-1 , and we set l -n = 0 for n > 0. We denote by b

i (Y ) the product (Y -θ) • • • (Y -θ q i-1 ) ∈ A[Y ] (for an indeterminate Y ) if i > 0 and we set b 0 (Y ) = 1.
We also write m = ⌊ s-1 q-1 ⌋ (the brackets denote the integer part so that m is the biggest integer ≤ s-1 q-1 ). We set

Π s,d = b d-m (t 1 ) • • • b d-m (t s ) l d-1 ∈ K[t s ], d ≥ max{1, m}.

Now, we quote [23, Theorem 1]:

Theorem 7. For all integers s ≥ 1, such that s ≡ 1 (mod q -1), there exists a non-zero rational fraction H s ∈ K(Y, t s ) such that, for all d ≥ m, the following identity holds:

F d (1; s) = Π s,d H s | Y =θ q d-m .
If s = 1, we have the explicit formula

H 1 = 1 t 1 -θ .
Further, if s = 1 + m(q -1) for an integer m > 0, then the fraction H s is a polynomial of A[Y, t s ] with the following properties:

(1) For all i, deg ti (H s ) = m -1, (2) deg Y (H s ) = q m -1 q-1 -m.

The polynomial H s is uniquely determined by these properties.

We apply this Theorem to compute explicitly the twisted power sums associated to the data we have chosen. For this, it suffices to choose s = q. In this case m = 1 and the polynomial H q of Theorem 7 has degree 0 in Y as well as in t 1 , . . . , t q . From [23, §2.6] we deduce that H q = 1 (the same result is given as an example in Florent Demeslay's thesis [START_REF] Demeslay | Formule de classes en caractéristique positive[END_REF]). In particular, to compute most of the twisted power sums associated to our data it suffices to analyze the polynomials

(5) F d+1 (1; q) = b d (t 1 ) • • • b d (t q ) l d .
The coefficients of (t

3 • • • t q ) d , (t 2 • • • t q ) d and (t 1 • • • t q ) d in F d+1
(1; q) are easily computed, and we get, for all d ≥ 0 (note that these are well known formulas; see [START_REF] Perkins | Explicit formulae for L-values in positive characteristic[END_REF]):

S d (1; 0) = 1 l d , (6)
S d (1; 1) = b d (t 1 ) l d , (7)
S d (1; 2) = b d (t 1 )b d (t 2 ) l d . ( 8
)
To compute S d (2; 0), S d (2, 1), S d (2, 2) (3 ) we observe that, replacing θ with θ q in (5):

F d+1 (q; q) = b d+1 (t 1 ) • • • b d+1 (t q ) l q d (t 1 -θ) • • • (t q -θ) .
Note that the former is a polynomial in t s , written as a reducible fraction. We get that

F d+1 (2; 2) = b d+1 (t 1 ) • • • b d+1 (t q ) l q d (t 1 -θ) • • • (t q -θ) t3=•••=tq=θ = b d+1 (t 1 )b d+1 (t 2 ) l 2 d (t 1 -θ)(t 2 -θ)
.

Calculating the coefficients of t d 2 and (t 1 t 2 ) d , and subtracting F d+1 (2; 2) -F d (2; 2), we easily obtain the formulas, valid for d ≥ 0 (the first one is well known):

S d (2; 0) = 1 l 2 d , S d (2; 1) = b d (t 1 ) (t 1 -θ)l 2 d (t 1 -θ q d ), (9)
S d (2; 2) = b d (t 1 )b d (t 2 ) (t 1 -θ)(t 2 -θ)l 2 d (t 1 t 2 -θ q d (t 1 + t 2 ) + 2θ 1+q d -θ 2 ) = b d (t 1 )b d (t 2 ) (t 1 -θ)(t 2 -θ)l 2 d [(t 1 -θ)(t 2 -θ) + (t 1 -θ)(θ -θ q d ) + (t 2 -θ)(θ -θ q d )]. ( 10
)
We will also need the next Lemma which is also well known, where τ :

A[t] → A[t] is the F q [t]-linear endomorphism such that τ (θ) = θ q .
Lemma 8. We have the following formula, which holds in

A[t]. τ (b d (t)) = l d d i=0 b i (t) l i , d ≥ 0.
Proof. We recall the proof for convenience of the reader. We proceed by induction over d.

If d = 0, the formula is obvious. If d > 0, it suffices to show that b d+1 (t) = (t -θ)l d d i=0 l -1 i b i (t).
Now, we compute easily, by using the induction hypothesis:

b d+2 (t) = (t -θ + θ -θ q d )b d+1 = (θ -θ q d )b d+1 + (t -θ)b d+1 = (t -θ)l d (θ -θ q d ) d i=0 l -1 i b i (t) + l -1 d+1 b d+1 (t) = (t -θ)l d+1 d+1 i=0 l -1 i b i (t).

