Christophe Delaunay

Emmanuel Fricain
email: fricain@math.univ-lyon1.fr

Elie Mosaki
email: mosaki@math.univ-lyon1.fr

Olivier Robert
email: olivier.robert@univ-st-etienne.fr

Zero free regions for Dirichlet series

Keywords: 2010 Mathematics Subject Classification. 11M26, 30H10 Dirichlet series, Beurling-Nyman criterion, Hardy spaces, zeros of Lfunctions

Introduction

In this article, we are interested in zero-free regions for functions that are obtained as meromorphic continuation of Dirichlet series s → n≥1 a n n -s . Such study arises naturally in various fields of mathematics such as functional analysis and number theory.

The particular case of the Riemann zeta function has been most studied, and is related to the Riemann Hypothesis, asserting that the zeta function does not vanish on the half-plane ℜ(s) > 1/2. Several attempts have been made in the direction of solving or reformulating this conjecture.

In his thesis B. Nyman [Nym50] gave a reformulation of the Riemann Hypothesis by means of functional analysis. More precisely let {•} denotes the fractional part and let N be the set of functions

f (x) = n j=1 c j θ j x ,
where 0 < θ j ≤ 1, c j ∈ C and n j=1 c j θ j = 0. (1.1) Then Nyman proved that the Riemann Hypothesis holds if and only if the characteristic function χ (0,1) of the interval (0, 1) belongs to the closure of N in L 2 (0, 1). Later on A. Beurling [START_REF] Beurling | A closure problem related to the riemann zeta-function[END_REF] extended this result by proving that if 1 < p < +∞, then the Riemann zeta function has no zeros in ℜ(s) > 1/p if and only if χ (0,1) belongs to the closure of N in L p (0, 1). The case p = 1 has been investigated in [START_REF] Bercovici | A real variable restatement of Riemann's hypothesis[END_REF]. After the works of Beurling and Nyman, several results occur in this direction, see for example [START_REF] Nikolski | Distance formulae and invariant subspaces, with an application to localization of zeros of the Riemann ζ-function[END_REF], [START_REF] Balazard | Notes sur la fonction ζ de Riemann. 1[END_REF], [START_REF] Báez-Duarte | Notes sur la fonction ζ de Riemann[END_REF], [START_REF] Báez-Duarte | A strengthening of the Nyman-Beurling criterion for the Riemann hypothesis[END_REF] and [dR07a].

In particular, N. Nikolski [START_REF] Nikolski | Distance formulae and invariant subspaces, with an application to localization of zeros of the Riemann ζ-function[END_REF] gave, in a way, an effective version of the Beurling-Nyman criterion and produces explicit zero-free regions for the Riemann zeta function: let r > 0 and λ ∈ C with ℜ(λ) > 0 be fixed parameters and let Kr be the subspace of L 2 ((0, 1), dx/x) spanned by functions

E α,r (x) = x r α x -α 1 x , 0 < x < 1,
where 0 ≤ α ≤ 1. Then the zero-free regions obtained by Nikolski are domains of the form

(1.2) r + µ ∈ C : µ -λ µ + λ 2 < 1 -2ℜ(λ) dr (λ) 2 ,
where dr (λ) = dist(x λ , Kr ) is the distance in L 2 ((0, 1), dx/x) between x λ and the subspace Kr . In the case λ = r = 1/2, if dr (λ) = 0 the region (1.2) corresponds to the half-plane ℜ(µ) > 1/2 and Nikolski recovers Nyman's result (note that in this case, d1/2 (1/2) = 0 if and only if χ (0,1) belongs to the closure of N in L 2 (0, 1); see Proposition 7.3). More recently, A. de Roton [dR07a] generalized Nyman's work to the Selberg class of Dirichlet functions and reformulated the Generalized Riemann Hypothesis. Finally let us mention that in [START_REF] Báez-Duarte | Notes sur la fonction ζ de Riemann[END_REF] and [START_REF] Burnol | A lower bound in an approximation problem involving the zeros of the Riemann zeta function[END_REF] interesting lower bounds are obtained. First the authors proved in [START_REF] Báez-Duarte | Notes sur la fonction ζ de Riemann[END_REF] that if 0 < λ ≤ 1 and N ♯ λ is the closed span in L 2 (0, +∞) of functions

f (x) = n j=1 c j θ j x , x > 0,
where λ < θ j ≤ 1, c j ∈ C, then the Riemann Hypothesis holds if and only if lim λ→0 d(λ) = 0, where d(λ) = dist(χ (0,1) , N ♯ λ ). In other words, we can drop the condition (1.1) in the Nyman's theorem. Furthermore they proved that lim inf

λ→0 d(λ) log 1 |λ| > 0.
Then this lower bound was improved by J.F. Burnol in [START_REF] Burnol | A lower bound in an approximation problem involving the zeros of the Riemann zeta function[END_REF] and generalized by de Roton in [dR06] and [dR09] for the Selberg class.

The aim of our article is to make a further investigation of Nikolski's work in a more general situation including, in particular, the Selberg class. Associated to some auxiliary function ϕ : [0, ∞) → C, we introduce a class of spaces K r (in particular, in the case of the Riemann zeta function, K r contains Kr for a special choice of ϕ). Then, for a large class of Dirichlet series, we obtain explicit zero-free regions that are larger than Nikolski's regions in the case of zeta. For some well chosen parameters, if dist(x λ , K r ) = 0 our zero-free regions correspond to ℜ(µ) > r and we recover (and in fact improve a little bit) the results of de Roton concerning the reformulation of the Generalized Riemann Hypothesis. Finally, we investigate several explicit applications. In particular, we give zero-free regions for the zeta function and Dirichlet L-functions ; it should be pointed out that the domains obtained have the property of being explicit, but they do not have the ambition of competing with the classical non-zero regions that are involved, for instance, in the prime number theorem. An other application we obtain is somehow a Beurling-Nyman's criterion for the Siegel's zero problem of Dirichlet L-functions. As far as we know, it seems to be a new criterion concerning this question.

Notations and statements of the main results

For a generic s ∈ C, we denote by σ (respectively t) its real (respectively imaginary) part so that we have s = σ + it. For s ∈ C, we denote by Π s the half-plane defined by Π s = Π σ = {z ∈ C : ℜ(z) > σ}. For the sequel, we fix a Dirichlet series

L(s) = n≥1 a n n s
satisfying the following conditions:

• For every ε > 0, we have a n ≪ ε n ε .

• There exists σ 0 < 1 such that the function s → L(s) admits a meromorphic continuation to ℜ(s) > σ 0 with a unique pole of order m L at s = 1. • The function s → (s -1) m L L(s) is analytic with finite order in Π σ 0 . The growth condition on the coefficients (a n ) n implies that L(s) is an absolutely convergent Dirichlet series for ℜ(s) > 1. The second condition tells us that the function (s -1) m L L(s) can be analytically continued in some half-plane Π σ 0 which contains Π 1 . Note that we do not require neither an Euler product nor a functional equation for L(s). We also consider an auxiliary function ϕ : [0, +∞[-→ C satisfying the following conditions:

• ϕ is supported on [0, 1]. • ϕ is locally bounded on (0, 1). • ϕ(x) = O(x -σ 0 ) when x → 0. • ϕ(x) = O((1 -x) -σ 1 ) when x → 1 -, for some σ 1 < 1/2.
The fact that σ 1 < 1 is sufficient for the integral 1 0 ϕ(t)t s dt t to be absolutely convergent for ℜ(s) > σ 0 . Hence, the Mellin transform ϕ of ϕ is analytic on Π σ 0 . The condition σ 1 < 1/2 will become clearer in Lemma 3.3. Recall that the (unnormalized) Mellin transform of a Lebesgue-measurable function ϕ : [0, +∞[→ C is the function ϕ defined by

ϕ(s) = +∞ 0 ϕ(t)t s dt t (s ∈ C),
whenever the integral is absolutely convergent. Let H 2 (Π σ ) be the Hardy space of analytic functions f : Π σ → C such that f 2 < ∞ where

f 2 = sup x>σ +∞ -∞ |f (x + it)| 2 dt 1 2
.

Then the (normalized) Mellin transform

M : L 2 * (0, 1), dt t 1-2σ -→ H 2 (Π σ ) ϕ -→ 1 √
2π ϕ is a unitary operator (use the Paley-Wiener's theorem and the change of variables going from the Fourier transform to the Mellin transform, for instance see [START_REF] Nikolski | Operators, functions, and systems: an easy reading[END_REF]p. 166]). Here, we write L 2 * (0, 1), dt t 1-2σ

for the subspace of functions in L 2 (0, +∞), dt t 1-2σ that vanish almost everywhere on (1, +∞). We define

ψ(u) = res (L(s) φ(s)u s , s = 1) - n<u a n ϕ n u (u ∈ R + ),
where res(F (s), s = 1) denotes the residue of the meromorphic function F at s = 1. The method that is going to be explored depends on the fact that the function ψ belongs to L 2 ((1, +∞), du u 1+2r ) for a certain real number r > σ 0 . The following gives a criterion for this property.

Theorem 2.1. Let r > σ 0 . If m L ≥ 1, we assume furthermore that r = 1. The following are equivalent:

(i) The function ψ belongs to L 2 ((1, +∞), du u 1+2r ). (ii) The function t -→ L(r + it) φ(r + it) belongs to L 2 (R).
We will see in Corollary 3.4 that the conditions (i) and (ii) above are satisfied whenever r > 1. Furthermore, Theorem 2.1 is a generalization of [dR07a, Proposition 3.3] when ϕ = χ (0,1) and r = 1/2.

We see that the second condition of the theorem above depends on some growth estimates of L in vertical strip which is, in its turn, linked with questions related to the convexity bound and to the Lindelöf hypothesis. In particular, if L is a function in the Selberg class, then one can prove that L satisfies the generalized Lindelöf hypothesis if and only if for every k ∈ N, we have

(2.1) t -→ L k ( 1 2 + it) 1 2 + it ∈ L 2 (R).
Note that with the special choice of ϕ = χ (0,1) , then φ(s) = 1/s and the condition (ii) above for r = 1/2 means exactly that (2.1) is satisfied for k = 1. Moreover, in [dR07a] using the functional equation, it is shown that the condition ψ ∈ L 2 ((1, +∞), du u 2 ) (in the case when ϕ = χ (0,1) ) is necessary for the generalized Riemann Hypothesis for L-functions in the Selberg class. In [dR07b], it is also shown that the condition on ψ (still with ϕ = χ (0,1) ) is satisfied for L-functions in the Selberg class of degree less than 4.

Fix an integer m ≥ 0 and let W = n≥1 (0, 1] n . We say that α ∈ W is of length n if α belongs to (0, 1] n . For each α in W , its length is denoted by ℓ(α). Now let α ∈ W and c ∈ C ℓ(α) , we say that A = (α, c) is an m-admissible sequence if

ℓ(α) j=1 c j α j (log α j ) k = 0 for all 0 ≤ k ≤ m -1 . (2.2)
Furthermore, A is said to be non-trivial if c ∈ C ℓ(α) \ {(0, . . . , 0)}.

It is easy to prove (see Lemma 4.2 and Lemma 4.1) that for any fixed ℓ > m, there are infinitely many non-trivial m-admissible sequences of length ℓ. We also easily see that every A = (α, c) is 0-admissible. Note that the notion of admissible sequences had been introduced in [dR07a] and it is a generalization of condition (1.1) which appears in Nyman's theorem and which corresponds to m = 1.

From now on, we assume that r is chosen such that ψ ∈ L 2 ((1, +∞), du u 1+2r ). Then we associate to each m L -admissible sequence A = (α, c) the function f A,r defined by

f A,r (t) = t r-σ 0 ℓ(α) j=1 c j ψ α j t , t > 0.
We will show that f A,r ∈ L 2 * ((0, 1), dt t 1-2σ 0 ) and we let

K r = span{f A,r : A a m L -admissible sequence} and d r (λ) = dist t λ-2σ 0 χ (0,1) , K r (λ ∈ Π σ 0 ),
where the (closed) span and the distance are taken with respect to the space L 2 * ((0, 1), dt t 1-2σ 0 ). Let us remark that we trivially have d 2 r (λ) ≤ 1/(2ℜ(λ) -2σ 0 ). One of our main theorems is the following which gives zero-free discs for the function L.

Theorem 2.2. Let λ ∈ Π σ 0 . Then L does not vanish on rσ 0 + D r (λ), where

D r (λ) := µ ∈ C : µ -λ µ + λ -2σ 0 < 1 -2(ℜ(λ) -σ 0 )d 2 r (λ) .
Note that the zero-free regions which appear in Theorem 2.2 are discs (empty

if d 2 r (λ) = 1/(2ℜ(λ) -2σ 0 )) or half-planes. More precisely, for λ = a + ib ∈ Π σ 0 (a > σ 0 , b ∈ R) and R ∈ [0, 1], then the set µ ∈ C : µ -λ µ + λ -2σ 0 < R is the open (euclidean) disc whose center is Ω = a+R 2 (a-2σ 0 ) 1-R 2
, b and radius is

2R(a-σ 0 ) 1-R 2 if R ∈ [0, 1[; if R = 1 this set is the half-plane Π σ 0 .
In both cases, we easily see that this set is contained in the half-plane Π σ 0 .

As a corollary of the proof of Theorem 2.2, we obtain an other explicit version.

Corollary 2.3. Let λ ∈ Π σ 0 . Then L does not vanish on the disc

r -σ 0 + µ ∈ C : µ -λ µ + λ -2σ 0 < 2(ℜ(λ) -σ 0 ) | f A,r (λ)| f A,r 2 ,
for any m L -admissible sequence A.

