Relations between values at $T$-tuples of negative integers of twisted multivariable zeta series associated to polynomials of several variables.

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Relations between values at T-tuples of negative integers of twisted multivariable zeta series associated to

polynomials of several variables.

Marc de Crisenoy, Driss Essouabri

To cite this version:

Marc de Crisenoy, Driss Essouabri. Relations between values atT-tuples of negative integers of twisted multivariable zeta series associated to polynomials of several variables.. 2005. �hal-00004986�

ccsd-00004986, version 1 - 26 May 2005

Relations between values at T-tuples of negative integers of twisted multivariable zeta series

associated to polynomials of several variables.

Marc de Crisenoy ^∗ & Driss Essouabri^† 26th May 2005

Abstract. We give a new and very concise proof of the existence of a holomorphic continuation for a large class of twisted multivariable zeta functions. To do this, we use a simple method of “decalage” that avoids using an integral representation of the zeta function. This allows us to derive explicit recurrence relations between the values at T−tuples of negative integers. This also extends some earlier results of several authors where the underlying polynomials were products of linear forms.

Mathematics Subject Classifications: 11M41; 11R42.

Key words: twisted Multiple zeta-function; Analytic continuation; special values.

1 Introduction

LetQ, P₁, . . . , P_T ∈R[X1, . . . , X_N] andµ₁, . . . , µ_N ∈C− {1},each of modulus 1. To this data we can associate the following “twisted” multivariable zeta series:

Z(Q;P₁, . . . , P_T;µ₁, . . . , µ_N;s₁, . . . , s_T) = X

m1≥1,...,mN≥1

(Q_N

n=1µ^m_nⁿ)Q(m1, . . . , m_N) Q_T

t=1P_t(m₁, . . . , m_N)^st where (s1, . . . , s_T)∈C^T.

In this article we will always assume that:

∀t∈ {1, . . . , T} ∀x∈[1,+∞[^N P_t(x)>0 and

T

Y

t=1

P_t(x)−−−−−→

|x|→+∞

x∈J^N

+∞ (#)

It is not difficult to see thatcondition (#)implies thatZ(Q;P₁, . . . , P_T;µ₁, . . . , µ_N;s₁, . . . , s_T) is an absolutely convergent series when ℜ(s1), . . . ,ℜ(sT) are sufficiently large.

Cassou-Nogus ([6]) and Chen-Eie ([7]) prooved in the case T = 1, and P = P₁ a polynomial with

∗Institut de Math´ematiques de Bordeaux (A2X UMR 5465), Universit´e Bordeaux 1, 351 cours de la Lib´eration 33405 Talence cedex, France. Email: mdecrise@math.u-bordeaux1.fr

†Universit´e de Caen, UFR des Sciences (Campus II), Laboratoire de Math´ematiques Nicolas Oresme (CNRS UMR 6139), Bd. Mal Juin, B.P. 5186, 14032 Caen, France. Email: essoua@math.unicaen.fr

positive coefficients, that the above series can be holomorphically continued to the whole complex plane and obtained very nice formulas for their values at negative integers. In [8] de Crisenoy extended these results by allowing T >1 and introducing the (HDF) hypothesis (see definition 2 in§2) that is much weaker than positivity of coefficients. The Main result of [8] is the following:

Theorem ([8]). Let Q, P₁, ..., P_T ∈R[X1, ..., X_N]and µ∈(T\ {1})^N. Assume that:

For each t= 1, . . . , T P_t satisfies the (HDF) hypothesis and that

T

Y

t=1

P_t(x)−−−−−→

|x|→+∞

x∈J^N

+∞.

Then:

• Z(Q;P₁, . . . , P_T;µ;·) can be holomorphically extended to C^T;

• For all k₁, ..., k_T ∈N if we set Q

T

Y

t=1

P_t^kt =X

α∈S

aαX^α, we have:

Z(Q;P₁, ..., P_T;µ;−k1, ...,−kT) = X

α∈S

aα N

Y

n=1

ζ_µn(−αn).

where for all µ∈T, ζ_µ(s) =P_+∞

m=1 µ^m m^s.

To obtain this theorem he used an integral representation for the zeta series. The proof of the holomorphic continuation of the resulting integral is long and complicated.