Proof of Theorem 6

Proof of Theorem 6,[START_REF] Anderson | Tensor powers of the Carlitz module and zeta values[END_REF]. We compute, by using ( 6) and ( 7):

S d (1; σ)S d (1; 1) = b d (t 1 ) l 2 d .
On the other hand, we have seen in ( 9) that

S d (2; σ) = b d (t 1 ) l 2 d t 1 -θ q d t 1 -θ = τ (b d (t 1 )) l 2 d ,
where τ (b d (t 1 )) has the obvious meaning. We compute (the third identity follows from Lemma 8):

S d 1 σ 1 1 = S d (1; 1) d-1 i=0 S i (1; σ) = 1 l d d-1 i=0 l -1 i b i (t 1 ) = τ (b d-1 (t 1 )) l d l d-1 = b d (t 1 ) l 2 d θ -θ q d t 1 -θ .
Combining the above formulas, we see that

(11) S d (1; σ)S d (1; 1) = S d (2; σ) -S d 1 σ 1 1 .
With the obvious meaning of some new notations introduced below, we deduce:

F d (1; σ)F d (1; 1) = d-1 i=0 S i (1; σ) d-1 j=0 S j (1; 1) = F d 1 σ 1 1 + F d σ 1 1 1 + d-1 i=0 S d (1; σ)S d (1; 1) = F d 1 σ 1 1 + F d σ 1 1 1 + d-1 i=0 S i (2; σ) -S i 1 σ 1 1 = F d 1 σ 1 1 + F d σ 1 1 1 -F d 1 σ 1 1 + F d (2; σ) = F d σ 1 1 1 + F d (2; σ),
where we have used [START_REF] Carlitz | On certain functions connected with polynomials in a Galois field[END_REF] in the third equality. We rewrite the resulting formula:

(12) F d (1; σ)F d (1; 1) = F d σ 1 1 1 + F d (2; σ).
Taking the limit d → ∞ in ( 12) we obtain the required multiple zeta identity.

The above also implies the formula (3) of Theorem 6 by interchanging t 1 and t 2 . To continue, we notice the next Lemma.

Lemma 9. We have that

S d (2; σψ) = b d (t 1 )b d (t s ) (t 1 -θ)(t 2 -θ)l 2 d + S d ψ σ 1 1 + S d σ ψ 1 1 .
Proof. We compute:

S d σ ψ 1 1 = S d (1; σ) d-1 i=0 S i (1; ψ) = b d (t 1 ) l d d-1 i=0 b i (t 2 ) l i = b d (t 1 )b d (t s ) (t 1 -θ)(t 2 -θ)l 2 d [(t 1 -θ)(θ -θ q d )],
in virtue of [START_REF] Anglès | Arithmetic of positive characteristic L-series values in Tate algebras Preprint[END_REF] and Lemma 8. Similarly, we have

S d ψ σ 1 1 = b d (t 1 )b d (t s ) (t 1 -θ)(t 2 -θ)l 2 d [(t 2 -θ)(θ -θ q d )].
The lemma follows from [START_REF] Brown | Mixed Tate motives over Z[END_REF].

Proof of Theorem 6, (4). As we have already mentioned, this is a well known formula but we want to provide a new proof. We note, by [START_REF] Anglès | Arithmetic of positive characteristic L-series values in Tate algebras Preprint[END_REF], that

S d (1; σ)S d (1; ψ) = b d (t 1 )b d (t 2 ) (t 1 -θ)(t 2 -θ)l 2 d [(t 1 -θ)(t 2 -θ)].
Hence, Lemma 9 implies the formula

(13) S d (1; σ)S d (1; ψ) = S d (2; σψ) -S d ψ σ 1 1 -S d σ ψ 1 1 .
We deduce:

F d (1; σ)F d (1; ψ) = F d ψ σ 1 1 + F d σ ψ 1 1 + d-1 i=0 S d (1; σ)S d (1; ψ) = F d ψ σ 1 1 + F d σ ψ 1 1 + d-1 i=0 S d (2; σψ) -S d ψ σ 1 1 -S d σ ψ 1 1 = F d ψ σ 1 1 -F d ψ σ 1 1 + F d σ ψ 1 1 -F d σ ψ 1 1 + F d (2; σψ) = F d (2; σψ),
where we have applied the formula [START_REF] Chang | Linear relations among double zeta values in positive characteristic[END_REF] in the third equality. The formula of the theorem follows by letting d → ∞.

Proof of Theorem 6,[START_REF] Anglès | Functional identities for L-series values in positive characteristic[END_REF]. The identities (6) (8) imply that

S d (1; 1)S d (1; σψ) = b d (t 1 )b d (t 2 ) (t 1 -θ)(t 2 -θ)l 2 d [(t 1 -θ)(t 2 -θ)].
Lemma 9 then implies that also:

(14) S d (1; 1)S d (1; σψ) = S d (2; σψ) -S d ψ σ 1 1 -S d σ ψ 1 1 .
We deduce:

F d (1; 1)F d (1; σψ) = F d 1 σψ 1 1 + F d σψ 1 1 1 + d-1 i=0 S d (1; σ)S d (1; ψ) = F d 1 σψ 1 1 + F d σψ 1 1 1 + d-1 i=0 S d (2; σψ) -S d ψ σ 1 1 -S d σ ψ 1 1 = F d (2; σψ) + F d 1 σψ 1 1 + F d σψ 1 1 1 -F d ψ σ 1 1 -F d σ ψ 1 1 , so we have reached the formula (15) F d (1; 1)F d (1; σψ) = F d (2; σψ) + F d 1 σψ 1 1 + F d σψ 1 1 1 -F d ψ σ 1 1 -F d σ ψ 1 1 .
Letting d tend to ∞ in (15) concludes the proof.