Note that taking L(s) = ζ(s), ℓ(α) = 2 and ϕ = χ (0,1) (so that σ 0 = σ 1 = 0) we recover exactly the results of Nikolski ([Nik95]). Now, taking ϕ(t) = (1t) -σ 1 χ (0,1) (t), we obtain larger zero-free discs whenever ℑ(λ) is large enough. We refer the reader to Section 7 for further discussions about the Rieman zeta function and other examples.

We will see in Theorem 4.3 that

f A,r (λ) = L(λ + r -σ 0 ) φ(λ + r -σ 0 ) n j=1 c j α λ+r-σ 0 j , λ ∈ Π σ 0 .
Hence Corollary 2.3 can be understood as follows: let λ ∈ Π σ 0 such that L(λ + rσ 0 ) = 0; then there is a small neighborhood of λ + rσ 0 free of zeros for L. Of course, this is an obvious consequence of the continuity of the function L but the interest of Corollary 2.3 is that it gives an explicit neighborhood where the function L does not vanish; moreover, this explicit neighborhood is expressed in terms of the values of L and in particular it does not use any estimates of the derivatives. Finally, we obtain a general Beurling-Nyman type theorem.

Theorem 2.4. Suppose that the function φ does not vanish on the half-plane Π r , that lim sup

x→+∞ log | φ(x+r-σ 0 )| x
= 0 and that a 1 = 0. Then the following assertions are equivalent:

(1) The function L does not vanish on the half-plane Π r .

(2) There exists

λ ∈ Π σ 0 such that d r (λ) = 0. (3) For all λ ∈ Π σ 0 , we have d r (λ) = 0. (4) We have K r = L 2 * ((0, 1), dt/t 1-2σ 0 ). Taking L(s) = ζ(s), ℓ(α) = 2, r = λ = 1/2 and ϕ = χ (0,1
) (so that we can take σ 0 = σ 1 = 0), we obtain Nyman's theorem. Now, taking ϕ(t) = (1t) -σ 1 χ (0,1) (t) we extend the results of [dR07a] (see also Section 7 for more details).

An other consequence of Theorem 2.2 is, in a way, a Beurling-Nyman criterion for Dirichlet L-functions. More precisely, let χ be a Dirichlet character with conductor q and L(χ, s) its L-function. Then, for 1/2 ≤ r < 1, we define d r by

d r = min ℓ,c,α   1 0 t 1-r -t r ℓ j=1 c j n<α j /t χ(n) 2 dt t   1 2
where the minimum is taken over all ℓ ≥ 0, c = (c j ) ∈ C ℓ and α = (α j ) ∈ (0, 1] ℓ . One can show (see Proposition 7.6) that d 2 r < 1/(2 -2r). We have Theorem 2.5. If

d 2 r ≤ 1 2 -2r - C 2 2(log q) 2 (1 -r) 3
for some (absolute) constant C and some 1/2 ≤ r ≤ 1, then L(χ, σ) does not vanish in the real-interval σ > 1 -C/ log q.

In order to obtain the criterion for the Siegel zero problem, then we consider all Dirichlet characters χ and an absolute constant C independant of χ.

The next section is devoted to the proof of Theorem 2.1 and to the study of the function ψ. Section 4 will focus on the admissible sequences and the functions f A,r . Theorem 2.2 and Corollary 2.3 will be proven in Section 5 and Theorem 2.4 will be proven in Section 6. Some explicit examples will be studied in Section 7 in which we will prove Theorem 2.5.

The function ψ and proof of theorem 2.1

We define the functions ψ 1 and ψ 2 by

ψ 1 (u) = res (L(s) φ(s)u s , s = 1) (u ∈ R + ), ψ 2 (u) = n<u a n ϕ n u (u ∈ R + ), so that ψ(u) = ψ 1 (u) -ψ 2 (u) by definition.
The function s → φ(s) is analytic in Π σ 0 and the meromorphic continuation of L(s) has a pole (of order m L ) only at s = 1, hence we can write

(3.1) L(s) φ(s) = m L k=1 p -k (s -1) k -H(s), with H analytic in Π σ 0 . Lemma 3.1. We have (3.2) H(s) = 1 0 ψ 1 t t s-1 dt = φ(t), ℜ(s) > 1,
where

φ(t) = ψ(1/t)χ (0,1) (t).
Proof of Lemma 3.1. On the one hand, we have

(3.3) L(s) φ(s) = +∞ 1 ψ 2 (u)u -s-1 du, ℜ(s) > 1.
Indeed, this equality comes from the following computation and Fubini's theorem:

L(s) φ(s) = n≥1 a n n s 1 0 ϕ(t)t s-1 dt = n≥1 a n 1 0 ϕ(t) t n s dt t = n≥1 a n +∞ n ϕ n u u -s-1 du = +∞ 1 n<u a n ϕ n u u -s-1 du = +∞ 1 ψ 2 (u)u -s-1 du.
Note that Fubini's theorem can be applied here because σ = ℜ(s) > 1 and then

n≥1 |a n | n σ 1 0 |ϕ(t)|t σ-1 dt < +∞.
On the other hand, for each 1 ≤ k ≤ m L , we have

(3.4) res u s (s -1) k , s = 1 = u(log u) k-1 (k -1)! ,
and an easy induction argument gives

1 (k -1)! (log u) k-1 u -s du = 1 (s -1) k .
Hence, with the notation of (3.1), the equality

(3.5) +∞ 1 ψ 1 (u)u -s-1 du = m L k=1 p -k (s -1) k .
follows by linearity.

Equations (3.3) and (3.5) imply

H(s) = m L k=1 p -k (s -1) k -L(s) φ(s) = +∞ 1 ψ(u)u -s-1 du = 1 0 ψ 1 t t s-1 dt.
Remark 3.2. From (3.1) and (3.4), we easily see that we can write ψ 1 (t) = tP (log t) where P is a polynomial of degree < m L (P ≡ 0 if m L = 0). More precisely we have

ψ 1 (t) = t m L k=1 p -k (k -1)! (log t) k-1 .
Lemma 3.3. The function s → H(s) is of finite order on Π r . Moreover, for all σ > 1, the function t → H(σ + it) belongs to L 2 (R) and

R |H(σ + it)| 2 dt = O 1 σ 1-2σ 1 for σ → +∞. Proof of Lemma 3.3. The function s → H(s) is bounded on some neighborhood V 1 of s = 1. The functions s → j p -j (s-1) j and (by assumption) s → L(s) are of finite order on Π σ 0 V 1 . Moreover, s → φ(s) is bounded on the closure of Π r since | φ(s)| ≤ 1 0 |ϕ(x)|x r dx x , ℜ(s) ≥ r,
and the last integral is convergent because r > σ 0 and ϕ(x) = O(x -σ 0 (1x) -σ 1 ).

Hence we can conclude that H is of finite order on Π r , which proves the first part of the lemma.

Let σ > 1 be a fixed real number. We easily check that, for every 1 ≤ j ≤ m L , the function t -→ (σ -1 + it) -j belongs to L 2 (R) and we have

R 1 (σ + it -1) j 2 dt = O 1 σ 2j-1 , σ → +∞. (3.6)
On the other hand, using Plancherel's formula and the fact that σ 1 < 1/2 and σ > 1 > σ 0 , we have

R | φ(σ + it)| 2 dt = 2π 1 0 |ϕ(x)| 2 x 2σ dx x < +∞. (3.7) More precisely 1 0 |ϕ(x)| 2 x 2σ dx x = O 1 σ 1-2σ 1 , σ → +∞. (3.8) Indeed, we have ϕ(x) = O(x -σ 0 (1 -x) -σ 1 ) and 1 0 |x -σ 0 (1 -x) -σ 1 | 2 x 2σ dx x = β(2(σ -σ 0 ), 1 -2σ 1 )
where β(s, z) is the beta function. Using Stirling's formula, we have

β(2(σ -σ 0 ), 1 -2σ 1 ) ∼ Γ(1 -2σ 1 ) (2σ) 1-2σ 1 , σ → +∞.
So we get the estimate (3.8). Therefore t -→ φ(σ + it) belongs to L 2 (R) and we have

(3.9) R | φ(σ + it)| 2 dt = O 1 σ 1-2σ 1 , σ → +∞.
It remains to notice that s → L(s) is bounded in Π σ (since by hypothesis σ > 1) and then the second part of the lemma follows immediately from (3.6), (3.9) and (3.1). Now we can prove Theorem 2.1.

Assume that (i) is satisfied. Then the function φ(t) = ψ 1 t χ (0,1) (t) belongs to L 2 * ((0, 1), dt t 1-2r ) and thus the function G := φ belongs to H 2 (Π r ). The analytic continuation principle implies that the equality (3.2) is satisfied for every s ∈ Π r , that is

H(s) = φ(s) = G(s), s ∈ Π r . Since G ∈ H 2 (Π r ), we know that the function G * , defined by G * (t) := lim σ→ > r G(σ + it),
exists almost everywhere on R and belongs to L 2 (R). But

G * (t) = lim σ→ > r H(σ + it) = H(r + it),
because H is continuous on the closed half-plane ℜ(s) ≥ r. Therefore t -→ H(r+it) belongs to L 2 (R). It remains to notice that for every 1 ≤ k ≤ m L , the function t -→ (r -1 + it) -k belongs to L 2 (R) (this is where we have to assume that r = 1 whenever m L ≥ 1). Thus according to (3.1), we get that t → L(r + it) φ(r + it) belongs to L 2 (R).

Conversely assume that (ii) is satisfied. Then the function t -→ H(r

+ it) belongs to L 2 (R). Let σ 2 > max(1, r).
The function H is analytic on Ω := {s : r < ℜ(s) < σ 2 } and continuous on the closure of Ω. Using the Hardy, Littlewood and Ingham's theorem [HIP27, Theorem 7], we deduce from Lemma 3.3 that sup

r≤σ≤σ 2 R |H(σ + it)| 2 dt < +∞.
Hence, the second part of Lemma 3.3 gives that H belongs to H 2 (Π r ). Since we have

H 2 (Π r ) = M L 2 * (0, 1), dt t 1-2r
, there exists

φ 1 ∈ L 2 * (0, 1), dt t 1-2r
such that H(s) = φ1 (s), for every s ∈ Π r . Thus for ℜ(s) > max(1, r), we have φ1 (s) = H(s) = φ(s).

By injectivity of the Mellin transform, we get that

ψ 1 t χ (0,1) (t) = φ(t) = φ 1 (t).
Thus t -→ ψ 1 t χ (0,1) (t) belongs to L 2 * (0, 1), dt t 1-2r , which implies that ψ belongs to L 2 (1, +∞), du u 1+2r and that concludes the proof of Theorem 2.1.

Corollary 3.4. Let r > 1. Then ψ ∈ L 2 (1, +∞), du u 1+2r and we have

ψ 2 = O r σ 1 -1/2 , as r → +∞.
Proof. Let r > 1 be a fixed real number. By Lemma 3.3, the function t -→ H(r +it) belongs to L 2 (R) and for every 1 ≤ j ≤ m L , the function t -→ (r -1 + it) -j belongs also to L 2 (R). Therefore it follows from (3.1) that t -→ L(r + it) φ(r + it) belongs to L 2 (R) and Theorem 2.1 implies that ψ ∈ L 2 ((1, +∞), du u 1+2r ). Moreover, if we let as before φ(t) = ψ(1/t)χ (0,1) (t), we have by Plancherel's formula

ψ 2 2 = +∞ 1 |ψ(u)| 2 du u 1+2r = 1 0 ψ 1 t 2 dt t 1-2r = φ 2 L 2 * ((0,1), dt t 1-2r ) = 1 2π R | φ(r + it)| 2 dt = 1 2π R |H(r + it)| 2 dt,
and Lemma 3.3 gives the result.

Remark 3.5. In fact, if L(s) has some nice arithmetical properties (e.g. satisfying a Wiener-Ikehara type theorem), one can expect that the main contributions of ψ 1 will be compensated by the main contributions of ψ 2 and so that the function ψ will belong to the space L 2 (1, +∞), du u 1+2r for smaller value of r. For example, if L(s) = ζ(s) and if ϕ(t) = χ 0,1 (t) for all t ∈ (0, 1), then ψ(u) = u -⌈u⌉ + 1 (so ψ(u) = {u} for almost all u ∈ (1, ∞)) and ψ ∈ L 2 (1, +∞), du u 1+2r for all r > 0. In this case φ(s) = 1/s and t → |ζ(r + it)|/|r + it| ∈ L 2 (R) for all r > 0 as expected by Theorem 2.1. We will discuss about other examples in Section 7.

Remark 3.6. Assume that ϕ is bounded at t = 1 (which corresponds to σ 1 = 0). Thus, for every α > 1, we have ψ(t) = ψ 1 (t)ψ 2 (t) = O(t α ), t → +∞. Indeed, on the one hand, as we have seen ψ 1 (t) = tP (log t), where P is a polynomial of degree < m L . On the other hand, let α > 1 be a fixed real number. Then there exists a constant C > 0 such that |ϕ(t)| ≤ Ct -α , t ∈ (0, 1).

Hence we get

|ψ 2 (t)| ≤ n<t |a n | ϕ n t ≤ C n<t |a n | t n α ≤ C ′ t α n<t 1 n α-ε , which gives that ψ 2 (t) = O(t α ) as t → +∞.
It is now easy to check that ψ belongs to L 2 ((1, +∞), du u 1+2r ), for every r > 1 and ψ 2 = O(r -1/2 ), r → +∞. We recover Corollary 3.4.

Admissible sequences and the functions f A,r

The following results will be useful in the sequel.

Lemma 4.1. Let (α, c) be an m-admissible sequence.

(1) For any polynomial P of degree d < m, we have ℓ(α) j=1 c j α j P (log α j ) = 0.

