On a conjecture of Montgomery-Vaughan on extreme values of automorphic L-functions at 1

functions at 1.

§ 1. Introduction

The automorphic L-functions constitute a powerful tool for studying arithmetic, algebraic or geometric objects. For squarefree integer N and even integer k, denote by H * k (N ) the set of all newforms of level N and of weight k. It is known that

(1.1) |H * k (N )| = k -1 12 ϕ(N ) + O (kN ) 2/3 ,
where ϕ(N ) is the Euler function and the implied constant is absolute. Let m 1 be an integer and let L(s, sym m f ) be the mth symmetric power L-function of f ∈ H * k (N ) normalised so that the critical strip is given by 0 < ℜe s < 1. The values of these functions at the edge of the critical strip contain information of great interest. For example, Serre [START_REF] Serre | Abelian ℓ-adic representation and elliptic Curves[END_REF] showed that the Sato-Tate conjecture is equivalent to L(1 + iτ, sym m f ) = 0 for all m ∈ N and τ ∈ R. The distribution of the values L(1, sym m f ) has received attention of many authors, including Goldfeld, Hoffstein & Lieman [START_REF] Goldfeld | An effective zero-free region[END_REF], Hoffstein & Lockhart [START_REF] Hoffstein | Coefficients of Maass forms and the Siegel zero[END_REF], Luo [START_REF] Luo | Values of symmetric square L-functions at 1[END_REF], Royer [START_REF] Royer | Statistique de la variable aléatoire L(1, sym 2 f )[END_REF][START_REF] Royer | Interprétation combinatoire des moments négatifs des valeurs de fonctions L au bord de la bande critique[END_REF], Royer & Wu [START_REF] Royer | Taille des valeurs de fonctions L de carrés symétriques au bord de la bande critique[END_REF][START_REF] Royer | Special values of symmetric power L-functions and Hecke eigenvalues[END_REF], Cogdell & Michel [START_REF] Cogdell | On the complex moments of symmetric power L-functions at s = 1[END_REF], Habsieger & Royer [START_REF] Habsieger | L-functions of automorphic forms and combinatorics : Dyck paths[END_REF] and Lau & Wu [START_REF] Lau | A density theorem on automorphic L-functions and some applications[END_REF][START_REF] Lau | Extreme values of symmetric power L-functions at 1[END_REF]. In particular, Lau & Wu ([10], [START_REF] Lau | Extreme values of symmetric power L-functions at 1[END_REF]) proved the following results:

(i) For every fixed integer m 1, there are four positive constants A ± m and B ± m such that for any newform f ∈ H * k (1), under the Great Riemann Hypothesis (GRH) for L(s, sym m f ), we have, for k → ∞,

(1.2) {1 + o(1)}(2B - m log 2 k) -A - m L(1, sym m f ) {1 + o(1)}(2B + m log 2 k) A + m .
Here (and in the sequel) log j denotes the j-fold iterated logarithm. For most values of m, the constants A ± m and B ± m can be explicitly evaluated, for example,

         A + m = m + 1, B + m = e γ (m ∈ N), A - m = m + 1, B - m = e γ ζ(2) -1 (odd m), A - 2 = 1, B - 2 = e γ ζ(2) -2 , A - 4 = 5 4 , B - 4 = e γ B ′- 4 ,
where ζ(s) is the Riemann zeta-function, γ denotes the Euler constant and B ′- 4 is a positive constant given by a rather complicated Euler product ( [START_REF] Lau | A density theorem on automorphic L-functions and some applications[END_REF], Theorem 3).

(ii) In the opposite direction, it was shown unconditionally that for m ∈ {1, 2, 3, 4} there are newforms f ± m ∈ H * k (1) such that for k → ∞ ([10], Theorem 2),

(1.3) L(1, sym m f + m ) {1 + o(1)}(B + m log 2 k) A + m , L(1, sym m f - m ) {1 + o(1)}(B - m log 2 k) -A - m .
(iii) In the aim of removing GRH and closing up the gap coming from the factor 2 in (1.2) (comparing it with (1.3)), an almost all result was established. Let ε > 0 be an arbitrarily small positive number, m ∈ {1, 2, 3, 4} and 2 | k. Then there is a subset