By restricting to hypoelliptics polynomials (see definition 1 in §2) we can avoid the integrals:

the first result of this article gives a new proof of the holomorphic continuation of these series under the assumption that P₁, . . . , P_T are hypoelliptic. Our proof uses the ”decalage” method of Essouabri ([12]). Of particular interest for this paper is that this method doesnotuse an integral representation for the zeta series. And the resulting proof is very concise and much simple.

The second result is relations between the values at T−tuples of negative integers of these series, relations that are true under the HDF hypothesis, and that give a mean to calculate by induction the values.

These relations are very simple in the case of linear forms, particularly interesting because of its link with the zeta functions of number fields ([5]). Similar results have been obtain by severals authors in particular cases of linear forms (see [1], [3] and [4]). Our method allows one also to obtain new relations even in the cases of linear forms.

2 Notations and preliminaries:

First, some notations:

1. SetN={0,1,2, . . .},N^∗=N− {0}, J = [1,+∞[, and T={z∈C| |z|= 1}.

2. The real part ofs∈Cwill be denoted ℜ(s) =σ and its imaginary partℑ(s) =τ. 3. Set0= (0, . . . ,0) ∈R^N and 1= (1, . . . ,1)∈R^N.

4. Forx= (x₁, . . . , x_N)∈R^N we set|x|=|x1|+. . .+|xN|.

5. Forz= (z₁, . . . , z_N)∈C^N andα= (α₁, . . . , α_N)∈R^N₊ we set z^α=z₁^α1. . . z_N^αN.

6. The notation f(λ,y,x) ≪y g(x) (uniformly inx ∈ X and λ ∈ Λ) means that there exists A=A(y)>0, that depends neither onxnorλ, but could a priori depend on other parameters, and in particular ony, such that: ∀x∈X ∀λ∈Λ |f(λ,y,x)| ≤Ag(x).

When there is no ambiguity, we will omit the word uniformly and the indexy.

7. The notationf ≍g means that we have bothf ≪g and g≪f.

8. Fora∈R^N₊ and P ∈R[X₁, . . . , X_N], we define ∆_aP ∈R[X₁, . . . , X_N] by

∆aP(X) =P(X+a)−P(X). If no ambiguity, we will note simply ∆aP by ∆P.

9. Leta∈N^N,N ∈N^∗,I ⊂ {1, . . . , N},q = #I,I^c ={1, . . . , N} \I and b= (bi)i∈I^c ∈Q

i∈I^c{0, . . . , ai}. SetI ={i1, . . . , i_q} andI^c ={j1, . . . , j_N−q}. Then:

(a) For all d= (d₁, . . . , d_q)∈N^q we define m^a,I,b(d) = (m₁, . . . , m_N)∈N^N by

∀k= 1, . . . , q,m_ik =a_ik+d_k and∀k= 1, . . . , N−q,m_jk =b_jk;

(b) For all f :N^N →Cwe definef^a,I,b:N^q→C byf^a,I,b(d) =f m^a,I,b(d) .

Convention: In this work we will say that a series defined by a sum over N ≥1 variables is conver- gent when it is absolutely convergent.

Let us recall some definitions:

Definition 1. P ∈R[X1, . . . , X_N]is said to be hypoelliptic if:

∀x∈J^N P(x)>0 and ∀α∈N^N \ {0} ∂^αP

P (x)−−−−−→

|x|→+∞

x∈J^N

0.

Definition 2. Let P ∈R[X1, . . . , X_N].

P is said to satisfy the weak decreasing hypothesis (denoted HDF in the rest of the article) if:

• ∀x∈J^N P(x)>0,

• ∃ǫ0>0 such that for α∈N^N and n∈ {1, . . . , N}: α_n≥1⇒ ∂^αP

P (x)≪x^−ǫ_n ⁰ (x∈J^N).

Remark 1. It follows from H¨ormander ([13], p.62) (see also Lichtin ([14], P.342) that if P ∈R[X1, . . . , X_N]is hypoelliptic then there exists ǫ >0 such that ∀α∈N^N \ {0}

∂^αP

P (x)≪x^−ǫ1 (x∈J^N). Therefore P satisfy also the HDF hypothesis.

3 Main results:

We first give a new and concise proof of the existence of a holomorphic continuation for our series when each polynomial is hypoelliptic. The procedure uses the ”decalage” method, and does not require an integral representation.