(2) For any positive real number λ 1 and for any real number λ 2 ≥ max j (α j ), both ((α λ 1 j ) j , (c j α 1-λ 1 j ) j ) and ((α j /λ 2 ) j , c) are m-admissible sequences.

The proof is easy and follows immediately from the definition and so we omit it. For an m-admissible sequence A = (α, c) we define the entire function

g A (s) = ℓ(α) j=1 c j α s j (s ∈ C).
Using (2.2), we notice that these functions satisfy (4.1)

g (k) A (1) = 0 (0 ≤ k ≤ m -1).
Lemma 4.2. Let m be a nonnegative integer. Then we have the followings

(1) For every integer ℓ ≥ m + 1 and every

0 < α 1 < α 2 < • • • < α ℓ ≤ 1, there exists c ∈ C ℓ such that A = (α, c
) is an m-admissible sequence and g (m)

A (1) = 0. Furthermore we can choose c ℓ = 0. (2) For every s 1 ∈ C {1}, there exists an m-admissible sequence A such that g A (s 1 ) = 0.

Proof.

(1) Let ℓ ≥ m + 1, then there is a unit vector

(b 1 , b 2 , . . . , b m , b ℓ ) ∈ C m+1 such that     1 . . . . . . 1 1 (log α 1 ) . . . . . . (log α m ) (log α ℓ ) . . . . . . . . . (log α 1 ) m . . . . . . (log α m ) m (log α ℓ ) m         b 1 . . . b m b ℓ     =     0 . . . 0 1     , (4.2)
since the corresponding van der Monde matrix is invertible. Setting

c j =      b j α j , 1 ≤ j ≤ m 0, m + 1 ≤ j ≤ ℓ -1 b ℓ α ℓ , j = ℓ.
Then A = (α, c) is an m-admissible sequence and g (2) Let A = (α, c) be a non trivial m-admissible sequence. For 0 < λ < 1, consider

g A λ (s) = g A (1 -λ + λs). Note that g A λ (s) = ℓ j=1 c j α 1-λ j (α λ j ) s ,
and, by Lemma 4.1, A λ = ((α λ j ) j , (c j α 1-λ j ) j ) is an m-admissible sequence. Now let s 1 = 1; then there necessarily exists 0 < λ < 1 such that g A λ (s 1 ) = 0, since g A is analytic and non identically zero.

Theorem 4.3. Let r > σ 0 such that ψ ∈ L 2 ((1, +∞), du u 1+2r ) and let A = (α, c) be an m L -admissible sequence. We define

f A,r (t) = t r-σ 0 ℓ(α) j=1 c j ψ α j t (t > 0).

Then we have:

(1) f A,r (t) = 0, if t > max j α j .

(2) f A,r (t) ∈ L 2 * ((0, 1), dt t 1-2σ 0 ). (3) For ℜ(s) > σ 0 , we have

f A,r (s) = -L(s + r -σ 0 ) φ(s + r -σ 0 )g A (s + r -σ 0 ). (4.3) Proof.
(1) As before, we write ψ = ψ 1ψ 2 . From the remark 3.2, we have ψ 1 (u) = uP (log u) where P is a polynomial of degree < m L . Then

ℓ(α) j=1 c j ψ α j t = ℓ(α) j=1 c j ψ 1 α j t - ℓ(α) j=1 c j ψ 2 α j t = ℓ(α) j=1 c j α j t P log( α j t ) - ℓ(α) j=1 c j n<α j /t
a n ϕ nt α j .

If t > max j (α j ) the second sum is 0 because α j /t < 1. Furthermore, by Lemma 4.1, (α/t, c) is an m L -admissible sequence and the first sum is also zero.

(2) Each of the terms t r-σ 0 ψ α j t belongs to the space L 2 ((0, 1), dt t 1-2σ 0 ). Indeed, if α := max j α j , we have

α 0 t 2r-2σ 0 ψ α j t 2 dt t 1-2σ 0 = α 0 t 2r-1 ψ α j t 2 dt = α 2r j +∞ α j α |ψ(u)| 2 du u 1+2r = α 2r j 1 α j α |ψ(u)| 2 du u 1+2r + α 2r j +∞ 1 |ψ(u)| 2 du u 1+2r = α 2r j 1 α j α |ψ 1 (u)| 2 du u 1+2r + α 2r j +∞ 1 |ψ(u)| 2 du u 1+2r ,
since ψ(u) = ψ 1 (u) for u < 1. Now, by hypothesis, the second integral is finite and the first integral is also finite because ψ 1 (u) = uP (log u) and

α j α > 0. Furthermore, we see that (4.4) f A,r L 2 * ((0,1), dt t 1-2σ 0 ) ≤ ℓ(α) j=1 |c j α r j |      1 min j α j max j α j |ψ 1 (u)| 2 du u 1+2r   1/2 + ψ L 2 ((1,+∞), du u 1+2r )    .
(3) For ℜ(s) > 1, we have by (3.3)

L(s) φ(s) = +∞ 1 ψ 2 (u)u -s-1 du. Hence L(s) φ(s) ℓ(α) j=1 c j α s j = ℓ(α) j=1 c j +∞ 1 u α j -s ψ 2 (u) du u = ℓ(α) j=1 c j α j 0 ψ 2 α j t dt t 1-s = ℓ(α) j=1 c j 1 0 ψ 2 α j t dt t 1-s , the last equality follows from ψ 2 (α j /t) = 0 if α j < t ≤ 1.
Using once more Lemma 4.1, we have

ℓ(α) j=1 c j ψ α j t = - ℓ(α) j=1 c j ψ 2 α j t , whence L(s) φ(s) ℓ(α) j=1 c j α s j = - 1 0   ℓ(α) j=1 c j ψ α j t   dt t 1-s = - 1 0 t σ 0 -r f A,r (t) dt t 1-s = - 1 0 f A,r (t)t s+σ 0 -r-1 dt = -f A,r (s + σ 0 -r), ℜ(s) > 1. So, for ℜ(s) > 1 + σ 0 -r, we have f A,r (s) = -L(s + r -σ 0 ) φ(s + r -σ 0 )g A (s + r -σ 0 ).
By the analytic continuation principle, the equality holds for ℜ(s) > σ 0 . (Note that from (4.1) the pole of L(s + rσ 0 ) is killed by the zero of g A (s + rσ 0 ) at s = 1r + σ 0 .)

Zeros free regions for Dirichlet series

Before proving Theorem 2.2 we need some well known tools concerning the Hardy space H 2 (Π σ 0 ). For these we refer to [START_REF] Hoffman | Banach spaces of analytic functions[END_REF]Chapter 8]. Actually, in [START_REF] Hoffman | Banach spaces of analytic functions[END_REF], the following facts are stated for the space H 2 (Π 0 ) but it is easy to obtain the corresponding results for H 2 (Π σ 0 ), for instance using the unitary map h(z)

-→ h(z -σ 0 ) from H 2 (Π σ 0 ) onto H 2 (Π 0 ). Recall that if h ∈ H 2 (Π σ 0 ), then h * (σ 0 + it) := lim σ→σ 0 > h(σ + it)
exists for almost every t ∈ R (with respect to the Lebesgue measure). Moreover we have h * ∈ L 2 (σ 0 + iR) and h 2 = h * 2 . We can therefore identify (unitarely) H 2 (Π σ 0 ) with a (closed) subspace of L 2 (σ 0 + iR). In the following we use the symbol h not only for the function in H 2 (Π σ 0 ) but also for its "radial " limit (in other words we forget the star). This identification enables us to consider H 2 (Π σ 0 ) as an Hilbert space, with scalar product given by

h, g 2 = +∞ -∞ h(σ 0 + it)g(σ 0 + it) dt, h, g ∈ H 2 (Π σ 0 ). (5.1)
Now for λ ∈ Π σ 0 , we have the following integral representation

h(λ) = 1 2π +∞ -∞ h(σ 0 + it) λ -σ 0 -it dt,
so that we can write, using (5.1),

h(λ) = h, k λ 2 , (5.2)
where k λ is the function in H 2 (Π σ 0 ) defined by

k λ (z) := 1 2π 1 z -2σ 0 + λ , z ∈ Π σ 0 . (5.3)
The function k λ is called the reproducing kernel of H 2 (Π σ 0 ) and we have

k λ 2 2 = k λ , k λ 2 = k λ (λ) = 1 4π(ℜ(λ) -σ 0 ) . (5.4)
Recall now a useful property of factorization for functions in the Hardy space.

Lemma 5.1. Let h ∈ H 2 (Π σ 0 ) and µ ∈ Π σ 0 such that h(µ) = 0. Then there exists g ∈ H 2 (Π σ 0 ) such that h 2 = g 2 and h(z) = z -µ z + μ -2σ 0 g(z), z ∈ Π σ 0 .
Proof. See the results in [Hof62, Chapter 8, pp. 132] and apply the transform

h(z) -→ h(z -σ 0 ).
Proof of Theorem 2.2 Denote by E r = MK r . Since the Mellin transform is a unitary map from L 2 * ((0, 1), dt t 1-2σ 0 ) onto H 2 (Π σ 0 ), we have E r = span H 2 (Πσ 0 ) (h A,r : A a m L -admissible sequence) , where h A,r (s) = Mf A,r (s), for ℜ(s) > σ 0 . It follows from Theorem 4.3 that

h A,r (s) = - 1 √ 2π L(s + r -σ 0 ) φ(s + r -σ 0 )g A (s + r -σ 0 ), ℜ(s) > σ 0 .
Now assume that there is µ ∈ Π σ 0 such that L(µ + rσ 0 ) = 0. Since µ ∈ Π σ 0 , we get that h A,r (µ) = 0, for every m L -admissible sequence A. Thus, for all h ∈ E r with h 2 = 1, we have h(µ) = 0. Since h ∈ H 2 (Π σ 0 ), we know from Lemma 5.1 that there is g ∈ H 2 (Π σ 0 ) such that g 2 = h 2 = 1 and

h(z) = z -µ z + μ -2σ 0 g(z) (z ∈ Π σ 0 ).
Hence with Cauchy-Schwarz inequality and (5.2), we deduce that

|h(λ)| = λ -µ λ + μ -2σ 0 |g(λ)| ≤ λ -µ λ + μ -2σ 0 g 2 k λ 2 = λ -µ λ + μ -2σ 0 k λ 2 , so sup h∈Er h 2 =1 |h(λ)| ≤ µ -λ µ + λ -2σ 0 k λ 2 .
By contraposition, we have proved that L does not vanish on

r -σ 0 +    µ ∈ C : µ -λ µ + λ -2σ 0 < sup f ∈Er h 2 =1 |h(λ)| k λ 2    .
It remains to prove that

sup h∈Er h 2 =1 |h(λ)| 2 k λ 2 2 = 1 -2(ℜ(λ) -σ 0 )d 2 r (λ).
To show this equality, first remark that sup

h∈Er h 2 =1 |h(λ)| = sup h∈Er h 2 =1 | h, k λ 2 | = P Er k λ 2 ,
where P E denotes the orthogonal projection onto the closed subspace E of H 2 (Π σ 0 ). Then Pythagoras' Theorem implies that sup

h∈Er h 2 =1 |h(λ)| 2 = P Er k λ 2 2 = k λ 2 2 -P E ⊥ r k λ 2 2 . Hence sup h∈Er h 2 =1 |h(λ)| 2 k λ 2 2 = 1 - dist 2 (k λ , E r ) k λ 2 2
.

But for ℜ(s) > σ 0 , we have

M t λ-2σ 0 χ (0,1) (t) (s) = 1 √ 2π 1 0 t λ-2σ 0 t s-1 dt = 1 √ 2π 1 λ -2σ 0 + s = √ 2πk λ (s),
according to (5.3). Since the Mellin transform is an isometry from L 2 * ((0, 1),

dt t 1-2σ 0 ) onto H 2 (Π σ 0 ), we obtain dist(k λ , E r ) = 1 √ 2π dist(t λ-2σ 0 χ (0,1) (t), K r ) = d r (λ) √ 2π .
We conclude the proof of Theorem 2.2 using k λ

2 2 = 1 4π(ℜ(λ)-σ 0 ) .
Proof of Corollary 2.3 It follows from the proof of Theorem 2.2 and (5.4) that L

does not vanish on

r -σ 0 +      µ ∈ C : µ -λ µ + λ -2σ 0 < 4π(ℜ(λ) -σ 0 ) sup h∈Er h 2 =1 |h(λ)|      .
So in particular (with h = h A,r ), we get that L does not vanish on

r -σ 0 + µ ∈ C : µ -λ µ + λ -2σ 0 < 4π(ℜ(λ) -σ 0 ) |h A,r (λ)| h A,r 2 , which proves corollary 2.3 because h A,r = Mf A,r = 1 √ 2π f A,r .
The following could be interesting in applications.

Corollary 5.2. Let M be a subspace of K r and λ ∈ Π σ 0 . Then L does not vanish on the disc

r -σ 0 + µ ∈ C : µ -λ µ + λ -2σ 0 2 < 1 -2(ℜ(λ) -σ 0 )dist 2 (t λ-2σ 0 χ (0,1) , M) .
Proof. It is sufficient to note that dist(t λ-2σ 0 χ (0,1) , M) ≥ dist(t λ-2σ 0 χ (0,1) , K r ) = d r (λ) and then apply Theorem 2.2.

Adapting the proof of Theorem 2.2, we could obtain immediately the following generalization.

Theorem 5.3. Let k ∈ N and λ ∈ Π σ 0 . The function L does not have any zero of order greater or equal to k on

r -σ 0 + µ ∈ C : µ -λ µ + λ -2σ 0 < 1 -2(ℜ(λ) -σ 0 )d 2 r (λ) 1 2k
.