E * k of H * k (1) such that |E * k | ≪ H * k (1)e -(log k) 1/2-ε and for each f ∈ H * k (1) E * k , we have, for k → ∞, (1.4
) {1 + O(ε k )}(B - m log 2 k) -A - m L(1, sym m f ) {1 + O(ε k )}(B + m log 2 k) A + m ,
where ε k := (log k) -ε and the implied constants depend on ε only ( [START_REF] Lau | Extreme values of symmetric power L-functions at 1[END_REF], Corollary 2). By comparing (1.3) with (1.4), the extreme values of L(1, sym m f ) seem to be given by (1.3). Clearly it is interesting to investigate further the size of exceptional set E * k . In the case of quadratic characters L-functions, Montgomery & Vaughan [START_REF] Montgomery | Extreme values of Dirichlet L-functions at 1[END_REF] proposed, based on a probabilistic model, three conjectures on the size of exceptional set. The first one has been proved recently by Granville & Soundararajan [START_REF] Granville | The distribution of values of L(1, χ d )[END_REF]. As Cogdell & Michel indicated in [START_REF] Cogdell | On the complex moments of symmetric power L-functions at s = 1[END_REF], it would be interesting to try to get, as close as possible, the analogues of the conjectures of Montgomery-Vaughan for automorphic L-functions. The analogue of Montgomery-Vaughan's first conjecture for the automorphic symmetric power L-functions can be stated as follows.

Conjecture. Let m 1 be a fixed integer and

F k (t, sym m ) := 1 |H * k (1)| f ∈H * k (1), L(1,sym m f ) (B + m t) A + m 1, G k (t, sym m ) := 1 |H * k (1)| f ∈H * k (1), L(1,sym m f ) (B - m t) -A - m 1.
Then there are positive constants c i = c i (m) (i = 1, 2) such that for k → ∞,

(1.5)    e -c1(log k)/ log 2 k ≪ F k (log 2 k, sym m ) ≪ e -c2(log k)/ log 2 k , e -c1(log k)/ log 2 k ≪ G k (log 2 k, sym m ) ≪ e -c2(log k)/ log 2 k .
The aim of this paper is to prove a weaker form of this conjecture for m = 1. In this case, we write, for simplification of notation,

L(s, f ) = L(s, sym 1 f ), F k (t) = F k (t, sym 1 ), G k (t) = G k (t, sym 1 ).
In view of the trace formula of Petersson ([8], Theorem 3.6), it is more convenient to consider the weighted arithmetic distribution function. As usual, denote by

ω f := Γ(k -1) (4π) k-1 f
the harmonic weight in modular forms theory and define the weighted arithmetic distribution functions

F k (t) := f ∈H * k (1) ω f -1 f ∈H * k (1), L(1,f ) (e γ t) 2 ω f , G k (t) := f ∈H * k (1) ω f -1 f ∈H * k (1), L(1,f ) (6π -2 e γ t) -2 ω f .
By using (1.1), the classical estimate

(1.6) f ∈H * k (1)
ω f = 1 + O k -5/6
and the bound of Goldfeld, Hoffstein & Lieman [START_REF] Goldfeld | An effective zero-free region[END_REF]:

(1.7) 1/(k log k) ≪ ω f ≪ (log k)/k,
we easily see that

(1.8) F k (t)/ log k ≪ F k (t) ≪ F k (t) log k, G k (t)/ log k ≪ G k (t) ≪ G k (t) log k.
This shows that in order to prove (1.5) it is sufficient to establish corresponding estimates of the same quality for F k (t) and G k (t). Our main result is the following one.

Theorem 1. For any A 1 there are two positive constants c = c(A) and C = C(A) such that the estimate

(1.9) F k (t) = {1 + ∆ k (t)} exp - e t-γ0 t 1 + O 1 t
holds uniformly for k 16, 2 | k and t T (k), where γ 0 is given by (1.24) below, |θ| 1 and

(1.10) ∆ k (t) := θe t-T (k)-C (t/T (k)) 1/2 + O A e -ce t/5 + (log k) -A , T (k) := log 2 k -5 2 log 3 k -log 4 k -3C.
In particular there are two positive constants c 1 and c 2 such that

(1.11) e -c1(log k)/{(log 2 k) 7/2 log 3 k} ≪ F k (T (k)) ≪ e -c2(log k)/{(log 2 k) 7/2 log 3 k} .
The similar estimates for G k (t) and G k (T (k)) hold also.