A consequence of this method is found in corollaries 1 and 2. This gives simple recurrent relations

between the values at T−tuples of negative integers when the polynomials P_i are products of linear polynomials or a class of quadratic polynomials. This procedure is different from that in [8] where the calculation of the special values was not an immediate consequence of the holomorphic continuation of the zeta series.

Theorem 1. Let µ∈(T\ {1})^N andQ, P₁, . . . , P_T ∈R[X₁, . . . , X_N].

We assume that P₁, . . . , P_T are hypoelliptic and that at least one of them is not constant.

Then Z(Q;P₁, . . . , P_T;µ;·) can be holomorphically extended to C^T. Let a∈N^N. We setE(a) ={m∈N^∗N |ma+1}=

m∈N^∗N | ∃n∈ {1, . . . , N} m_n≤a_n . Then for all k∈N^T we have the following relation:

(1−µ^a)Z(Q;P₁, . . . , P_T;µ;−k) =µ^a X

0≤u<k

k u

Z Q(X+a)

T

Y

t=1

(∆_aP_t)^kt−ut;P₁, . . . , P_T;µ;−u

!

+µ^aZ(∆Q;P₁, . . . , P_T;µ;−k) +Z_N−1^a (−k).

Where Z_N−1^a (s) = X

m∈E(a)

µ^mQ(m)

T

Y

t=1

P_t(m)^−st. By using notation (9) above it’s easy to see that:

Z_N−1^a (s) =

N−1

X

q=1

X

I⊂{1,...,N}

#I=q

X

b=(bi)∈Q

i∈Ic{0,...,ai}

Y

i∈I^c

µ^b_iⁱ

! Z

Q^a,I,b;P₁^a,I,b, . . . , P_T^a,I,b; (µ_i)_i∈I;s .

In particular, it is clear thatZ_N−1^a is a finite linear combination of zeta series Z associated to hypoel- liptic polynomials of at most N −1 variables.

Remark 2. When µ^a 6= 1 (and we can of course choose a such that this is satisfied, this formula allows us to compute the values at T−tuples of negative integers of the series Z by recurrence because in each term on the right a integral value is strictly less than the corresponding value on the left:

|u|<|k|, deg(∆Q)<degQ, N−1< N.

Now, using theorem A of [8](that gives the existence of the holomorphic continuation under the HDF hypothesis) and theorem B of [8] (that gives closed formula for the values, still under HDF) and the preeceding theorem, we show that the relations remain true under the HDF hypothesis:

Theorem 2. Let µ∈(T\ {1})^N andQ, P₁, . . . , P_T ∈R[X1, . . . , X_N].

We assume that P₁, . . . , P_T satisfies the HDF hypothesis and that

T

Y

t=1

P_t(x)−−−−−→

|x|→+∞

x∈J^N

+∞.

Then, the relations of theorem 1 are still true.

Now we deal with the particular case of linear forms. In this case the relations become particularly simple!

Corollary 1. Let µ∈(T\ {1})^N and L₁, . . . , L_T linear forms with positive coefficients.

We assume that for each nthere exists t such thatL_t really depends onX_n. We set Z(µ;·) =Z(1;L₁, . . . , L_T;µ;·).

Let a∈N^N. We setE(a) ={m∈N^∗N |ma+1}=

m∈N^∗N | ∃n∈ {1, . . . , N} m_n≤a_n . For all t∈ {1, . . . , T} we set: δ_t=L_t(X+a)−L_t(X). δ_t∈R.

Then, for all k∈N^T we have the following relation:

(1−µ^a)Z(µ;−k) =µ^a X

0≤u<k

δ^k−u k

u

Z(µ;−u) +Z_N^a₋₁(µ,−k)

Where Z_N−1^a (s) = X

m∈E(a)

µ^m

T

Y

t=1

L_t(m)^−st. By using notation (9) above it’s easy to see that:

Z_N−1^a (s) =

N−1

X

q=1

X

I⊂{1,...,N}

#I=q

X

b=(bi)∈Q

i∈Ic{0,...,ai}

Y

i∈I^c

µ^b_iⁱ

! Z

1;L^a,I,b₁ , . . . , L^a,I,b_T ; (µi)i∈I;s .

In particular, it is clear that Z_N^a₋₁ is a finite linear combination of zeta series Z associated to linear forms of at most N −1 variables.

Remark 3. Of course, here as well this formulaes allows a calculus by induction.

Since the works of Cassou-Nogus and Shintani, we know that the case of linear forms is particulaly interesting for algebraic number theory because of the link with the zetas functions of number fields.