A Beurling-Nyman type theorem for Dirichlet series

One of the main steps for proving Theorem 2.4 is to prove that the space E r is a closed subspace of H 2 (Π σ 0 ) that is invariant under multiplication operator τ v , v ≥ 0, and then to apply Lax-Beurling's theorem. Before recalling this theorem, we give some notations and results. We refer to [Nik02, Part A, Chap. 2 & 6] for more details.

For v ∈ R, let τ v be the operator of multiplication on L 2 (σ 0 + iR) defined by

(τ v f )(σ 0 + it) = e -ivt f (σ 0 + it) (f ∈ L 2 (σ 0 + iR)).
We also denote by H ∞ (Π σ 0 ) the Hardy space of bounded analytic functions on Π σ 0 ; as in H 2 (Π σ 0 ), functions in H ∞ (Π σ 0 ) admit "radial" limits at almost every points of the boundary ℜ(s) = σ 0 and if f ∈ H ∞ (Π σ 0 ) and f * is its boundary limits, then

f * ∈ L ∞ (σ 0 +iR) and f * ∞ = f ∞ (= sup z∈Πσ 0 |f (z)|). A function Θ in H ∞ (Π σ 0 ) is said to be inner if |Θ(σ 0 + it)| = 1 for almost every point t ∈ R.
It is easy to see that if Θ is inner and E = ΘH 2 (Π σ 0 ), then E is a closed subspace of H 2 (Π σ 0 ) invariant by τ v , v ≥ 0. Lax-Beurling's theorem gives the converse. Theorem 6.1 (Lax-Beurling). Let E be a closed subspace of H 2 (Π σ 0 ) such that τ v E ⊂ E, ∀v ≥ 0. Then there is an inner function Θ ∈ H ∞ (Π σ 0 ) unique (up to a constant of modulus one) such that E = ΘH 2 (Π σ 0 ). Remark 6.2. When E is spanned by a family of functions, we can precise a little bit more the conclusion of Theorem 6.1. Indeed, let E be a closed subspace of H 2 (Π σ 0 ) spanned by a family of functions (f i ) i∈I , f i ∈ H 2 (Π σ 0 ) and let f i = h i g i be the factorization of f i in inner factor h i and outer factor g i . If τ v E ⊂ E, ∀v ≥ 0, then E = ΘH 2 (Π σ 0 ), where Θ = gcd(h i : i ∈ I) is the greatest common inner divisor of the family (h i ) i∈I .

Proof of Theorem 2.4 First of all, note that

h∈Er h -1 ({0}) = {s ∈ Π σ 0 : L(s + r -σ 0 ) = 0}, (6.1)
where we recall that

E r = MK r = span H 2 (Πσ 0 ) (h A,r : A a m L -admissible sequence) , and h A,r (s) = - 1 √ 2π L(s + r -σ 0 ) φ(s + r -σ 0 )g A (s + r -σ 0 ), ℜ(s) > σ 0 .
Indeed, the first inclusion

{s ∈ Π σ 0 : L(s + r -σ 0 ) = 0} ⊂ h∈Er h -1 ({0})
is trivial. For the converse inclusion, it is sufficient to notice that according to Lemma 4.2, the only common zeros to functions g A (s + rσ 0 ) is s = 1r + σ 0 but this zero is compensated by the singularities of L(s + rσ 0 ) at this point. The proofs of (4) =⇒ (3) and (3) =⇒ (2) are trivial.

(2) =⇒ (1): let λ ∈ Π σ 0 such that d r (λ) = 0. Then according to Theorem 2.2, the function L does not vanish on

r -σ 0 + µ ∈ C : µ -λ µ + λ -2σ 0 < 1 ,
and it follows from the paragraph after Theorem 2.2 that this region is precisely the half-plane Π r .

(1) =⇒ (4): denote by S β the "shift" on L 2 * ((0, 1), dt t 1-2σ 0 ) defined by

(S β f )(t) := β -σ 0 f t β , 0 < β ≤ 1, 0 ≤ t ≤ 1.
It is clear that S β is a unitary operator on L 2 * ((0, 1), dt t 1-2σ 0 ). We will show that S β K r ⊂ K r (0 < β ≤ 1). (6.2) c) is a m L -admissible sequence and 0 ≤ t ≤ 1, we have

If A = (α,
(S β f A,r )(t) =β -σ 0 f A,r t β =β -σ 0 t r-σ 0 β r-σ 0 ℓ(α) j=1 c j ψ k α j β t =β -r f A ′ ,r (t),
where A ′ = (α ′ , c) is the m L -admissible sequence with α ′ = (α ′ j ) j = (βα j ) j and so f A ′ ,r ∈ K r . Hence for every admissible sequence A, we have proved that S β f A,r ∈ K r . Since S β is a bounded operator and the functions f A,r span the subspace K r , we deduce (6.2). Therefore we obtain that E r = MK r is a closed subspace of H 2 (Π σ 0 ) which is invariant under the semi-group of operators MS β M -1 , 0 < β ≤ 1. Now let us show that

MS β M -1 = τ v , (6.3) with v = -log β. If h ∈ H 2 (Π σ 0 ) and if f ∈ L 2 * ((0, 1), dt t 1-2σ 0 ) is such that M -1 h = f , we have for ℜ(s) > σ 0 , (MS β M -1 h)(s) = 1 √ 2π ∞ 0 (S β f )(t)t s-1 dt = 1 √ 2π +∞ 0 β -σ 0 f t β t s-1 dt =β s-σ 0 1 √ 2π +∞ 0 f (u)u s-1 du =β s-σ 0 (Mf )(s) = β s-σ 0 h(s).
Hence we have

(MS β M -1 h)(σ 0 + it) = β it h(σ 0 + it) = e it log β h(σ 0 + it) = (τ v h)(σ 0 + it),
with v =log β, which proves (6.3). We therefore obtain that τ v E r ⊂ E r , for all v ≥ 0 and then Lax-Beurling's Theorem (see Theorem 6.1) implies that there is an inner function Θ in the half-plane Π σ 0 such that E r = ΘH 2 (Π σ 0 ). Moreover we know that Θ = BS, where B is a Blaschke product and S is a singular inner function [START_REF] Nikolski | Operators, functions, and systems: an easy reading[END_REF]). But according to Remark 6.2, the zeros of B coincide with the set of common zeros of functions h ∈ E r . Hence it follows from (6.1) and the hypothesis that B has no zeros. In other words, B ≡ 1. We will now show that S ≡ 1. First note that since h A,r is analytic on ℜ(s) > 2σ 0r and since S is a common inner divisor of all functions in E r , it follows that S can be continued analytically through the axis σ 0 + iR. In particular, this forces S to be of the form S(s) = e -a(s-σ 0 ) , for some a ≥ 0 (see for instance [Nik02, Part A, Chap. 4 & 6]). Now let h A,r be a function in E r and write h A,r = Sh, with h ∈ H 2 (Π σ 0 ).

Lemma 6.3. Let h ∈ H 2 (Π σ 0 ). Then lim sup σ→+∞ log |h(σ)| σ ≤ 0.
Proof of lemma 6.3. Using (5.2) and (5.4), we get

|h(σ)| ≤ h 2 k σ 2 = h 2 2 √ π(σ -σ 0 ) 1/2 , for every σ > σ 0 . Hence log |h(σ)| σ ≤ O(1) σ - 1 2 log(σ -σ 0 ) σ ,
which gives the result letting σ → +∞.

On the one hand, according to the previous lemma, we have

h A,r (x) = L(x + r -σ 0 ) φ(x + r -σ 0 ) ℓ(α) j=1 c j α x+r-σ 0 j , we can assume, by Lemma 4.2, that 0 < α 1 < • • • < α ℓ(α) = 1 and that c ℓ(α) = 0. Then we have |h A,r (x)| ∼ |a 1 || φ(x + r -σ 0 )c ℓ(α) |, x → +∞.
Hence we can find x 0 > 1 such that

|h A,r (x)| ≥ |a 1 | 2 | φ(x + r -σ 0 )c ℓ(α) |, (x > x 0 ), which gives log |h A,r (x)| ≥ -log 2 + log |a 1 | + log |c ℓ(α) | + log | φ(x + r -σ 0 )|, (x > x 0 ).
Therefore we obtain that lim sup

x→+∞ log |h A,r (x)| x ≥ lim sup x→+∞ log | φ(x + r -σ 0 )| x = 0.
Using (6.4), we conclude that a = 0. Hence S ≡ 1, which gives E r = H 2 (Π σ 0 ), that is K r = L 2 * ((0, 1), dt t 1-2σ 0 ). Remark 6.4. In fact, with the hypothesis on ϕ, one can easily show that we always have lim sup x→+∞ log | φ(x+r-σ 0 )| x ≤ 0. Note that there exists some ϕ for which the lim sup is negative.

ϕ(t) = (1 -t) -σ 1 if 0 ≤ t < 1 0 if t ≥ 1 ,
where σ 1 < 1/2 is fixed. Then an elementary computation shows that φ(s) = Γ(s)Γ(1-σ 1 ) Γ(s+1-σ 1 ) , ℜ(s) > 0. Hence,

(7.1) | φ(s)| ∼ Γ(1 -σ 1 ) |t| 1-σ 1 as |t| = |ℑ(s)| → ∞.
We now illustrate 1 the use of Theorem 2.1 in order to determine the values of r such that

ψ ∈ L 2 (1, ∞), du u 1+2r .
An obvious computation shows that

ψ(u) = u 1 -σ 1 - n<u 1 1 -n u σ 1 , u > 0.
From this, we deduce that for r > σ 1

ψ 2 2 = ∞ k=1 k+1 k u 1-σ 1 1 -σ 1 - k n=1 (u -n) -σ 1 2 du u 1+2r-2σ 1 ≤ ∞ k=1 1 k 1+2r-2σ 1 k+1 k u 1-σ 1 1 -σ 1 - k n=1 (u -n) -σ 1 2 du ≤ C(σ 1 ) ∞ k=1 1 k 1+2r-2σ 1 < +∞.
As far as C(σ 1 ) is concerned, we have the following bound

C(σ 1 ) = 1 0 (u + 1) 2-2σ 1 (1 -σ 1 ) 2 du + 2 1 -σ 1 1 0 (1 + u) 1-σ 1 u σ 1 du + 1 0 du u 2σ 1 ≤ 1 0 (u + 1) 2-2σ 1 (1 -σ 1 ) 2 du + 2 2-σ 1 1 -σ 1 1 0 du u σ 1 + 1 0 du u 2σ 1 ≤ 2 3-2σ 1 (3 -2σ 1 )(1 -σ 1 ) 2 + 2 2-σ 1 (1 -σ 1 ) 2 + 1 1 -2σ 1 = C(σ 1 ),
which proves the first part of the theorem. On the other hand, suppose that σ 1 > 0. (If σ 1 ≤ 0 there is nothing to prove since by hypothesis r > σ 0 and σ 0 = 0 in our case).

It sufficies to find U(σ 1 ) > 0 such that

k+1 k u 1-σ 1 1 -σ 1 - k n=1 (u -n) -σ 1 2 du ≥ U(σ 1 )
uniformely in k. In that case, we shall have

ψ 2 2 = ∞ k=1 k+1 k u 1-σ 1 1 -σ 1 - k n=1 (u -n) -σ 1 2 du u 1+2r-2σ 1 ≥ ∞ k=1 1 (k + 1) 1+2r-2σ 1 k+1 k u 1-σ 1 1 -σ 1 - k n=1 (u -n) -σ 1 2 du ≥ U(σ 1 ) ∞ k=1 1 (k + 1) 1+2r-2σ 1 hence r > σ 1 . It now remains to prove k+1 k u 1-σ 1 1 -σ 1 - k n=1 (u -n) -σ 1 2 du ≥ U(σ 1 )
uniformely in k. From (7.2), we see that we have in fact

u 1-σ 1 1 -σ 1 - k n=1 (u -n) -σ 1 = - 1 (u -k) σ 1 + κ(u) 1 -σ 1 u ∈ (k, k +1], k ≥ 1, σ 1 < 1/2 , where 0 ≤ κ(u) ≤ (u -k + 1) 1-σ 1 for all u ∈ (k, k + 1].
We deduce that for any 0 < δ < 1

k+1 k u 1-σ 1 1 -σ 1 - k n=1 (u -n) -σ 1 2 du ≥ k+δ k u 1-σ 1 1 -σ 1 - k n=1 (u -n) -σ 1 2 du ≥ k+δ k 1 (u -k) 2σ 1 -2 (u -k + 1) 1-σ 1 (1 -σ 1 )(u -k) σ 1 du ≥ k+δ k 1 (u -k) 2σ 1 -2 (1 + δ) 1-σ 1 (1 -σ 1 )(u -k) σ 1 du ≥ δ 1-2σ 1 1 -2σ 1 - 2(1 + δ) 1-σ 1 δ 1-σ 1 (1 -σ 1 ) 2 ≥ δ 1-2σ 1 1 1 -2σ 1 - 2(1 + δ) 1-σ 1 δ σ 1 (1 -σ 1 ) 2 .
We have the expected result by choosing U(σ 1 ) := δ 1-2σ 1 1 1 -2σ 1 -2(1 + δ) 1-σ 1 δ σ 1 (1σ 1 ) 2 with δ > 0 small enough so that 2(1 + δ) 1-σ 1 δ σ 1 (1σ 1 ) 2 < 1 1 -2σ 1 .

7.1.1. Zero-free discs for ζ. Let r > max(0, σ 1 ), r = 1. Then according to Theorem 7.2, the function ψ ∈ L 2 (1, ∞), du u 1+2r . Let A = (α, c) be a 1-admissible sequence, which means that ℓ(α) j=1 c j α j = 0, for all A = (α, c) where ℓ ≥ 0, α ∈ (0, 1] ℓ and c ∈ C ℓ . Let r such that 1/2 ≤ r < 1 and λ = 1r. For simplicity, we write d r = d r (λ) so that Furthermore, we will write d r,χ if we need to specify the dependance on χ. Remark that we have trivially d 2 r ≤ 1/(2 -2r) (just take all the c j to be 0). In fact, we can be a little bit more precise.