Remark 1. The estimates (1.11) of Theorem 1 can be considered as a weaker form of Montgomery-Vaughan's conjecture (1.5) for m = 1, since T (k) ∼ log 2 k as k → ∞. Moreover, if we could take T (k) = log 2 k in (1.11) then (1.9) would lead to the Montgomery-Vaughan's conjecture (1.5). Hence we fail from a shift 5 2 log 3 k + log 4 k + 3C.

It seems however to be rather difficult to resolve completely this conjecture. One of the main difficulties is that there are no analogues of the quadratic reciprocity law and Graham-Ringrose's estimates for short characters sums of friable moduli [START_REF] Graham | Lower bounds for least quadratic nonresidues[END_REF], which have been exploited by Granville & Soundararajan [START_REF] Granville | The distribution of values of L(1, χ d )[END_REF].

In order to prove Theorem 1, we need to introduce a probabilistic model as in [START_REF] Cogdell | On the complex moments of symmetric power L-functions at s = 1[END_REF]. Consider a probability space (Ω, µ), with measure µ. Let SU(2)

♮ be the set of conjugacy classes of SU [START_REF] Goldfeld | An effective zero-free region[END_REF].

The group SU( 2) is endowed with its Haar measure µ H and

SU(2) ♮ = e iθ 0 0 e -iθ : θ ∈ [0, π] ∼
is endowed with the Sato-Tate measure dµ st (θ) := (2/π) sin 2 θ dθ, i.e., the direct image of µ H by the canonical projection SU(2) → SU(2) ♮ . On the space (Ω, µ), define a sequence indexed by the prime numbers, g ♮ (ω) = {g ♮ p (ω)} p of random matrices taking values in SU(2) ♮ , given by

g ♮ p (ω) := e iϑp(ω) 0 0 e -iϑp(ω) ♮ .
We assume that each function g ♮ p (ω) is distributed according to the Sato-Tate measure. This means that, for each integrable function φ : SU(2)

♮ → R, the expected value of φ • g ♮ p is E(φ • g ♮ p ) := Ω φ • g ♮ p (ω) dµ(ω) = π 0 φ e iθ 0 0 e -iθ
• (2/π) sin 2 θ dθ.

Moreover, we assume that the sequence g ♮ (ω) is made of independent random variables. This means that, for any sequence of integrable functions {G p : SU(2) ♮ → R} p , we have

E p G p • g ♮ p := Ω p G p • g ♮ p (ω) dµ(ω) (1.12) = p Ω G p • g ♮ p (ω) dµ(ω) = p π 0 G p e iθ 0 0 e -iθ
• (2/π) sin 2 θ dθ.

Let I be the identity matrix. Then for ℜe s > 1 2 , the random Euler product L(s, g ♮ (ω)) :=

p det I -p -s g ♮ p (ω) -1 =: p L p (s, g ♮ (ω))
turns out to be absolutely convergent a.s. Now we define our probabilistic distribution functions

Φ(t) := Prob L(1, g ♮ (•)) (e γ t) 2 , Ψ(t) := Prob L(1, g ♮ (•)) (6π -2 e γ t) -2 .
We shall prove Theorem 1 in two steps. The first one is to compare F k (t) with Φ(t) (resp. G k (t) with Ψ(t)).

Theorem 2. For any A 1 there are two positive constants c = c(A) and C = C(A) such that the asymptotic formulas

(1.13) F k (t) = Φ(t){1 + ∆ k (t)} and G k (t) = Ψ(t){1 + ∆ k (t)}
hold uniformly for k 16, 2 | k and t T (k), where ∆ k (t) and T (k) are defined by (1.10).

The second step of the proof of Theorem 1 is the evaluation of Φ(t) (resp. Ψ(t)). For this, we consider a truncated random Euler product

L(s, g ♮ (ω); y) := p y L p (s, g ♮ (ω))
and the corresponding distribution functions

Φ(t, y) := Prob L(1, g ♮ (ω); y) (e γ t) 2 , Ψ(t, y) := Prob L(1, g ♮ (ω); y) (6π -2 e γ t) -2 .
We have