Remark 10. In fact, it is trivial that

ζ A (1)ζ A (1; σψ) -ζ A (1; σ)ζ A (1, ψ) = ζ d 1 σψ 1 1 + ζ d σψ 1 1 1 -ζ d ψ σ 1 1 -ζ d σ ψ 1 1 ,
and that the corresponding identity for the sums F d holds as well. Indeed, setting

α i = S i (1; 1) = l -1 i , β i = S i (1; σ) = b i (t 1 )l -1 i , γ i = S i (1; ψ) = b i (t 2 )l -1 i and δ i = S i (1; σψ) = b i (t 1 )b i (t 2 )l -1 i , we see that i≥0 α i j≥0 δ j - i≥0 β i j≥0 γ j = = i>j≥0 α i δ j + j>i≥0 α i δ j - i>j≥0 β i γ j - j>i≥0 β i γ j + + i≥0 α i δ i - i≥0 β i γ i .
But, of course, α i δ i = β i γ i for all i, from which the identity follows. Up to this simple trick, the identity (5) of Theorem 6 can be considered as equivalent to the identity (4) of the same result.

Some consequences

In virtue of Proposition 4 or by direct verification, the identities of Theorem 6 involve entire functions in two variables t 1 , t 2 . Hence, specializing the variables, we are able to recover identities in C ∞ or in some intermediate Tate algebra. We are going to show several results arising from the formula (2). We recall the formula (2) of Theorem 6 for convenience:

(16) ζ A (1; σ)ζ A (1) = ζ A σ 1 1 1 + ζ A (2; σ).
First of all, we can replace t 1 by θ, but this does not give anything interesting; we mention it only to show how the substitution works. We recall the following formula that the author proved in [START_REF] Pellarin | Values of certain L-series in positive characteristic[END_REF]:

(17) ζ A (1; σ) = π (θ -t 1 )ω(t 1 )
,

where ω(t 1 ) = (-θ)

1 q-1 i≥0 1 - t 1 θ q i -1 ∈ T × 1 ,
is the Anderson-Thakur function (note that Ω(t 1 ) = 1 (t1-θ)ω(t1) is an entire function; see [START_REF] Anglès | Universal Gauss-Thakur sums and L-series[END_REF], containing a recent overview on the known properties of this function). The function ω(t 1 ) having a simple pole of residueπ at t 1 = θ, we see that ζ A (1; σ)| t1=θ = 1. Now, it is easy to see that [START_REF] Goss | Basic Structures of Function Field Arithmetic[END_REF] ζ

A 1 σ 1 1 = d≥0 S d (1; σ) d-1 i=0 l -1 i = d≥0 l -1 d b d (t 1 ) d-1 i=0 l -1 i vanishes at t 1 = θ. Further, ζ A (2; σ) = d≥0 S d (2; σ)
takes the value ζ A (1) at t 1 = θ. Hence, with this evaluation, we only get the tautological identity ζ A (1) = ζ A (1).

5.1.

A family of multiple zeta identitities. We can also evaluate this identity at t 1 = θ q -k with k > 0 and raise the obtained identity to the power q k . Working out the intermediate details, the reader will easily recover the following sum shuffle formula:

ζ A (q k )ζ A (q k -1) = ζ A (2q k -1) + ζ A (q k -1, q k ), k ≥ 1.

5.2.

Evaluation at trivial zeros. Now, we evaluate the second identity of Theorem 6 at t 1 equal to a trivial zero of the function ζ A (1; σ) which, as it appears from the computation of the poles of the gamma factor of ( 17), means that we replace t 1 with θ q d with d > 0. This implies the following result.

Theorem 11. The following formula holds

(19) BG q d -2 = - d≥i>j≥0 b i (θ q d ) l i l j , d ≥ 1.
Proof. In fact, to make things more transparent, we make a step back to the identity (12) that we rewrite as

F k (2; σ) = F k (1; σ)F k (1; 1) -F k σ 1 1 1 , k ≥ 0.
By Proposition 4, we see that all the sequences F k (• • • ) involved tend, for k → ∞, to entire functions of the variable t 1 . However, we already know from [5, Proposition 6] that F k (2; σ) and F k (1; σ)F k (1; 1) tend to entire functions, so we immediately obtain that F k ( σ 1 1 1 ) tends to an entire function as k → ∞ without using Proposition 4 (in fact, this can be also seen directly). Replacing t 1 = θ q d with d > 0 yields the value zero for the limit

lim k→∞ F k (1; σ)F k (1; 1) = ζ A (1; σ)ζ A (1)
evaluated at t 1 = θ q d . Indeed, after [START_REF] Goss | L-series of t-motives and Drinfeld Modules[END_REF], θ q d is a trivial zero of ζ A (1; σ). Further, we see that

ζ A (2; σ)| t1=θ q d = lim k→∞ F k (2; σ)| t1=θ q d = k≥0 a∈A + (k) a q d -2 = BG q d -2 ∈ A.
Moreover, by ( 18) and evaluating at

t 1 = θ q d , ζ A σ 1 1 1 t1=θ q d = lim k→∞ F k σ 1 1 1 t1=θ q d = i>j≥0 b i (θ q d ) l i l j = d≥i>j≥0 b i (θ q d ) l i l j ,
because b i (θ q d ) vanishes for all i > d.

One nice aspect of the formula [START_REF] Kaneko | Finite multiple zeta values[END_REF] is that it is easy to reduce it modulo an irreducible polynomial of A of degree d. The following family of congruences is an immediate consequence of our result, and was first observed by Thakur in [START_REF] Thakur | Iwasawa theory and cyclotomic function fields[END_REF], and Anglès and Ould Douh in [START_REF] Anglès | Arithmetic of "units[END_REF]: Corollary 12. For all P an irreducible polynomial of A + (d), we have (recall that |P | = q d ):

BG |P |-2 ≡ d-1 i=0 1 l i ≡ F d (1; 1) (mod P ).
Proof. For all i, j with d ≥ i > j ≥ 0, the fraction b i (θ q d )(l i l j ) -1 is P -integral for any P irreducible of degree d.