Proposition 7.6. We have d 2 r < 1/(2 -2r). Proof. Let c ∈ C and 0 < α ≤ 1. A short calculation gives

d 2 r ≤ 1 0 t 1-r -ct r n<α/t χ(n) 2 dt t = 1 2 -2r -2αℜ(cL(χ, 1)) + |c| 2 α 2r+1 ∞ 1 n<u χ(n) 2 du u 1+2r .
Since we know that L(χ, 1) = 1 (see for example [IK04, p. 37]), one can choose a suitable c such that the right hand side of the equation above is less than 1/(2 -2r). Theorem 2.2 asserts that L(χ, s) does not vanish on the disc (7.4) r + µ ∈ C : µ + r -1 µr + 1 < 1 -2(1r)d 2 r .

Note that the proposition above implies that the disc is not empty. We deduce, in particular, that L(χ, σ) does not vanish on the real-interval

σ > 1 - 1 -2(1 -r)d 2 r -(1 -2(1 -r)d 2 r ) d 2 r .
(We can easily check that the disc (7.4) contains σ = 1 and it is known that L(χ, σ) = 0 for σ ≥ 1.) Hence, L(χ, σ) does not vanish on

σ > 1 -(1 -r) 1 -2(1 -r)d 2 r . (7.5)
We expect (by the Riemann hypothesis for Dirichlet functions) that d r = 0 which would imply that L(χ, s) does not vanish on Π r . If we had "only"

d 2 r ≤ 1 2 -2r - C 2 2(log q) 2 (1 -r) 3
for some (absolute) constant C, it would imply (using (7.5)) that L(χ, σ) does not vanish in the real-interval σ > 1 -C/ log q. That proves Theorem 2.5 and we also get immediately the following.

Theorem 7.7. If there exists an absolute constant C such that for all (real) character χ there exists r ∈ [1/2, 1) with d 2 r,χ < 1 2 -2r -C 2 2(log cond(χ)) 2 (1r) 3 , then there is no Siegel's zero for Dirichlet L-functions.

The previous theorem can be seen as a Beurling-Nyman criterion for Siegel's conjecture. 7.2.1. Explicit zero-free discs for L(χ, s). We now apply Corollary 2.3. Note that for the special choice of ϕ we have done (ϕ = χ (0,1) ), we have φ(s) = 1/s. Moreover, since the function ψ is bounded, say ψ ≤ B, we have ψ L 2 ((1,∞),du/u 1+2r ) ≤ B/ √ 2r.

Then Corollary 2.3 asserts that s → L(χ, s) does not vanish in the disc

r + µ ∈ C : µ -λ µ + λ < 2ℜ(λ) |L(χ, λ + r)| |λ + r| √ 2r B .
Let us remark that the Pólya-Vinogradov's theorem implies that B ≤ 2 √ q log q where q = cond(χ) is the conductor of χ. Note also that there are some improvements of the Pólya-Vinogradov's inequality for some characters (see for example [START_REF] Granville | Large character sums: pretentious characters and the Pólya-Vinogradov theorem[END_REF]). ). We denote by d its degree and by m L the order of its pole at s = 1. As for the Riemann zeta function, we take

ϕ(t) = (1 -t) -σ 1 if 0 ≤ t < 1 0 if t ≥ 1
where σ 1 < 1/2. By the Phragmen-Lindelöf principle, we have that L(r + it) = O ε (t d 2 (1-r)+ε ) for 0 < r ≤ 1/2. Then we deduce from Theorem 2.1 and (7.1) that ψ ∈ L 2 (1, ∞), du u 1+2r if the following inequality holds

σ 1 < 1 2 - 1 -r 2 d.

A

  (1) = 1. Furthermore we easily see that b ℓ = 0. Indeed if b ℓ = 0 then we have  1 ) m-1 . . . . . . (log α m ) more that the corresponding van der Monde matrix is invertible, we get b 1 = b 2 = • • • = b m = b ℓ = 0, which contradicts (4.2). Therefore b ℓ = 0 and then c ℓ = 0.

7.

  Some examples and applications 7.1. The Riemann zeta function. Let ζ(s) = n≥1 1 n s (ℜ(s) > 1), be the Riemann zeta function. Then it is well known that ζ can be meromorphically continued in the whole plane C, with a unique pole of order 1 at the point s = 1 ([Tit86]). Thus, the function ζ satisfies our hypothesis with m L = 1 (and for instance, σ 0 = 0). Now let us consider the function ϕ defined on [0, +∞[ by

7. 3 .

 3 The Selberg class. Let L(s) = n≥1 a n n -s be a L-function in the Selberg class S ([dR07a]

CHRISTOPHE DELAUNAY, EMMANUEL FRICAIN, ELIE MOSAKI, AND OLIVIER ROBERT

It is an illustration since here Theorem

2.1 is useless as it can be seen in Theorem 7.2.

This work was supported by the ANR project no. 07-BLAN-0248 "ALGOL", the ANR project no. 09-BLAN-005801 "FRAB" and the ANR project no. 08-BLAN-0257 "PEPR".

Let µ = µ(1/2) be a convexity bound for ζ(1/2 + it) ( [START_REF] Tenenbaum | Introduction à la théorie analytique et probabiliste des nombres[END_REF]), so that we have for all ε > 0,

We know that µ < 1/4. We get from (7.1)

So a direct application of Theorem 2.1 gives:

Proposition 7.1. With the notation above, then ψ ∈ L 2 (1, ∞), du u 1+2r

if one the following holds:

By a result of Huxley ([Hux05]), we can take µ = 32 205 . In particular, one may have r = 1/2 if σ 1 is chosen such that σ 1 < 141 410 ≈ 0.343 . . . . If we assume the Lindelöf Hypothesis, we have µ(1/2) = 0 and a similar computation as above implies that if r = 1 and r > max(0, σ 1 ), then

In fact, the work above is useless for the Riemann zeta function since we can prove directly the following:

) if and only if r > max(0, σ 1 ). Moreover, if the condition holds we have

Proof. In the case of the zeta function and for the previous choice of ϕ, we have

Hence

and so

In particular, one has

Moreover, if 2σ 1 < 1 these integrals are uniformly bounded with respect to k because using (7.2), we have

It follows from Theorem 2.2 that the Riemann zeta function s -→ ζ(s) does not vanish in the disc

where d r (λ) = dist(t λχ (0,1) , K r ), with K r = span(f A,r : A a 1-admissible sequence).

If we take only admissible sequences of length 2, we can easily prove the following.

Proposition 7.3. We have

In particular, taking ϕ = χ (0,1) (that is σ 1 = 0), we recover the subspace Kr (see the Introduction) and the result of Nikolski [Nik95, Theorem 0.1].

We will now try to give more explicit zero free discs for the Riemann zeta function. Applying Corollary 2.3 gives that s → ζ(s) does not vanish in the disc

But the estimate (4.4) gives

and thus we deduce (note that the case r > 1 is less useful):

Proposition 7.4. Let max(0, σ 1 ) < r < 1 and let A = (α, c) be a 1-admissible sequence. If ℜ(λ) > 0, then s → ζ(s) does not vanish in the disc

In particular, if we consider only admissible sequences of length 2, we obtain zero-free discs of the form (7.3)

) we obtain larger disc than in ([Nik95]) whenever ℑ(λ) is large enough and σ 1 > 0. We emphasize on the fact that our zero-free discs are explicit. For example, if we take α = 1/4 in (7.3), we obtain the following zero-free region for zeta which can be improved but has the merit to be quite explicit: Corollary 7.5. Let max(0, σ 1 ) < r < 1 and let λ ∈ C such that ℜ(λ) > 0. Then the Riemann zeta function does not vanish in the disc

where

Recall that the discs in Corollary 7.5 are euclidean discs of center (x, y) and radius R where:

1 -F (λ, r, σ 1 ) 2 . Taking, for example, λ = 0.01 + 50i, r = 0.49 and σ 1 = 0.4 then a simple evaluation of F (λ, r, σ 1 ) implies that ζ has no zero in the disc of center 1 2 + 50i and radius 3.75 × 10 -6 . Note that ζ has a zero at s = 1 2 + 49.773 . . . i and at s = 1 2 + 52.970 . . . i. 7.2. Dirichlet L-functions. Let χ be a non trivial Dirichlet character of conductor cond(χ) = q and L(χ, s) = n≥1 χ(n)n -s be the (degree 1) associated Dirichlet function. This function has an analytic continuation to the whole complex plane. For simplicity, we take σ 1 = σ 0 = 0 and

So we have ψ(u) = -n<u χ(n). Since ψ is bounded, it belongs to L 2 ((1, ∞), du u 1+2r ) for all r > 0. The admissibility condition for the sequence A is empty and we may take In particular, for r = 1/2, this gives σ 1 < 1/2d/4 which is better than σ 1 < -d/4 obtained in [dR07a]. Nevertheless, we should mention that for d < 4, we can in fact take σ 1 = 0 (see [dR07b]).


Marco Gaetani
email: m.gaetani@ibimet.cnr.it

Bernard Fontaine
email: bernard.fontaine@u-bourgogne.fr

Pascal Roucou
email: pascal.roucou@u-bourgogne.fr

Marina Baldi
email: m.baldi@ibimet.cnr.it

Influence of the Mediterranean Sea on the West African Monsoon: intraseasonal variability in numerical simulations

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Introduction

The understanding and predictability of the West African monsoon (WAM) activity are fundamental for the Sudano-Sahelian countries, whose economy, mainly based on rainfed agriculture, is vulnerable to climate variability. The analysis of rainfall variability at interannual and decadal timescales is therefore the basis for any short/long-term agricultural planning, while the correct description of the WAM seasonal cycle and of the mechanisms ruling the intraseasonal variability can be considered as the main factor to be taken into consideration for agricultural and food stocks management [START_REF] Sultan | Agricultural impacts of largescale variability of the West African monsoon[END_REF].

The WAM activity spans over a wide range of time-scales, from intraseasonal [Sultan et al., 2003] to decadal [START_REF] Lebel | Recent trends in the Central and Western Sahel rainfall regime (1990-2007)[END_REF], and is sensitive to both local forcing and remote influences. In this respect, [START_REF] Giannini | Oceanic forcing of Sahel rainfall on interannual to interdecadal time scales[END_REF] emphasized the action of the ocean basins surrounding the African continent on the West Africa rainfall decadal trend, while [START_REF] Eltahir | Dynamics of wet and dry years in West Africa[END_REF] and, more recently, [START_REF] Joly | Role of the Gulf of Guinea in the inter-annual variability of the West African monsoon: what do we learn from CMIP3 coupled simulations?[END_REF] pointed out the role of the Gulf of Guinea in the interannual modulation of the monsoonal circulation driven by the moist static energy content in the lower troposphere. Additionally, using numerical experiments, [START_REF] Losada | A multimodel approach to the Atlantic Equatorial mode: impact on the West African monsoon[END_REF] described the negative impact of the positive phase of the Atlantic Equatorial mode (the so called Atlantic Niño) on the WAM precipitation, while [START_REF] Raicich | Teleconnections between Indian Monsoon and Sahel rainfall and the Mediterranean[END_REF] found a positive correlation with the Indian monsoon, through a connection with the eastern Mediterranean basin. Furthermore, the relationship with the ENSO has been extensively explored and, in particular, periods of strong and weak correlation between WAM and ENSO have been identified by [START_REF] Janicot | Summer Sahel-ENSO teleconnection and decadal time scale SST variations[END_REF], who showed a significant negative impact of positive ENSO phases on the WAM precipitation after the 70s.

The role of the Mediterranean sea has been underestimated until the seminal work of [START_REF] Rowell | The Impact of Mediterranean SSTs on the Sahelian Rainfall Season[END_REF], who found a positive correlation between the Mediterranean SST and the WAM precipitation and explored the underlying dynamical mechanisms, finding a significant increase in the lower troposphere moisture content, advected toward the Sudano-Sahelian belt by the northeasterly flow crossing the Mediterranean. Also [START_REF] Fontaine | Atmospheric water cycle and moisture fluxes in the West African monsoon: mean annual cycles and relationship using NCEP/NCAR reanalyses[END_REF] have shown that a sizeable moisture advection from the Mediterranean feeds the monsoonal front. More recently, empirical and numerical studies [START_REF] Jung | Response to the Summer of 2003 Mediterranean SST Anomalies over Europe and Africa[END_REF][START_REF] Peyrillé | An idealized two-dimensional framework to study the West African Monsoon. Part I: validation and key controlling factors[END_REF][START_REF] Peyrillé | An idealized two-dimensional framework to study the West African Monsoon. Part I: validation and key controlling factors[END_REF] confirmed the connection between Mediterranean and WAM, a connection intensified in last years [START_REF] Joly | Rôle des océans dans la variabilité climatique de la mousson Africaine[END_REF]. In the framework of the African Monsoon Multidisciplinary Analysis (AMMA-FP6 program; [START_REF] Redelsperger | African Monsoon Multidisciplinary Analysis: An International Research Project and Field Campaign[END_REF]) the WAM-Mediterranean connection has been investigated using atmospheric general circulation models (AGCMs). In this context, [START_REF] Fontaine | Impacts of warm and cold situations in the Mediterranean basins on the West African monsoon: observed connection patterns (1979-2006) and climate simulations[END_REF] provided evidence that anomalous Mediterranean warm events are associated with a more intense monsoon, enhanced flux convergence and upward motion along the ITCZ, a strengthening of low level moisture advection, and a more northward location of ascending motion in West Africa. Furthermore, they found that in the western Mediterranean, surface temperature is statistically associated with the atmospheric deep convection over the Gulf of Guinea, while in the eastern Mediterranean the SST variability forces the atmospheric circulation over the North African subcontinent. Thus, anomalous eastern Mediterranean warm conditions are linked to a northward migration of the monsoon system accompanied by enhanced southwesterly flow and weakened northeasterly climatological wind.