See also [16] for more motivations.

Remark 4. Let us assume that P ∈ R[X1, . . . , X_N] is a product of linear forms: P =

T

Y

t=1

L_t where L₁, . . . , L_T have real positive coefficients, and that we want to evaluate the numbers Z(1;P;µ;−k) where k∈N.

We could use theorem I, but it seems more interesting to note that

∀s∈C Z(1;P;µ;s) =Z(1;L₁, . . . , L_T;µ;s, . . . , s) and then to consider the numbers

Z(1;P;µ;−k) = Z(1;L₁, . . . , L_T;µ;−k, . . . ,−k) inside the family Z(1;L₁, . . . , L_T;µ;−k₁, . . . ,−k_T) because of the simple relations between these numbers given by the preceeding proposition.

The particular case of linear forms is not the only case when relations of theorems 1 and 2 become particularly simple. The great flexibility in the assumptions of this theorems, allowed one to obtain also very simple relations in somes others cases as in the following:

Corollary 2. Letµ∈(T\ {1})^N and letP₁, . . . , P_T ∈R[X1, . . . , X_N]be polynomials of degree at most 2. Suppose that for all t= 1, . . . , T:

P_t(X) =P_t(X₁, . . . , X_N) =

rt

X

k=1

hα^t,k,Xi2

+

N

X

n=1

c_t,nX_n+d_t where α^t,k∈R^N, c_t,i∈R^∗₊ and d_t∈R+ ∀t, k, i.

Assume that there exist a∈N^N\ {0} such that hα^t,k,ai= 0 for allt, k.

Then:

1. for all t∈ {1, . . . , T}, δ_t:=P_t(X+a)−P_t(X)∈R;

2. the relations of Corollary 1 are still true.

4 Proof of theorem 1

Let t ∈ {1, . . . , T}. It’s clear that there exists α ∈ N^N such that ∂^αP_t is a no vanishing constant polynomial. So the hypoellipticity of P_t implies that P_t(x) →+∞ when |x| →+∞ (x∈[1,+∞[^N).

By using Tarski-Seidenberg (see for example [11], Lemme 1), we know that there exists δ > 0 such that P_t(x) ≫ (x₁. . . x_N)^δ uniformly in x ∈ [1,+∞[^N. Therefore, its clear that there exists σ₀ such that if σ₁, . . . , σ_T > σ₀ thenZ(Q, P₁, . . . , P_T,µ,s) converges.

By Remark 2.1 above, there exist also a fixedǫ >0 such that∀t∈ {1, . . . , T} we have

∀α∈N^N \ {0} ∂^αP

P (x)≪x^−ǫ1 (x∈J^N).

Step 1: we establish a formula (∗).

Proof of step 1:

P₁, . . . , P_T are fixed in the whole proof so we will denoteZ(Q,µ,·) instead ofZ(Q, P₁, . . . , P_T,µ,·).

With this notation for all s such thatσ₁, . . . , σ_T > σ₀ we have:

Z(Q,µ,s) = X

m∈N∗N

µ^mQ(m)

T

Y

t=1

P_t(m)^−st

= X

m≥a+1

µ^mQ(m)

T

Y

t=1

P_t(m)^−st+Z_N−1^a (s)

=µ^a X

m∈N∗N

µ^mQ(m+a)

T

Y

t=1

P_t(m+a)^−st +Z_N−1^a (s) U ∈Nis set for the whole step.

We have g_U:C×C\]− ∞,−1]→C holomorphic and satisfying:

∀s∈C∀z∈C\]− ∞,−1] (1 +z)^s =

U

X

u=0

s u

z^u+z^U+1g_U(s, z).

∀k∈N vrifiantk≤U et∀z∈C\]− ∞,−1],g_U(k, z) = 0.

Fort∈ {1, . . . , T} we define ∆_t= ∆_aP_t.

For t∈ {1, . . . , T} and m∈N^∗N, we define H_t,m,U:C→Cby:

H_t,m,U(st) =

U

X

ut=0

−st

u_t

∆t(m)^utP_t(m)^−ut.