If i < d, we have that b i (θ q d )l -1 i ≡ 0 (mod P ), because b i (θ q d ) is divisible, in A, by θ q d -θ. Further, b d (θ q d )/l d ≡ -1,
from which the congruence follows.

Remark 13. The reader can do similar computations with other formulas; more results will appear elsewhere. Observe, however, that manipulating in the same way the formula (5) of Theorem 6 returns relatively less identities. The reason seems to be that this formula is trivially equivalent to the formula (4), as it follows from Remark 10. At least, we deduce, specializing t 1 = θ q -k and t 2 = θ q -h , the following strange sum shuffle identity, valid for h, k ≥ 0 with h + k > 0:

ζ A (1) q k ζ A (q k+h -q h -1) = = ζ A (2q h+k -q h -1) + ζ A (q k+h , q k+h -q h -1) + ζ A (q k+h -q h -1, q k+h ) - -ζ A (q k+h -q h , q k+h -1) -ζ A (q k+h -1, q k+h -q h ).
The formula (2) of Theorem 6 and especially the formula of Theorem 19 can be seen as some kind of analogue of Euler's formula ζ(3) = ζ(2, 1) (for classical Euler multiple zeta values).

A degree computation.

In contrast with the universal formulas obtained in [START_REF] Pellarin | On twisted A-harmonic sums and Carlitz finite zeta values[END_REF] for the sums F d (n; s) in the case s ≡ n (mod q -1), there seems to be no such a formula for BG q d -2 , for d ≥ 1.

At least, we have a "universal formula" for its degree, and this can be deduced from [START_REF] Kaneko | Finite multiple zeta values[END_REF] as we are going to see in the next result, where it is supposed, again (as we did until now), that q > 2.

Theorem 14. We have

deg θ (BG q d -2 ) = (d -1)q d - 2q(q d-1 -1) q -1 .
This result should be compared with more classical degree computations by Wan, Diaz-Vargas, Poonen, Sheats, as well as Böckle's [START_REF] Böckle | The distribution of the zeros of the Goss zeta-function for A = F 2 [x, y]/(y 2 + y + x 3 + x + 1)[END_REF]Theorem 1.2] where the interested reader can find all the necessary references to the work of these authors on this topic.

Theorem 14 seems to be new. Thomas [START_REF] Thomas | On the zeta function for function fields over Fp[END_REF]Theorem 2] already proved an explicit formula to compute, not only the degree of BG q d -2 for d > 0 but also the degree of BG n for any n > 0 with q -1 ∤ n in case q is a prime number ( 4). It follows from Thomas' Theorem 1 and Corollary 1 ibid. that if q -1 ∤ n, and q is a prime, then BG n = 1 if and only if ℓ q (n) < q, where ℓ q (n) is the sum of the digits of the base-q expansion of n. However, Thomas formula contains an iterative process and for this reason, the identity of Theorem 14 is not immediately recognizable in it, and even if it was, it would have been valid only for q a prime number. Bruno Anglès has communicated to the author a simple proof of Theorem 14 in the case of q = p > 2 a prime number by using Sheats' method. Also, Dinesh Thakur has pointed out to the author that this result, for general q, can be more simply deduced from an application of his duality formula [START_REF] Thakur | Power sums of polynomials over finite fields and applications: A survey[END_REF]Theorem 2,[START_REF] Anglès | Functional identities for L-series values in positive characteristic[END_REF]].

Remark 15. Note that for all d ≥ 1, q d -2 is a dual magic number in the sense of [17, §5.7] (see also [START_REF] Goss | Basic Structures of Function Field Arithmetic[END_REF]). In this paper, Goss points out a result of Thomas which exhibits a computation of the degree of BG n when q -1 ∤ n and when n is a magic number ([18, §8.22]), in terms of the the degree of the Carlitz factorial. Our computation involves certain dual magic numbers which are not magic numbers, and this could also be a new instance of the conjectural functional equation for the Goss zeta function associated to the algebra A.

More results of the type of Theorem 14 can be obtained from more general consequences of the sum shuffle relations for our multi-zeta values in the Tate algebras, but they will be described in another work (the present paper can be considered as a first of more general results that will appear elsewhere). Before proving the Theorem, we need some notation and Lemmas. For commodity, we set

α i = b i (θ q d ) l i , β j = l -1 j ,
4 He obtained a more general result in this direction, also involving "first derivatives" of the Goss zeta function of A at its "trivial zeroes," the negative integers divisible by q -1.

so that the formula (19) rewrites as

BG q d -2 = - d≥i>j≥0 α i β j .
Then, we have

(20) δ i,j := deg θ (α i β j ) = iq d - i n=1 q n - j m=1 q m .
We recall the convention that an empty sum is zero. Moreover, the degree of 0 in θ is set to be -∞. We have the following Lemma.

Lemma 16. Assuming that d ≥ i > j ≥ 0, d ≥ i ′ > j ′ ≥ 0, we have that δ i,j = deg θ (α i β j ) = deg θ (α i ′ β j ′ ) = δ i ′ ,j ′
if and only if the following cases occur.