While decadal and interannual WAM variability has been widely investigated, the intraseasonal variability has been for a long time less documented. The day-to-day rainfall variability has been first related to the African easterly waves regime [START_REF] Diedhou | Easterly wave regimes and associated convection over West Africa and tropical Atlantic: Results from the NCEP/ NCAR and ECMWF reanalyses[END_REF]Gu et al., 2004] and to mesoscale convective systems [START_REF] Mathon | Mesoscale convective system rainfall in the Sahel[END_REF]. Then the northward installation of the WAM rainbelt has been analysed: it can be divided into four sub-periods, each composed of a pause and an active phase of precipitation, as described by [START_REF] Louvet | Active phases and pauses during the installation of the West African monsoon through 5-day CMAP rainfall data (1979-2001[END_REF]. Furthermore, Sultan et al. [2003] showed that precipitation and convection are mainly modulated by 10-25 day and 25-60 day periodicities. The mechanisms controlling this variability is still debated, although connections to the Madden-Julian oscillation [START_REF] Lavender | Response of the West African Monsoon to the Madden-Julian Oscillation[END_REF] and to the intraseasonal variability of the Asian summer monsoon [START_REF] Janicot | The dynamics of the West African monsoon. Part IV: Analysis of 25-90-day variability of convection and the role of the Indian monsoon[END_REF] have been demonstrated.

The previous cited studies on the WAM-Mediterranean relationship focused on the interannual variability at a seasonal scale. In contrast, the sub-seasonal scale of the interaction has not been yet documented, although the variability internal to the WAM seasonal cycle is a crucial issue. The aim of this work is hence to explore the WAM-Mediterranean co-variability and connection at intraseasonal time-scale, by analysing the daily outputs from dedicated AGCMs sensitivity experiments prescribing warm and cold SST anomalies over the Mediterranean and performed in the framework of the AMMA-FP6 program for exploring the impact on WAM. This analysis will allow us to investigate (i) the major sub-seasonal features of the WAM in response to the Mediterranean SST forcing, i.e., the monsoon onset, and the African easterly waves activity; and also (ii) the timing of the response of major dynamical monsoonal features, i.e., the moisture transport and divergence, the moist static energy meridional gradient, and the Saharan heat-low and African easterly jet intensity and displacement.

In Section 2 the observational evidence of the WAM-Mediterranean connection is presented; in Section 3 the sensitivity experiments are described and the outputs are analysed, depicting the impact of the SST forcing on the monsoonal precipitation and the underlying dynamical mechanisms; in Section 4 the results are discussed and the conclusions are drawn.

Observed Mediterranean/West African monsoon interannual relationship

In this section the observed WAM-Mediterranean field relationship in the period 1979-2007 is first presented through a singular value decomposition (SVD) analysis [START_REF] Von Storch | Statistical Analysis in Climate Research[END_REF] of Mediterranean skin temperature and precipitation fields over West Africa at the interannual timescale. The SVD approach is used in order to identify covariance modes between surface conditions over the Mediterranean and the WAM activity. Then a few basic associated signals in atmospheric dynamics are analysed using composite analyses of wind and specific humidity to highlight the statistical relationship.

Data

We use the rainfall data from Climate Prediction Center Merged Analysis of Precipitation (CMAP; [START_REF] Xie | Global precipitation: A 17-year monthly analysis based on gauge observations, satellite estimates, and numerical model outputs[END_REF]) and Global Precipitation Climatology Project (GPCP; [START_REF] Xie | GPCP Pentad Precipitation Analyses: An Experimental Dataset Based on Gauge Observations and Satellite Estimates[END_REF]), and the atmospheric variables from the NCEP-DOE AMIP-II Reanalysis (R-2; [START_REF] Kanamitsu | NCEP-DOE AMIP-II Reanalysis (R-2)[END_REF]).

The CMAP and GPCP data are obtained as a merging of raingauges observations and satellite estimations. Monthly and pentad values are available from 1979 to 2008, on a global domain at 2.5° resolution starting from (1.25°E, 88.75°S). The R-2 dataset is derived from the NCEP-NCAR Reanalysis, and it contains a wide range of atmospheric variables obtained as a merging of in situ and satellite observations and modeling. Monthly, daily and 6-hours values are available from 1979 to present, on a global domain at different resolutions (2.5° regular grid and T62 Gaussian grid) and vertical levels (pressure and "sigma" levels).

SVD analysis

The SVD analysis has been performed between skin temperature over the Mediterranean basin [10°W-40°E; 30°N-45°N] and WAM precipitation over the 0°-15°N belt in May-June (MJ) and the 5°N-20°N belt in July-September (JAS). Different precipitation domains have been chosen in order to follow the meridional migration of the rainbelt embedded in the ITCZ, which is positioned along the coast of Guinea (5°N) in MJ, and abruptly shifts in late June to 10°N (preonset and onset phases; Sultan and Janicot [2003]). After the onset phase, in JAS, precipitation and convection persist over the Sudano-Sahelian belt [Gu and Adler, 2004]. In order to give the same weight to the different parameters describing the phenomenon and to focus on the interannual variability, we applied the SVD analysis to the standardized data after removing any linear trend at each grid-point in the domains.

In May-June, the first SVD mode (Fig. 1ab) explains only 33% of the squared covariance fraction (SCF), although the correlation between the time series of the expansion coefficient (ECs) is high and significant (r = 0.65): negative anomalies of surface temperature in the eastern Mediterranean are related to negative precipitation anomalies along the coast of Guinea and positive anomalies inland, but the pattern shows low significance. By contrast, in JAS, the pattern is largely significant: the first SVD mode (Fig. 1cd) explains 59% of the SCF and the correlation between the ECs time series is r = 0.64, indicating that warmer temperatures in the eastern Mediterranean are related to a wetter Sudan-Sahel. Thus, a significant statistical relationship exists between surface temperatures in the Mediterranean and precipitation over West Africa on an interannual time-scale: this relationship is active in JAS, when the monsoon is fully developed inland. However, part of the WAM-Mediterranean common variability can be ascribed to the Indian monsoon circulation that affects both the eastern Mediterranean basin [Raicich et al., 2006] and the north African summer circulation [START_REF] Chen | Maintenance of the Midtropospheric North African Summer Circulation: Saharan High and African Easterly Jet[END_REF]. The modelling approach described in this work is helpful in determining and quantifying the direct action of the Mediterranean on the WAM variability.

Atmospheric dynamics in July-September

In order to investigate the atmospheric dynamics associated to the statistical WAM-Mediterranean relationship, we analyse the wind and humidity composite difference between the warm and cold situations in the Mediterranean, identified using the skin temperature ECs time series of the first SVD mode in JAS. The time series is divided in terciles, with the cold (warm) years in the first (third) tercile (Tab. 1) to amplify the skin temperature anomaly pattern in the first SVD mode, that explains the highest fraction of covariance.

In the lower troposphere, the JAS mean wind field (Fig. 2a) is characterized by a clear monsoonal southwesterly flow penetrating from the Gulf of Guinea inland into West Africa, while an anticyclonic (a cyclonic) circulation occurs over the western (eastern) Mediterranean. These inverse cells are associated with a northeasterly flow crossing the Sahara desert, limiting the inland penetration of the WAM but feeding the ITF convergence. In the upper troposphere, the circulation contrasts the extratropical westerlies and the easterlies in the Tropics (Fig. 2b). Composite results in Fig. 2c show that a warmer Mediterranean has an impact on the wind field in the lower troposphere, enhancing both the northeasterly flow from the eastern Mediterranean and the southwesterly flow from the Atlantic Ocean. This mechanism increases the flux convergence and the cyclonic circulation in the eastern part of North Africa, around 20°N, while a significant southwesterly flow is observed at a latitude of 10°N. In the upper troposphere, a weakening of the westerly flow along the Mediterranean coast of Africa and a strengthening along the Alps are observed (Fig. 2d). This is consistent with the correlation between the midlatitudes jetstream variability and the occurrence of anomalous summers in the Mediterranean [START_REF] Baldi | Heat-waves in the Mediterranean: a local feature or a larger scale effect?[END_REF].

The JAS atmospheric moisture field is displayed at 850 hPa because specific humidity is higher close to the surface (Fig. 3a). It shows a maximum in the Tropical belt and a minimum over the Sahara desert. Moisture is transported towards the monsoonal front both by the southwesterly flow from the Atlantic and across the desert from the Mediterranean (Fig. 3b). The warm-cold composite (Fig. 3c) displays significant positive humidity anomalies over the Mediterranean basin, due to enhanced evaporation from the warmer sea surface, and more southward along the monsoonal front.

Thus, additional moisture is available for precipitation where northeasterlies and southwesterlies converge (Fig. 3d), while the enhanced southwesterly moisture transport from the Atlantic ocean is able to feed the rains on the western coast of North Africa (Fig. 3cd).

Simulated Mediterranean/West African monsoon intraseasonal relationship

The picture described in the previous section is fully consistent with other studies [START_REF] Rowell | The Impact of Mediterranean SSTs on the Sahelian Rainfall Season[END_REF][START_REF] Raicich | Teleconnections between Indian Monsoon and Sahel rainfall and the Mediterranean[END_REF][START_REF] Jung | Response to the Summer of 2003 Mediterranean SST Anomalies over Europe and Africa[END_REF][START_REF] Fontaine | Impacts of warm and cold situations in the Mediterranean basins on the West African monsoon: observed connection patterns (1979-2006) and climate simulations[END_REF] showing that a warmer Mediterranean sea is favourable to the Sudan-Sahel precipitation. In this paragraph we discuss the potential impact of Mediterranean SST anomalies on the intraseasonal variability over the WAM region in May-September, by analysing the daily outputs from the AMMA-FP6 numerical sensitivity experiments performed forcing the AGCMs with prescribed warm and cold SST anomalies over the Mediterranean sea.

Sensitivity experiments and analysis methodology

The numerical sensitivity experiments simulating normal conditions and warm/cold anomalies over the Mediterranean sea have been performed using three AGCMs:

-ARPEGE-CLIMAT [START_REF] Déqué | The ARPEGE/IFS atmosphere model: A contribution to the French community climate modelling[END_REF] Version 3, at T42 Gaussian resolution with 45 vertical levels, run at CNRM (National Center for Meteorological Research, France); -ECHAM [START_REF] Roeckner | The atmospheric general circulation model ECHAM-4: model descriptions and simulation of present-day climate[END_REF] Mediterranean SST patterns to the observed SST climatology. The SST anomaly patterns have been constructed using the expansion coefficients of the leading extended maximum covariance mode between the WAM monthly precipitation (CMAP) and the Mediterranean monthly SST (NOAA Extended Reconstructed SST; [START_REF] Smith | Improved extended reconstruction of SST (1854-1997)[END_REF]) on the period 1979-2005, as described in [START_REF] Polo | Part I: time-evolving SST modes related to West African rainfall[END_REF]. This mode shows a strong positive relationship between the Sahelian precipitation and the spring to summer evolution of the eastern Mediterranean SST, therefore warm/cold SST composites were computed by averaging years for which the SST-precipitation coupling was maximal, i.e., choosing those years in which the SST and precipitation ECs are higher than 1 standard deviation (1984,1987,1991,1992,1997) and those in which the expansion coefficients are less than -1 standard deviation (1979,1994,1999,2001). The May to October monthly SST anomaly fields used to force the warm experiments are shown in Fig. 4. For each model and each experiment (control, warm, cold) we use 10 simulations, for a total of 10 × 3 × 3 = 90 simulations from 15 th of April to 15 th of October, following the AMIP protocol [START_REF] Gates | AMIP: The Atmospheric Model Intercomparison Project[END_REF],

with initial conditions taken from years 1979 to 1988. We have to specify that using a prescribed SST as a lower boundary condition implies that the sea acts as an infinite reservoir of heat capacity and this tends to damp the surface heat fluxes and therefore to decrease the amplitude of the results.

We analyse the outputs of each individual model and the multi-model mean (ensemble-mean). In order to limit the influence of models biases, for each model the control run is subtracted to the sensitivity experiment before to compute the warm/cold ensemble-mean anomalies. Because of differences in models horizontal resolution, the ensemble-mean is computed after linearly interpolating the individual models outputs over a 2.5° resolution grid, with the starting point at (0°E, 0°N). In order to describe the impact of the thermal anomalies in the Mediterranean sea on the WAM at a sub-seasonal time-scale, the differences between warm and cold simulations are computed and the results are displayed in time-latitude diagrams. The purpose of this approach is the determination of the timing of the relationship and the amplitude of the WAM modifications in terms of latitudinal position, exploiting the quasi-zonal displacement of the monsoonal front. The intraseasonal variability is investigated after filtering the daily outputs using a Butterworth low-pass filter [START_REF] Murakami | Large scale aspects of deep convective activity over the Gate area[END_REF] with a 10 day cut-off, in order to remove the synoptic variability.

WAM seasonal cycle in control experiments

In order to evaluate the model capability to simulate the WAM seasonal cycle, we discuss the timelatitude diagrams of the May to September precipitation averaged between 10°W and 30°E (Fig. 5), for each model, for the ensemble-mean, and for the observations (CMAP and GPCP rainfall estimates). In addition, the seasonal evolution of the zonal wind at 850 and 600 hPa in the models is compared to R-2, in order to verify the correct description of the main dynamical features associated with the WAM. The simulations outputs are compared with the 1979-1988 observations after filtering the daily model outputs and the R-2 data using a 10 day low-pass filter, while the CMAP and GPCP pentad data are filtered using a 2.5 pentad low-pass filter.