∀t∈ {1, . . . , T}we have:

P_t(m+a)^−st = [P_t(m) + ∆_t(m)]^−st

=P_t(m)^−st

1 + ∆_t(m)P_t(m)⁻¹−st

=P_t(m)^−sth

H_t,m,U(s_t) + ∆_t(m)^U+1P_t(m)^−(U+1)g_U −s_t,∆_t(m)P_t(m)⁻¹i

For x₁, . . . , x_T, y₁, . . . , y_T ∈R, we have

T

Y

t=1

(xt+y_t) = X

ǫ∈{0,1}^T T

Y

t=1

x^1−ǫ_t ^ty^ǫ_t^t, so:

T

Y

t=1

P_t(m+a)^−st = X

ǫ∈{0,1}^T T

Y

t=1

H_t,m,U(st)^1−ǫt∆t(m)^ǫt(U+1)P_t(m)^{−st−ǫt(U+1)}g_U −st,∆t(m)Pt(m)⁻¹ǫt

Form∈N^∗N and ǫ∈ {0,1}^T we define f_m,U,ǫ:C^T →C thanks to the following formula:

f_m,U,ǫ(s) =

T

Y

t=1

H_t,m,U(st)^1−ǫt∆t(m)^ǫt(U+1)P_t(m)^{−st−ǫt(U+1)}g_U −st,∆t(m)Pt(m)⁻¹ǫt

So for all m∈N^∗N and s∈C^T we have:

T

Y

t=1

P_t(m+a)^−st = X

ǫ∈{0,1}^T

f_m,U,ǫ(s).

We define Z_U(Q,µ,·) by: Z_U(Q,µ,s) := X

ǫ∈{0,1}^T\{0}

X

m∈N∗N

µ^mQ(m+a)f_m,U,ǫ(s).

We will see in step 2 that for U large enoughZ_U(Q,µ,·) exists and is holomorphic on s∈C^T |σ₁, . . . , σ_T > σ₀ .

It’s clear thatZ_U(Q,µ,s) = X

m∈N∗N

µ^mQ(m+a) X

ǫ∈{0,1}^T\{0}

f_m,U,ǫ(s).

So Z(Q,µ,s) =µ^a X

m∈N∗N

µ^mQ(m+a)f_m,U,0(s) +µ^aZ_U(Q,µ,s) +Z_N^a₋₁(s).

By definition f_m,U,0(s) =

T

Y

t=1

H_t,m,U(st)Pt(m)^−st so:

f_m,U,0(s) =

T

Y

t=1 U

X

ut=0

−s_t u_t

∆_t(m)^utP_t(m)^−(st+ut)

= X

0≤u1,...,uT≤U T

Y

t=1

−st

u_t

∆t(m)^utP_t(m)^−(st+ut)

= X

u∈{0,...,U}^T

−s u

T

Y

t=1

∆_t(m)^utP_t(m)^−(st+ut)

then, for all s such thatσ₁, . . . , σ_T > σ₀, we have:

X

m∈N∗N

µ^mQ(m+a)f_m,U,0(s) = X

m∈N∗N

µ^mQ(m+a) X

u∈{0,...,U}^T

−s u

T

Y

t=1

∆_t(m)^utP_t(m)^−(st+ut)

= X

u∈{0,...,U}^T

−s u

X

m∈N∗N

µ^mQ(m+a)

T

Y

t=1

∆_t(m)^utP_t(m)^−(st+ut)

= X

u∈{0,...,U}^T

−s u

Z Q(X+a)

T

Y

t=1

∆^u_t^t,µ,s+u

!

The combination of the precedings results gives us:

Z(Q,µ,s) =µ^a X

u∈{0,...,U}^T

−s u

Z Q(X+a)

T

Y

t=1

∆^u_t^t,µ,s+u

!

+µ^aZ_U(Q,µ,s) +Z_N−1^a (s) The following (∗) formula is now clear:

(∗) (1−µ^a)Z(Q,µ,s) =µ^a X

u∈{0,...,U}^T\{0}

−s u

Z Q(X+a)

T

Y

t=1

∆^u_t^t,µ,s+u

!

+µ^aZ(∆Q,µ,s) +Z_N^a₋₁(s) +µ^aZ_U(Q,µ,s)

Step 2: For all a ∈ R+ there exists U₀ ∈ N such that for all U ≥ U₀, Z_U(Q,µ,·) exists and is holomorphic on

s∈C^T | ∀t∈ {1, . . . , T} σ_t>−a . Proof of step 2:

Let a∈R. LetK be a compact of C^T included in

s∈C^T | ∀t∈ {1, . . . , T} σ_t>−a .