(

1) i = i ′ , j = j ′ , (2) i = d, i ′ = d -1 and j = j ′ , (3) i ′ = d, i = d -1 and j = j ′ .
Proof. For symmetry of the roles of i, i ′ , we can assume that i ≥ i ′ . First of all, if i = i ′ we have that

δ i,j -δ i ′ ,j ′ = j ′ m ′ =1 q m ′ - j m=1 q m , which equals zero if and only if j = j ′ ; if i = i ′ , δ i,j = δ i ′ ,j ′ if and only if j = j ′ . Now, let us suppose that i > i ′ .
From [START_REF] Papanikolas | Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms[END_REF] we deduce that

δ i,j -δ i ′ ,j ′ = (i -i ′ )q d -ψ i,i ′ ′ ,
where

ψ i,i ′ ,j,j ′ = i n=1 q n - i ′ n ′ =1 q n ′ + j m=1 q m - j ′ m ′ =1 q m ′ ∈ Z.
Since i > i ′ > j ′ and i > j, we have that ψ i,i ′ ,j,j ′ ≥ 0 and we can find integers c r ∈ {0, 1, 2}, unique, such that ψ i,i ′ ,j,j ′ = i r=0 c r q r , so we see that 0 ≤ ψ i,i ′ ,j,j ′ < q i+1

(recall that q > 2). If i < d, we see that

δ i,j -δ i ′ ,j ′ = (i -i ′ )q d -ψ i,i ′ ,j,j ′ > q d -q i+1 ≥ 0 so that δ i,j = δ i ′ ,j ′ in this case.
It remains to study the case in which i = d. In this case, j ′ < i ′ ≤ d -1 and we can write:

δ i,j -δ i ′ ,j ′ = (d -i ′ -1)q d -ρ i ′ ,j,j ′ ,
where

ρ i ′ ,j,j ′ = d-1 n=i ′ +1 q n + j m=1 q m - j ′ m ′ =1 q m ′ ∈ Z.
This number is obviously ≥ 0 and the following estimate holds

0 ≤ ρ i ′ ,j,j ′ < q d . If i ′ ≤ d -2, we thus have that δ d,j > δ i ′ ,j ′ for any choice of j < d and j ′ < i ′ . If i ′ = d -1, we have δ i,j -δ i ′ ,j ′ = j ′ m ′ =1 q m ′ - j m=1 q m
which equals zero if and only if j = j ′ .

In view of the above Lemma, to compute the degree of BG q d -2 , we rearrange the sum [START_REF] Kaneko | Finite multiple zeta values[END_REF] in the following way:

BG q d -2 = - U α d β d-1 -(α d + α d-1 ) d-2 j=0 β j V - d-2 i=0 i-1 j=0 α i β j W =: -(U + V + W ).
Lemma 17. We have:

(1)

deg θ (U ) = (d -1)q d -2(q + • • • + q d-1 ), (2) deg θ (V ) = (d -2)q d -(q + • • • + q d-2 ), (3) deg θ (W ) = (d -2)q d -(q + • • • + q d-2 ).
Proof. We compute the degree of U :

deg θ (U ) = dq d - d n=1 q n - d-1 m=1 q m = (d -1)q d -2 d-1 n=1 q n .
To compute the degree of V , we observe:

α d + α d-1 = b d (θ q d ) l d - b d-1 (θ q d ) l d-1 = b d (θ q d ) -(θ q d -θ)b d-1 (θ q d ) l d = b d-1 (θ q d )[θ q d -θ q d-1 -θ q d + θ] l d = (θ -θ q d-1 )b d-1 (θ q d ) l d .
Hence,

deg θ (V ) = deg θ (α d + α d-1 ) + deg θ   d-2 j=0 β j   = deg θ (α d + α d-1 ) = q d-1 + (d -1)q d - d n=1 q n = (d -2)q d - d-2 n=1 q n .
To compute the degree of W , we first notice, by Lemma 20, that all the terms involved in the double sum have different degrees. The term with the largest degree is the one corresponding to i = d -2 and j = 0, which is equal to α d-2 , and which has the expected degree.

Proof of Theorem 14. By Lemma 17 and by the assumption q > 2, we have

deg θ (U ) > deg θ (V ), deg θ (W ),
and

deg θ (BG q d -2 ) = deg θ b d (θ q d ) l d l d-1 = (d -1)q d - 2q(q d-1 -1) q -1 .
Remark 18. Anglès and Ould Douh have proved, in [START_REF] Anglès | Arithmetic of "units[END_REF], that there exist infinitely many irreducible elements of [START_REF] Kaneko | Finite multiple zeta values[END_REF] and that q > 2 all along the present note). As a consequence we see, by the fact that

A + such that BG |P |-2 ≡ 0 (mod P ) (recall that |P | = q d in
F deg θ (P ) (1; σ) ≡ 0 (mod P )
for all irreducible P of A + (easily checked), that the right-hand sides of ( 12) and (15) determine non-zero elements of the ring A s defined in [START_REF] Pellarin | On twisted A-harmonic sums and Carlitz finite zeta values[END_REF]. This result is an easy consequence of their formula that we have re-obtained in our Corollary 12.