Observed rainfall data (Fig. 5ef) show clearly the installation of the WAM, following the seasonal meridional migration of the ITCZ, through a sequence of active phases and pauses [START_REF] Louvet | Active phases and pauses during the installation of the West African monsoon through 5-day CMAP rainfall data (1979-2001[END_REF]. In May and June the maximum of the rainbelt is located around 5°N, and, after a pause in late June/early July, it shifts northward to around 8°/10°N, until mid September. In August, at the peak of the monsoonal season, the northernmost edge of the rainbelt is around 18°N. The two datasets show similar rainfall amounts in JAS, although CMAP values are larger than GPCP in May-June. The ARPEGE model (Fig. 5a) reproduces the WAM seasonal cycle, with comparable rainfall amounts in June-September, it overestimates rainfall in May and it shows, in early June, an anticipation of the pause occurring before the inland installation of the monsoon. The ECHAM model (Fig. 5b) generally underestimates the JAS rainfall, with higher amounts in May-June, it does not reproduce the pause in June, and it shows a progressive northward penetration of the monsoon (with no abrupt northward shift), while the retreat in September is not well reproduced. By contrast, the UCLA model (Fig. 5c) largely overestimates the observations, simulates a pause before the inland shift in mid June, and locates the JAS rainfall maximum around 12°N. However observations are partly reproduced in the ensemble-mean (Fig. 5d), since the UCLA and ARPEGE model overestimations well balance the too weak signal coming from the ECHAM model.

The observed 850 hPa zonal wind shows the change in the low-levels circulation from the easterly regime to the westerly monsoonal flow in June, up to 15°N and with a maximum at the peak of the rainy season in July-August (Fig. 6i). The ensemble-mean reproduces the seasonal cycle and the intensity of the westerly flow (Fig. 6g), except in May, when positive values are observed as a consequence of the ARPEGE and UCLA anticipation of the flow change (Fig. 6ae). ECHAM produces a weak monsoonal flow, balanced in the ensemble mean by the large UCLA overestimation (Fig. 6ce).

In the mid troposphere, the circulation is dominated by the African easterly jet (AEJ), generated by the meridional thermal gradient between the Equatorial Africa moist surface and the dry and hot Sahara desert. In boreal summer the AEJ, fully developed around 600 hPa between 10°N and 15°N, blows from 30°E to the Atlantic coast and concentrates rainfall maxima on its southern flank [START_REF] Cook | Generation of the African easterly jet and its role in determining West African precipitation[END_REF]. The observed seasonal cycle (Fig. 6j) is quite well reproduced in the ensemble-mean (Fig. 6h), even if the mean position is northward shifted because of the ECHAM and UCLA responses (Fig. 6df). Also the intensity is underestimated due to the ARPEGE and ECHAM weak responses (Fig. 6bd).

WAM precipitation and onset dates

The ensemble-mean response of model precipitation to SST forcing over the Mediterranean sea is now presented through time-latitude diagrams of the warm-cold composite (Fig. 7). The rainfall is averaged in two zonal domains, [10°W-10°E] and [10E°-30°E], after removing the synoptic variability (< 10 days). The longitudinal domains have been chosen in order to separate the region in which the sea-land thermal gradient and the monsoonal circulation develop approximatively in the meridional direction (west of 10°E). We first verify that, as in the observations [Sultan et al. 2003], the individual simulations present a typical 15 day variability around 5°N in May-June and around 10°N in JAS, since out-of-phase combinations in ensemble-means and composites tend to suppress intraseasonal periodicities (Fig. 7). Rainfall anomalies are often positive, from May to September, from the Equator to 20°N, with significant wet differences over western Sudan-Sahel in August-September (Fig. 7a; Tab. 3), also observed in each individual model, and eastward in late July, August and September (Fig. 7b; Tab. 3): the mean response is here dominated by the UCLA model (not shown).

The WAM onset has been defined by Sultan and Janicot [2003] as the abrupt latitudinal shift of the ITCZ from its quasi-stationary May-June position around 5°N to the JAS quasi-stationary position around 10°N. Using OLR data in the period 1968-1990, they found that the WAM onset occurs in mean on June 24 th , with an 8 day standard deviation (STD). Moreover, the end of June has been confirmed as the period for the mean occurrence of the WAM onset by [START_REF] Fontaine | Definition and predictability of an OLR based West African monsoon onset[END_REF], using another OLR-based methodology, and by [START_REF] Dalu | A hydrological onset and withdrawal index for the West African monsoon[END_REF], using a hydrological index. A precipitation-based onset index has been also developed by [START_REF] Fontaine | Sudan-Sahel rainfall onset: Definition of an objective index, types of years, and experimental hindcasts[END_REF]. They The onset dates in numerical simulations are estimated using the WAMOI (Tab. 2). The WAMOI is computed for each run, then the mean and STD are computed. It is noteworthy the substantial agreement between the models despite a general anticipation of the onset date: June 18 th for the control run ensemble-mean, with a 5.9 day STD. WAM onset dates in sensitivity experiments are delayed in both warm and cold simulations, with a "warm onset" occurring later than a "cold onset". In the ARPEGE model the mean onset occurs earlier than in the ECHAM and UCLA models, but the shifts are not significant, except for a later "warm onset" in the UCLA model (June 29 th with 5.3 day STD). The ensemble-mean shows no significant difference between the warm and cold simulations.

Atmospheric dynamics

Low and mid troposphere circulation

The low troposphere circulation over the Mediterranean/North Africa region in summer is under the influence of the Azores system [Meteorological Office, 1962;Lionello et al., 2006], which produces an anticyclonic circulation to the west centred over the Maghreb; and of the Indian monsoon [START_REF] Rodwell | Subtropical anticyclones and summer monsoons[END_REF], which produces a cyclonic circulation and an equatorward flow to the east over the Middle East and the Levantine basin (Fig. 2a). This northerly wind regime, known as Etesian or Meltemi [Meteorological Office, 1962], is the geostrophic wind produced by the pressure gradient between eastern and western Mediterranean. It can be considered as the local low level branch of the northern Hadley meridional circulation connecting the Mediterranean basin and the ITCZ over Africa. [START_REF] Raicich | Teleconnections between Indian Monsoon and Sahel rainfall and the Mediterranean[END_REF] found statistical evidences that the Etesian winds regime teleconnects the Indian monsoon and the Sahel monsoonal rainfall. Computing the Mediterranean pressure index (MPI) as the SLP standardized difference at the stations of Mersa Matruh (Egypt) and Marseille (France), i.e., between the cyclonic and anticyclonic circulations, they showed that when the Indian monsoon is stronger than normal, a negative SLP anomaly affects the eastern Mediterranean, enhancing the Mediterranean pressure gradient and reinforcing the northerly flow east of 15°E. The consequent strengthened northerly flow favors change in low level convergence over Sahel with the contribution of the strengthening of the southwesterly monsoonal flow. We hypothesize that the prescribed SST anomalies over the Mediterranean sea have a sizeable influence on the Mediterranean SLP, modifying the circulation in the lower troposphere. In order to assert it, we compute the MPI as the difference between the SLP in the eastern basin [15°E-35°E; 30°N-40°N] and in the western basin [5°W-15°E; 35°N-45°N]. Similar indices have been used in the past as representative of the differences in the behaviour of the eastern and western Mediterranean, in terms of temperature field and hydrological cycle [START_REF] Conte | The Mediterranean oscillation. Impact on precipitation and hydrology in Italy[END_REF][START_REF] Palutikov | Climate and Climate Change, in Mediterranean desertification and land use[END_REF][START_REF] Palutikov | Analysis of Mediterranean climate data: measured and modelled[END_REF]. In our case a negative value of the index, corresponding to a negative pressure gradient toward the eastern Mediterranean, is related to a northerly wind. The behaviour of the MPI is strongly related to the seasonal evolution of the SST forcing field, in which the eastern basin is warmer than the western (Fig. 4). In warm-cold composites from the ensemble-mean (Tab. 3), the MPI anomaly shows negative values in JAS, with significant values in the first half of September.

In individual models, MPI warm-cold composites are significantly negative in June-July (UCLA model) and in September (ECHAM model), while the ARPEGE model produces a significant positive anomaly in late August (not shown). Such an increased pressure gradient is related to reinforced convergence in the Sahelian belt. Indeed, even though the MPI anomaly is significant only at the beginning of September, the dynamical response in the ensemble-mean is sizeable from August to mid September, with significant convergence anomalies at 850 hPa over the eastern Sahel (Tab. 3).

A warmer Mediterranean sea suggests the presence of additional moisture coming from the increased evaporation, mainly located in the lower troposphere. This statement is verified computing the time-latitude diagram of the specific humidity at 850 hPa for the warm-cold composite. In the ensemble-mean, moisture anomalies are positive both in the western and eastern domains, from May to September. West of 10°E, this is mainly observed between 20°N and 40°N from mid July to mid September, and between 10°N and 20°N in August-September (Fig. 8a; Tab.

3). East of 10°E, this occurs over the eastern Mediterranean over the whole season, and between 15°N and 30°N from late June to September (Fig. 8b).

A good indicator of the monsoonal activity is the vertical integrated moisture transport (VIMT; [START_REF] Fasullo | A Hydrological Definition of Indian Monsoon Onset and Withdrawal[END_REF]; [START_REF] Dalu | A hydrological onset and withdrawal index for the West African monsoon[END_REF]). In the Mediterranean/North Africa region it is defined as

∫ = 1 0 0 0 8 5 0 g d p q ρ U V I M T ,
where q is the specific humidity, U is the vector wind, p is the pressure, g is the gravity acceleration and ρ is the air density, considered constant in the lower levels (ρ = 1 kg / m 3 ). The warm-cold differences of the VIMT divergence show that there is no strong coherency over the western domain, except for a significant convergence anomaly around 20°N at the turn of August/September (Fig. 8c), while, over the eastern domain we notice a divergence above the Mediterranean sea from May to September, and a significant convergence anomaly (Fig. 8d; Tab. 3) over the Sahel from late

July to mid September.

The role of the moisture transport from the Mediterranean in determining the convergence anomalies in the Sahel is discussed through the warm-cold differences of the 850 hPa wind and VIMT fields, averaged in 15 day sub-periods between July and September (Fig. 9). When the VIMT field is considered, a persistent anomalous northerly flow from the eastern Mediterranean reaches the eastern Sahel, with maxima in the first half of August and September (Fig. 9fj). Furthermore, a reinforced southwesterly flow from the Atlantic ocean in August and September (Fig. 9fhjl) contributes to the anomalous moisture convergence in the eastern Sahel (Fig. 8d). On the other hand, significant wind anomalies in the southwesterly flow from the Atlantic are observed, while the northerly wind from the Mediterranean is unchanged or reduced (Fig. 9egik). Therefore, the moisture transport from both the Atlantic ocean and the Mediterranean sea contribute to the convergence over the Sahel, the former reinforced by the increased wind field, the latter by the augmented specific humidity.

The AEJ changes induced by the SST forcing are shown through the 600 hPa zonal wind warm-cold differences averaged between 10°W and 30°E, displayed in Fig. 10. From mid July to September, negative differences are observed between 15°N and 20°N, with significant values in late July, mid-August and September showing that when the Mediterranean is warmer than average, the AEJ is shifted northward.

African easterly waves

At synoptic scale, the West African convection is modulated by the AEWs (3-5 day and 6-9 day periodicity, around 5°N and 15°N, respectively), during the boreal summer, mainly in August-September [Diedhiou et al., 1999]. We use the variance of the 700 hPa meridional wind filtered at 2-10 day [START_REF] Sylla | Simulation of the intraseasonal and the interannual variability of rainfall over West Africa with RegCM3 during the monsoon period[END_REF] as a descriptor of the wave activity, and we test the effect of Mediterranean SST anomalies on the AEWs in August-September, after evaluating the numerical capability to reproduce AEWs mean activity. In fact ARPEGE and UCLA models reproduce part of the observed AEWs features. In August, the ensemble-mean signal is in the lee of the Ethiopian plateau and of the Ahaggar massif (Fig. 11ac), while, in September it is generally weaker and shifted southward to 5°N and northward to 20°N, respectively (Fig. 11eg).

The models response has been evaluated through the 'one model one vote' concept [START_REF] Santer | Incorporating model quality information in climate change detection and attribution studies[END_REF]. It consists in computing over the successive 15 day sub-periods of the bi-month August-September, the number of positive/negative occurrences in warm-cold differences, for each model and each run, 3 × 10 = 30 events. Therefore, the wave activity response can be defined by calculating at each grid point, a simple score ranging either between +16 and +30 or between -16 and -30: +16 (-16) means that 16 models out of 30 simulate an increasing (a decreasing) signal in wave activity, and +30 (-30) that all models produce an excess (a deficit) in this activity. The significance is then evaluated considering the field response in each sub-period: the response is significantly positive (negative) in the sub-period if the percentage of positive (null or negative) grid points in the field is significantly greater than 50% regarding a binomial distribution with only 30 degrees of freedom (confidence interval [33% 66%]). The ensemble warm-cold differences show a significant positive response (71% of positive occurrences) in the first half of August, with a large belt of increased wave activity between 5°N and 25°N (Fig. 11b). Notice also that after mid August the global field significance decreases (59%) except in the western domain where the wave activity continues (Fig. 11d). By contrast, in September (first and second half field significance 53% and 41%, respectively) and over the eastern domain (Fig. 11fh), this activity tends to augment in consistency with increased convergence over eastern Sahel (Tab. 3).