⋆ Lett∈ {1, . . . , T}.

We know that P_t(x)≫1 (x∈J^N) so it’s easy to seen that : P_t(x)^−st ≪P_t(x)^a (x∈J^N s∈K).

Let denotep= max

deg_XnP_t |1≤n≤N 1≤t≤T . From now we assume a >0. So P_t(x)^a ≪x^pa1 (x∈J^N).

As a conclusion we have: P_t(x)^−st ≪x^pa1 (x∈J^N s∈K).

Let U ∈N^∗ and ǫ∈ {0,1}^T. By definition H_t,m,U(s) =

U

X

ut=0

−st

u_t

∆_t(m) P_t(m)

ut

, so hypoellipticity of P_t implies that H_t,m,U(s)≪1 (m∈N^∗N,s∈K).

It implies also that there exists a compact of ]−1,+∞[ containing all the ∆_t(m)

P_t(m) wheremis inN^∗N so: g_U −st,∆_t(m)P_t(m)⁻¹ǫt

≪1 (m∈N^∗N s∈K).

Thanks to the Taylor formula and to the choice of ǫwe have ∆_t(m)

P_t(m) ≪m^−ǫ1 (m∈N^∗N).

From what preceeds we deduce that we have, uniformely in m∈N^∗N ands∈K:

H_t,m,U(s)^1−ǫt

∆_t(m) P_t(m)

ǫt(U+1)

P_t(m)^−stg_U −st,∆_t(m)P_t(m)⁻¹ǫt

≪m^pa1m^{−ǫǫt(U+1)1}.

⋆ By definition f_m,U,ǫ(s) =

T

Y

t=1

H_t,m,U(s)^1−ǫt

∆_t(m) P_t(m)

ǫt(U+1)

P_t(m)^−stg_U −st,∆_t(m)P_t(m)⁻¹ǫt

Sinceǫ6=0, we have: f_m,U,ǫ(s)≪m^{T pa1}m^−ǫ(U+1)1 (m∈N^∗N).

We denote q = max

deg_XnQ|1≤n≤N (obviously we can assume that Q6= 0).

We see thatQ(m+a)f_m,U,ǫ(s)≪m(q+T pa−ǫ(U+1))1 (m∈N^∗N).

So it is enough to have q+T pa−ǫ(U + 1)≤ −2, forU to fit.

Thus it is enough to choose U ∈Nsuch thatU ≥U₀ :=

q+T pa+ 2 ǫ

+ 1.

Convention:

let us take a convention, that we will use until the end of the proof. Let a ∈R. We will say that a functionY is an entire combination until aof the functionsY₁, . . . , Y_k if there exists:

⋆ entire functionsλ₁, . . . , λ_k:C^T →C,

⋆ one functionλ:

s∈C^T | ∀t∈ {1, . . . , T}σ_t> a →Cholomorphic, such that Y =λ+

k

X

i=1

λ_iY_i. A definition and a remark:

foru∈N^T andQ∈R[X1, . . . , X_N], we denoteEu(Q) the subspace ofR[X1, . . . , X_N] generated by the polynomials of the following form: ∂^βQ

T

Y

t=1

Y

k∈Ft

∂^ft(k)P_t where:

⋆β∈N^N,

⋆ F₁, . . . , F_T are finite subset of Nsatisfying |Ft|=u_t,

⋆∀t∈ {1, . . . , T} f_t is a function fromF_t intoN^N \ {0}.

We remark that Eu(Q) is stable under partial derivations.

We are now beginning the proof of the existence of an holomorphic continuation.

The proof is by recurrence on N; it will be clear that the proof that rankN −1 implies rankN −1 gives the result at rank N = 1.

Let N ≥1. We assume that the result is true at rankN −1.

Step 3:

Let Q∈R[X1, . . . , X_N] and a∈R+.

Then Z(Q,µ,s) is an entire combination until −a of functions of the type Z(R,µ,s+u) where u∈N^N\ {0}and R∈ Eu(Q).

Proof of step 3:

we are going to show by recurrence ond∈Nthat if degQ < d then the result is true.

For d= 0 it is clear.

Let us assume the result for d≥1.

Let Q∈R[X1, . . . , X_N] such that degQ < d+ 1.

Thanks to step 2 we setUsuch thatZ_U(Q,µ,·) is holomorphic on

s∈C^T | ∀t∈ {1, . . . , T} σ_t>−a .