Let us recall the elegant proof of this property in [START_REF] Anglès | Arithmetic of "units[END_REF]. Since BG q d -2 ≡ d-1 i=0 l -1 i (mod P ) (for P irreducible of degree d), we have BG q d -2 ≡ 0 (mod P ) if and only if P divides the polynomial

V (d) = l d-1 d-1 i=0 l -1 i ∈ A which has degree d-1 n=1 q n = q d -q
q-1 , so that we have at most q dq d(q -1) monic irreducible polynomials P of degree d dividing V (d). Now, the number of monic irreducible polynomials P of degree d is equal to the the necklace polynomial (where µ designates Moebius' function)

M d (q) = 1 d l|d µ(l)q d l ,
which is known to have an asymptotic behavior, as d → ∞, which is of a strictly bigger magnitude than that of the above fraction if q > 2. For example, if d = p ′ is a prime number, the necklace polynomial M p ′ (q) equals q p ′ -q p ′ and we have

q p ′ -q p ′ > q p ′ -q p ′ (q -1) , because q > 2.
The formula [START_REF] Kaneko | Finite multiple zeta values[END_REF] does not seem to immediately imply the result of Anglès and Ould Douh (without using the intermediate congruence with the polynomial V (d)), but we have not tried to rearrange the terms of the sum completely.

We also point out that the proof of the identity (6) can be easily generalized to give the identity

ζ A (1; σ)ζ A (n) = ζ A (n + 1; σ) + ζ A σ 1 1 n , 1 ≤ n ≤ q -1.
This almost immediately implies a generalization of Theorem 19, Corollary 12, Theorem 14 and the result of Anglès and Ould Douh. For all m = 2, . . . , q -1,

BG q d -m = - k≥d>i≥0 b d (θ q k ) l d l m-1 i .
In particular,

BG q d -m ≡ d-1 i=0 l 1-m i (mod P ), deg θ (P ) = d, m = 2, . . . , q -1
and the same proof as [START_REF] Anglès | Arithmetic of "units[END_REF] can be used to show that, for all 2 ≤ m ≤ q -1 we have

BG q d -m ≡ 0 (mod P ),
for infinitely many P .

Looking for more relations

We gave above some hints of a variant of the shuffle product for the multiple zeta values:

ζ A σ 1 σ 2 • • • σ r n 1 n 2 • • • n r
in the simplest non-trivial cases (weight 2). We shall complete our note by suggesting some other tools to develop, in order to compute other kinds of relations. We denote by K{τ } the skew polynomial ring of finite sums i c i τ i , with c i ∈ K, with the non-commutative product uniquely determined by the rule τ c = c q τ for c ∈ K. Additionally, let t be a variable (we can set t = t 1 to get compatibility with the first part of the note). We have an isomorphism of K-vector spaces:

K[t] η -→ K{τ } defined by η(t i ) = C θ i = (θ + τ ) i for i > 0 and η(1) = 1.
Here C θ = θ + τ is the multiplication by θ of the Carlitz module. The inverse of this isomorphism sends 1 to 1 and, for j > 0, τ j to b j (t), where we recall that b j (t) = (tθ) • • • (tθ q j-1 ).

To check that η is an isomorphism, one uses the evaluation at the Anderson-Thakur function. The evaluation f (ω) of an element

f = f 0 + f 1 τ + • • • + f r τ r ∈ K{τ } at ω is by definition the expression (f 0 + f 1 b 1 + • • • + f r b r )ω. It is easy to see that C a (ω) = a(t)ω, so that, for all f (t) ∈ K[t], we have η(f )(ω) = f (t).
This isomorphism η is useful to construct certain identities for finite sums. We recall, as a first example, the formula (7):

S d (1; 1) = S d (1; σ) = a∈A + (d) a -1 a(t) = b d (t) l d , d ≥ 0.
It is easy to show that η(a(t)) = C a ∈ K{τ }. Therefore, the isomorphism η yields the identity:

η(S d (1; 1)) = a∈A + (d) a -1 C a = l -1 d τ d , d ≥ 0.
This picture can be generalized. We can use variables t 1 , . . . , t s , indeterminates τ 1 , . . . , τ s , the rings K[t 1 , . . . , t s ] (commutative) and K{τ 1 , . . . , τ s } (non commutative, with multiplication rules: τ i τ j = τ j τ i and τ i c = c q τ i for c ∈ K), and the isomorphism uniquely defined by η(t j i ) = (θ + τ i ) j (we write η instead of the more precise expression η s we should have used, to simplify our notations). Then, any time we can show a formula for power sums in K[t 1 , . . . , t s ], we obtain a similar formula in the ring K{τ 1 , . . . , τ s }.

Florent Demeslay proved, in his Thesis [START_REF] Demeslay | Formule de classes en caractéristique positive[END_REF], the following result.

Theorem 19. Assume that s > 0. There exists a rational fraction Q k,s ∈ K(t 1 , . . . , t s )(Y ) such that

S d (k; s) = b d (t 1 ) • • • b d (t s ) l d Q k,s (θ q d-m ), d ≥ 0
where m = max{0, ⌊ s-1 q-1 ⌋}. The case s = 0 (no variables) was already known to Anderson and Thakur [START_REF] Anderson | Tensor powers of the Carlitz module and zeta values[END_REF]. We would like to apply this Theorem for s > 0 to produce identities in the non-commutative indeterminates τ 1 , . . . , τ s by means of the isomorphism η. For example, if k = 1 and s = 1, we are reduced to the formula (7) with Q 1,1 = 1. However, there is no reason to expect that Q k,s is a polynomial and in fact, in general, this is false. For example, it is easy to check that Q q,1 = t-Y t-θ , which is not a polynomial. It is of course possible to compute the rational fractions Q k,s by using the polynomials H s of Theorem 7, but even with that in mind, we cannot escape this problem.