WAM energetics

Models response to a warmer Mediterranean produces a moisture transport over the Sahara toward the Sahel in August-September (Fig. 8; Tab. 3), affecting the energy budget of the monsoon. Over the Sahara desert, the circulation is influenced by the shallow dry convection related to the diabatic heating in the Saharan heat-low (SHL) in the low troposphere, and by the Saharan high related to the diabatic cooling at higher levels. This circulation contributes to the maintenance of the AEJ through the Coriolis acceleration associated to the divergent northerlies flowing between the SHL and the Saharan high [START_REF] Chen | Maintenance of the Midtropospheric North African Summer Circulation: Saharan High and African Easterly Jet[END_REF]. Furthermore, it is important to evaluate the moist static energy (MSE) meridional gradient near the surface which forces the monsoon circulation and its inland excursion, since, as shown by [START_REF] Eltahir | Dynamics of wet and dry years in West Africa[END_REF], a stronger (weaker) gradient drives stronger (weaker) monsoonal circulation and northward excursion. In this paragraph we discuss the position and thickness of the SHL and the MSE meridional gradient at 1000 hPa over West Africa in the warm-cold composites.

Lavaysse et al. [2009] studied the SHL behaviour by computing the geopotential height differences between two levels (700hPa and 925hPa) and detected its location by searching the maximum, i.e.,

where the thermally induced dilatation of low atmospheric levels is largest. In summer the SHL climatological position is between the Ahaggar and the Atlas. The warm-cold differences observed in the ensemble-mean show a general north-eastward migration with significant values during particular subperiods of the rainy season: up to 5° in longitude in late July, up to 7° in longitude and 7° in latitude in late August and up to 8° in longitude in late September (Fig. 12a; Tab. 3). This is associated with a significant thickness increase in August-September (Fig. 12b; Tab. 3). The changes in the SHL are consistent with the northward shift and intensification of the AEJ (Fig. 10) and with a more inland penetration of the monsoonal front and consequent wet anomalies over western Sahel (Fig. 7a).

The MSE is defined as

L T C g z M S E p + + = ,
where z is the geopotential height, C p the specific heat of dry air at constant pressure, T the temperature, L the latent heat of evaporation and q the specific humidity. Regional contents are estimated by averaging the MSE at 1000hPa over four domains:

Atlantic [15°W-15°E; 0°N-5°N], Guinea [15°W-15°E; 5°N-10°N], Sudan [15°W-15°E; 10°N-15°N], Sahel [15°W-15°E; 15°N- 20°N].
Differences between these adjacent domains are computed following the northward direction in order to approximate the MSE meridional gradient between the main belts: the Atlantic and the Guinean domains (Guinea-Atlantic difference), the Guinean and Sudanian regions (Sudan-Guinea)

and the Sudanian and Sahelian belts (Sahel-Sudan). In the ensemble-mean, warm-cold differences in the Guinea-Atlantic gradient are weak (Fig. 13d). By contrast the MSE gradients are reinforced, while northward, in JAS, with significant values from mid August to mid September (Fig. 13bc). This is also observed when the land-sea contrast is taken into consideration: the Sahel-Atlantic MSE differences augment in July, and are significant in early August (Fig. 13a; Tab. 3). Those reinforcements, directly linked to a warmer Mediterranean, are a consequence of the moisture supply provided over the western domain in August-September (Fig. 8a and Fig. 9) and consistent with the intensification of the precipitation in August-September (Fig. 7a). Since the positive moisture anomaly peaks around 15°N, the reinforced Sudan-Guinea MSE gradient is also larger than local anomalies.

Summary and conclusions

In the present paper we explored the seasonal cycle of the WAM-Mediterranean relationship and focused on the intraseasonal variability using the R-2 surface and atmospheric variables, the CMAP and GPCP rainfall estimates and results from numerical forced experiments performed using three Tables Table 1:

  Version 4, at T30 Gaussian resolution with 32 vertical levels, run at ENEA (National Agency for New Technologies, Energy and Sustainable Economical Development, Italy); -UCLA model [Richter et al., 2008] Version 7.3, at 2.5° × 2° lon-lat resolution and 29 vertical levels, run at UCM (Complutense University of Madrid, Spain). The control experiments have been performed forcing the models with the 1979-2005 observed global SST climatology, while warm and cold experiments have been performed adding anomalous

  defined a northern index (NI) by averaging the precipitation in the domain [10°W-10°E; 7.5°N-20°N] and a southern index (SI) in the domain [10°W-10°E; 0°-7.5°N], and a WAM onset index (WAMOI) as the standardized NI minus SI difference, after eliminating the synoptic variability using a 15 day low-pass Butterworth filter. WAMOI increases (decreases) when the rainbelt migrates northward (southward) and the onset date is defined as the first pentad of a 20 day (or longer) period registering positive values of WAMOI, in order to determine a robust installation of the rainfall north of 7.5°N. Using the CMAP and GPCP pentad data, they found 28-29 June with a 8.5 day STD as the mean onset date for the period1979-2004.

  AGCMs: ARPEGE, ECHAM, UCLA.The observed WAM-Mediterranean interannual relationship over the period 1979-2007 has been investigated by analysing the Mediterranean surface temperatures, the rainfall over West Africa, the specific humidity and the wind field in order to better understand the associated atmospheric dynamics. The first SVD mode confirms the statistical relationship between the two domains mainly when the monsoon is fully developed over the Sudano-Sahelian belt. The warm-cold composites show that a warmer eastern Mediterranean produces a low pressure anomaly near the surface and feeds the lower troposphere with moisture: this generates a SLP gradient between the eastern and western basins which reinforces the low-levels northeasterly geostrophic flow towards the Sahara, transports additional moisture and favours moisture convergence in the WAM area along with its inland penetration from the Atlantic. Accordingly with[START_REF] Rowell | The Impact of Mediterranean SSTs on the Sahelian Rainfall Season[END_REF],[START_REF] Raicich | Teleconnections between Indian Monsoon and Sahel rainfall and the Mediterranean[END_REF],[START_REF] Jung | Response to the Summer of 2003 Mediterranean SST Anomalies over Europe and Africa[END_REF],[START_REF] Fontaine | Impacts of warm and cold situations in the Mediterranean basins on the West African monsoon: observed connection patterns (1979-2006) and climate simulations[END_REF], a warm Mediterranean sea positively affects the monsoonal precipitation over the Sudano-Sahelian belt in July-September.The May to September model outputs of experiments with prescribed warm/cold SST anomalies over the Mediterranean have been also analysed, showing a seasonal modulation: the second half of the monsoon season (August-September), is more affected by the Mediterranean SST anomalies. In short the succession of events is as follows. Warmer SSTs over the Mediterranean generate in July positive moisture anomalies due to increased evaporation. These anomalies reinforce in August and September and the higher moisture amount is advected by the northerlies (Etesian winds) across the Sahara desert toward the eastern Sahel. Moisture convergence over the monsoonal region is thus thermodynamically reinforced, with the contribution of a strengthened southwesterly monsoonal flow, while enhanced convection increases the AEWs activity across the Sudano-Sahelian belt in August. Over the western Sahel the behaviour of the SHL and of the MSE meridional gradient have been also analysed since part of anomalous moisture reaches West Africa north of 10°N after mid August, which affects the SHL and the MSE low-levels content. The response is a reinforcement and a northward shift of the SHL and an increase of the MSE gradient between the Sudan-Sahel and the Guinea coast, favouring rainfall excess in western Sahel. The resulting northward shift of the AEJ, consistent with the meridional displacement of the rainbelt, is related to changes in divergent circulation between the SHL and the Saharan high. Thus, in the eastern Sahel the rainfall anomalies from late July to the end of September are related to the additional moisture available, which is advected from the Mediterranean, and to the dynamical reinforcement of the monsoonal front. In the western Sahel rainfall excesses from early August to the end of September are related to the more inland penetration of the WAM rainbelt, the additional moisture producing here a clear northward shift of the monsoon air.The simulated monsoonal precipitation and dynamical responses in the models are closely linked to the SST forcing evolution but are not synchronous. Anomalies in the MPI start in July, when the eastern Mediterranean basin is slightly warmer than the western basin. Interestingly they do not reach any significant values before September, i.e., when the east-west SST gradient is stronger, although the convergence effects in the eastern Sudan-Sahel are sizeable since the beginning of August. A difference between observations and simulations arises: in the R-2 data, warm Mediterranean anomalies are liked to a significant intensification of the Etesian winds and moisture transport from the Mediterranean; while in the sensitivity experiments, the northerly moisture transport is strengthened by the additional moisture availability, with no significant changes in the wind from the Mediterranean. Changes in the R-2 northerly wind are consistent with the MPI-Etesian dynamics observed by[START_REF] Raicich | Teleconnections between Indian Monsoon and Sahel rainfall and the Mediterranean[END_REF] and they can be related to the variability of the Indian monsoon circulation, while moisture transport anomalies are directly related to the warming of the Mediterranean sea. This feature is confirmed by the sensitivity experiments, in which the Mediterranean forcing is not capable to induce significant changes in the MPI, whereas it supplies additional moisture, inducing large anomalies in the northerly moisture transport. So we make the hypothesis that rainfall anomalies in eastern Sudan-Sahel occur in late July as a consequence of the moisture-Etesian dynamics, while in western Sudan-Sahel they occur in early August in relation to this additional moisture transport favouring the inland penetration of the monsoon, through the intensification of the SHL and the reinforcement of the MSE meridional gradient. The contemporary strengthening of the southwesterly monsoonal flow suggests that the Mediterranean sea acts positively on the Sudan-Sahel precipitation when the thermodynamical processes are fully developed inland in the monsoon air.

Figures captions Figure 1 :

 captions1 Figures captions

Figure 2 :

 2 Figure 2: 1979-2007 JAS climatology of (a) 850 hPa and (b) 200 hPa wind (m/s). JAS warm-cold

Figure 3 :

 3 Figure 3: 1979-2007 JAS climatology of (a) 850 hPa specific humidity (g/kg) and (b) 850 hPa

Figure 4 :

 4 Figure 4: May to October forcing fields used in the AGCM warm experiments. SST anomalies are

Figure 5 :

 5 Figure 5: May to September time-latitude diagram of the 10 years control run precipitation

Figure 6 :

 6 Figure 6: May to September time-latitude diagram of the 10 years control run zonal wind at 850 and

Figure 7 :

 7 Figure 7: May to September time-latitude diagram of the ensemble-mean precipitation (mm/day),

Figure 8 :

 8 Figure 8: May to September time-latitude diagram of the ensemble-mean 850 hPa specific humidity

Figure 9 :

 9 Figure 9: July to September ensemble-mean 850 hPa wind (panels a, c, e, g, i, k) and VIMT (panels

Figure 10 :

 10 Figure 10: May to September time-latitude diagram of the ensemble-mean 600hPa zonal wind (m/s)

Figure 11 :

 11 Figure11: AEWs: control runs ensemble-mean of the variance of the 700hPa meridional wind filtered in the 2-10 day band (m 2 /s 2 ), in (a,c) August and (e,g) September; positive and negative

Figure 12 :

 12 Figure 12: May to September evolution of the ensemble-mean SHL (a) latitudinal (solid line) and

Figure 13 :

 13 Figure 13: May to September evolution of the ensemble-mean 1000hPa MSE (m 2 /s 2 ) difference

  Warm and cold JAS in the Mediterranean computed using the ECs of the first SVD mode skin temperature pattern. The years are sorted from the warmest and the coldest.

Table 2 :

 2 WAM onset dates and standard deviations in the Mediterranean experiments, control run, warm and cold experiments. The UCLA warm-control onset difference is 90% significant after a t-Student test (bold).

	Model run ARPEGE ECHAM	UCLA	Ensemble
	Control	18 Jun (6.8) 20 Jun (9.6) 23 Jun (3.4) 18 Jun (5.9)
	Warm	22 Jun (5.2) 28 Jun (20.2) 29 Jun (5.3) 20 Jun (4.5)
	Cold	21 Jun (8.7) 25 Jun (11.1) 26 Jun (6.5) 20 Jun (5.2)

Table 3 :

 3 Warm-cold differences: July to September evolution of the main indices used in the text and averaged over 15 day sub-periods. Precipitation is averaged in [10°N-20°N]; 850 hPa divergence is averaged in [15°N-25°N]; 850 hPa moisture is averaged in [15°N-30°N]. The western domain is between 10°W and 10°E, the eastern domain is between 10°E and 30°E. Bold values are 90% significant after a t-Student test.

	JUL	AUG	SEP

10.45 -11.18 -8.42 -4.73

	Western Sahara-Sahel moisture (g/kg)	+0.50 +0.43 +0.66 +0.84 +0.76 +0.37
	SHL thickness (m)	-1.35	0.96	+2.80	+3.77	+2.89	+4.13
	SHL migration (direction)	W	S E	N E	N E	N W	N E
	Sahel-Atlantic MSE gradient (m 2 /s 2 )	+1060	+139 +869 +2405 +1953 +1056

Acknowledgement

NCEP Reanalysis data are provided by the NOAA-CIRES ESRL/PSD Climate Diagnostics branch, Boulder, Colorado, USA (http://www.esrl.noaa.gov/). The GPCP data were developed and computed by the NASA/Goddard Space Flight Center's Laboratory for Atmospheres as a contribution to the GEWEX Global Precipitation Climatology Project. CMAP Precipitation data provided by the NOAA/OAR/ESRL PSD, Boulder, Colorado, USA, from their Web site at http://www.esrl.noaa.gov/psd/. This study was supported by the TELEMEDAF (Italian CNR / French CNRS joint project) and by the European Union AMMA-FP6 project. Based on the French initiative, AMMA was built by an international scientific group and is currently funded by a large number of agencies, especially from France, the United Kingdom, the United States, and Africa. It has been the beneficiary of a major financial contribution from the European Community's Sixth Framework Research Programme. (Detailed information on scientific coordination and funding is available on the AMMA International Web site, http://www.amma-international.org.).