A partial solution is given by Lemma 8. We now denote by τ (we do not want to mix it up with τ which is now an indeterminate!) the F q [t]-algebra endomorphism of K[t] defined by τ (c) = c q . Then, Lemma 8 and induction imply the following result.

Proposition 20. For all n > 0 and d ≥ 0, the following formula holds:

τ n (b d (t)) = l q n-1 d d≥i1≥i2≥•••≥in-1≥in≥0 l q n-2 -q n-1 i1 l q n-3 -q n-2 i2 • • • l 1-q in-1 l -1 in b in (t).
In particular, for all n > 0 and d ≥ 0:

S d (q n ; 1) = l q n-1 -q n d d≥i1≥i2≥•••≥in-1≥in≥0 l q n-2 -q n-1 i1 l q n-3 -q n-2 i2 • • • l 1-q in-1 l -1 in b in (t).
Although the rational fraction Q q,1 and more generally the fractions Q q n ,1 are certainly not polynomials, the above formulas in K[t] can be transferred to identities in the ring K{τ }. We obtain, by applying the map η:

Corollary 21. For all d ≥ 0, S d (q n ; 1) := a∈A + (d) a -q n C a = l q n-1 -q n d d≥i1≥i2≥•••≥in-1≥in≥0 l q n-2 -q n-1 i1 l q n-3 -q n-2 i2 • • • l 1-q in-1 l -1 in τ n .
Let σ 1 , . . . , σ r be semi-characters, let n 1 , . . . , n r be integers, and d a non-negative integer. We set, for convenience:

S ⋆ d σ 1 σ 2 • • • σ r n 1 n 2 • • • n r = S d (n 1 ; σ 1 ) d≥i2≥•••≥ir ≥0 S i2 (n 2 ; σ 2 ) • • • S ir (n r ; σ r ) ∈ F ac q ⊗ Fq K(t s )
(we have introduced non-strict inequalities in the sum). Further, we set:

ζ ⋆ A σ 1 σ 2 • • • σ r n 1 n 2 • • • n r := d≥0 S * d σ 1 σ 2 • • • σ r n 1 n 2 • • • n r ∈ K ∞ ⊗ Fq F ac q ⊗ Fq F s .
We observe that S d (j; 1) = l -j d if j = kq l with l ≥ 0 and k = 1, . . . , q -1. Hence, the second identity of Proposition 20 can be rewritten, with σ = χ t , in the following way:

S d (q n ; σ) = S ⋆ d 1 1 • • • 1 σ q n-1 (q -1) q n-2 (q -1) • • • q -1 1
.

Summing over d = 0, 1, . . ., we obtain the formula:

(22) ζ A (q n ; σ) = ζ ⋆ A 1 1 • • • 1 σ q n-1 (q -1) q n-2 (q -1) • • • q -1 1
.

We observe that the evaluation at t = θ q k gives:

. . . , θ q n-1 special values =0

, θ q n value one , θ q n+1 , θ q n+2 , . . .

trivial zeroes

.

Evaluating e.g. at t = θ returns us the following identity, with the obvious meaning of the second sum:

ζ A (q n -1) = ζ ⋆ A (q n-1 (q -1), q n-2 (q -1), . . . , q -1 n terms

).

We can rewrite the identity of our Corollary 21 as follows:

S d (q n ; 1) = a∈A + (d) a -q n C a = = S d (q n-1 (q -1)) d≥i1≥i2≥•••≥in-1≥in≥0 S i1 (q n-2 (q -1))S i2 (q n-3 (q -1)) • • • S in-1 (q -1)S in (1)τ n , with S d (n) := S d (n; 1). If f = f 0 + f 1 τ + • • • + f r τ r ∈ K{τ }, the evaluation at one f (1) of f is the element f 0 + f 1 + • • • + f r ∈ K.
It is easy to see that the series d≥0 a∈A + (d) a -q n C a (1) converges in K ∞ . We obtain the formula:

∞ d=0 S d (q n ; 1)(1) = ζ ⋆ C (q n-1 (q -1), q n-2 (q -1), . . . , q -1 n terms , (23)
).

These formulas can be easily related to Thakur's multiple zeta values ζ A (without the ⋆ mark), by means of simple manipulations. We illustrate this in the case n = 1. We observe that

∞ d=0 S d (q; 1)(1) = ζ ⋆ C (q -1, 1) = ζ A (q -1, 1) + i≥0 S i (q -1; 1)S i (1, 1). Now, since S i (q -1; 1) = l 1-q i and S i (1; 1) = l i , we get i≥0 S i (q -1; 1)S i (1; 1) = i≥0 l -q i = log C (1) q ,
where log C (z) = i≥0 l -1 i z q i is the Carlitz logarithm of z ∈ C ∞ , well defined for |z| < q q/(q-1) (we recall that | • | is the unique norm of C ∞ such that |θ| = q) and in particular, well defined at z = 1. It is plain that log C (1) = ζ A (1), an identity which was, essentially, first noticed by Carlitz. Thus we have that ζ ⋆ C (q -1, 1) = ζ A (q -1, 1) + ζ A (1) q . The shuffle product of ζ A (s 1 ) and ζ A (s 2 ) yields, for s 1 , s 2 ∈ N + such that s 1 + s 2 ≤ q (see Thakur, [27, Theorem 1]), the simple formula:

ζ A (1)ζ A (q -1) = ζ A (q) + ζ A (q -1, 1) + ζ A (1, q -1) = ζ A (1) q + ζ A (q -1, 1) + ζ A (1, q -1).
This means that ∞ d=0 S d (q; 1)(1) = ζ ⋆ C (q -1, 1) = ζ A (1)ζ A (q -1)ζ A (1, q -1).

We do not know how to evaluate the sum ∞ d=0 S d (q; 1)(1) (and more generally, similar sums we do not want to introduce in this paper) directly, and it would be nice to develop a technique to do so independently of the shuffle product, in order to re-obtain the shuffle product formula. Note also that Thakur demonstrated the formula (see [START_REF] Thakur | Relations Between Multizeta Values for Fq[t[END_REF]Theorem 5]):

ζ A (m, m(q -1)) =

ζ A (mq) (θθ q ) m , m = 1, . . . , q -1.

Hence, we compute easily, with m = 1:

ζ ⋆ A (q -1, 1) = ζ A (q -1, 1) + ζ A (1) q = ζ A (q -1)ζ A (1)ζ A (1, q -1)

= ζ A (1) ζ A (q -1) -ζ A (1) q-1 θθ q .

Remark 22.

In the examples we have studied above, the semi-characters are all of Dirichlet type but for no reason this should be considered as a necessary condition for the existence of shuffle-like formulas. For example, if ν : A + → F q [t] is the semi-character which which associates a ∈ A + to t deg θ (a) (this is not of Dirichlet type), then the following formula holds in the Tate algebra T, as the reader can easily check:

ζ A (1; ν)ζ A (1; 1) = ζ A (2; ν) + ζ A ν 1 1 1 + ζ A 1 ν 1 1 .
Evaluating at t = 1 we deduce the formula (1) of Theorem 6.

Remark 23. The referee pointed out a possible link between certain series modeled after the finite sums of Corollary 21 (that we also see in the left-hand side of [START_REF] Pellarin | On twisted A-harmonic sums and Carlitz finite zeta values[END_REF], before evaluation at one) and Chieh-Yu Chang's multiple polylogarithms (see [START_REF] Chang | Linear independence of monomials of multizeta values in positive characteristic[END_REF]) which can also be viewed as formal series:

Li (n1,...,ns) = i1>i2>•••>is≥0 l -n1 i1 • • • l -ns is τ i1 1 • • • τ is r ∈ K{{τ 1 , . . . , τ s }},
in the completion of the non-commutative polynomial ring K{τ 1 , . . . , τ s } with respect to the left ideal generated by τ 1 , . . . , τ s . It is easy to see that the K-isomorphism η of ( 21) induces an isomorphism between the K-vector space spanned by the multiple zeta values (4) with σ i = χ ti for all i and 1 ≤ n i ≤ q for all i, and the K-vector space spanned by the multiple polylogarithms Li (n1,...,ns) with this same condition 1 ≤ n i ≤ q. More precisely, we get: a -n a(t i )

η ζ A χ t1 χ t2 • • • χ ts n 1 n 2 • • • n s = Li (
  = l -n d τ d i , 1 ≤ n ≤ q, 1 ≤ i ≤ s, d ≥ 0.
Note also that the identity ( 22) is rewritten, by means of η, as Li q s (τ s ) = Li ⋆ q s-1 (q-1),...,q-1 (1, . . . , 1, τ s ) ∈ K{{τ s }}, with the obvious meaning of the symbol ⋆ and where the list of 1's means that we are replacing τ i = 1 formally for i = 1, . . . , s -1, which gives well defined series for the τ s -adic topology.

This suggests to explore, in parallel of the multiple zeta values (4), also the "non-commutative" counterparts provided by Drinfeld modules themselves, which can also be viewed, formally, as some kind of semi-characters A + → A{τ 1 , . . . , τ s } * . For example, with this viewpoint, we have, with the above conditions on the parameters, Li (n1,...,ns) = It seems that the image in K ∞ {τ 1 , . . . , τ s } of the K ∞ -vector space spanned by the multiple zeta values (4) requires more general functions than Chang's polylogarithms to be generated, and it is an interesting question to see if an appropriate notion of "non-commutative" multiple zeta values associated to the Drinfeld modules of rank one of [START_REF] Anglès | Arithmetic of positive characteristic L-series values in Tate algebras[END_REF] would suffice for this purpose.

( 21 )

 21 K[t 1 , . . . , t s ] η -→ K{τ 1 , . . . , τ s }

  n1,...,ns) , 1 ≤ n 1 , . . . , n s ≤ q

-n1 1 •

 1 i1>•••>is≥0 a 1 ,...,as ∈A + deg θ (a j )=i j a • • a -ns s C a1 (τ 1 ) • • • C as (τ s ).

The "evaluation at θ = t", in other words, the map which sends a polynomial a = a(θ) = a 0 + a 1 θ + • • • + arθ r with the coefficients a 0 , . . . , ar ∈ Fq to the polynomial a(t) = a 0 + a1 t + • • • + art r ∈ Fq[θ].

In the notations of[START_REF] Pellarin | On twisted A-harmonic sums and Carlitz finite zeta values[END_REF]; in our note, we should write S d (2; 1), S d (2; σ), S d (2; σψ).

Acknowledgements. The author is thankful to Bruno Anglès for fruitful discussions, to David Goss for having suggested some references, including [30] and [17], to Dinesh Thakur for fruitful discussions and for having pointed out several inaccuracies in a previous version of the present note, and to the referee, for interesting remark and useful suggestions with which it was possible to enrich the present note in substantial way. The work was finalized during a stay at the MPIM, Bonn and a stay at the IHES, Bures-sur-Yvette.