(1.14) Φ(t) = Φ(t, ∞) and Ψ(t) = Ψ(t, ∞).
We shall use the saddle-point method (introducted by Hildebrand & Tenenbaum [START_REF] Hildebrand | On integers free of large prime factors[END_REF]) to evaluate Φ(t, y) and Ψ(t, y). For this, we need to introduce some notation. For s ∈ C and y 2, define According to Lemmas 2.3 and 8.1 below, there is an absolute constant c 2 such that for t 4 log c and y ce t , the equation

(1.17) φ 1 (κ, y) = 2(log t + γ)
has a unique positive solution κ = κ(t, y) and for each integer J 1, there are computable constants γ 0 , γ 1 , . . . , γ J such that the asymptotic formula (1.18) κ(t, y) = e t-γ0 1 + J j=1 γ j t j + O J 1 t J+1 + e t t y log y holds uniformly for t 1 and y 2e t , the constant γ 0 beign given by (1.24) below.

Finally write σ n := φ n (κ, y).

Theorem 3. We have

Φ(t, y) = E(κ, y) κ √ 2πσ 2 (e γ t) 2κ 1 + O t e t
uniformly for t 1 and y 2e t .

Theorem 4. For each integer J 1, we have

(1.19) Φ(t, y) = exp -κ J j=1 a j (log κ) j + O J R J (κ, y)
uniformly for t 1 and y 2e t , where the error term R J (κ, y) is given by

(1.20) R J (κ, y) := 1 (log κ) J+1 + κ y log y and (1.21) a j := ∞ 0 h(u) u ′ (log u) j-1 du with (1.22) h(u) :=          log 2 π π 0 e 2u cos θ sin 2 θ dθ if 0 u < 1, log 2 π π 0 e 2u cos θ sin 2 θ dθ -2u if u 1.
As a corollary of Theorem 4, we can obtain an asymptotic developpment for log Φ(t, y) in t -1 . In particular we see that the probabilistic distribution function Φ(t) decays double exponentially as t → ∞.

Corollary 5. For each integer J 1, there are computable constants a * 1 , . . . , a * J such that the asymptotic formula

(1.23) Φ(t, y) = exp -e t-γ0 J j=1 a * j t j + O J R J (e t , y)
holds uniformly for t 1 and y 2e t . Further we have

(1.24) γ 0 := 1 2 ∞ 0 h ′ (u) u du, a * 1 := 1, a * 2 := γ 0 - γ 2 0 2 - ∞ 0 h(u) u 2 (log u) du.
In particular for each integer J 1, we have

(1.25) Φ(t) = exp -e t-γ0 J j=1 a * j t j + O J 1 t J+1
uniformly for t 1.

Remark 2. (i)

The same results hold also for Ψ(t, y). (ii) Taking t = log 2 k and J = 1 in (1.25) of Corollary 5, we see that the probabilistic distribution function Φ(t) (resp. Ψ(t)) verifies Montgomery-Vaughan's conjecture (1.5). But (1.13) is too weak to derive this conjecture for F k (t) (resp. G k (t)). This means that we must take T (k) = log 2 k in Theorem 2, which seems be rather difficult.

(iii) Our method can be generalized (with a little extra effort) to prove that Theorems 1 and 2 hold for L(1, sym m f ) for m 1 (unconditionally when m = 1, 2, 3, 4 and under Cogdell-Michel's hypothesis Sym m (f ) and LSZ m (1) [START_REF] Cogdell | On the complex moments of symmetric power L-functions at s = 1[END_REF] when m 5) and that Theorems 3, 4 and Corollary 5 are true for L(1, sym m g ♮ (ω); y) when m 1.

Acknowledgements. We began working on this paper in November 2004 during the visit of the first author to l'Institut Elie Cartan de Nancy, and finished in January 2006 when the third author visited School of Mathematics and System Sciences of Shandong University. We are indebted to both institutions for invitations and support. The second and third authors want to thank the CRM at Montréal for its invitation. Finally we would express our sincere gratitude to Y.-K. Lau of the University of Hong Kong for valuable discussion. § 2. Expression of E(s, y) and existence of saddle-point

The aim of this section is to prove the existence of the saddle-point κ(t, y), defined by equation (1.17). The first step is to give an explicite expression of E(s, y), which is (1.24) of [START_REF] Cogdell | On the complex moments of symmetric power L-functions at s = 1[END_REF]. For the convenience of readers, we state it here as a lemma.

Lemma 2.1. For prime p, real θ and complex number s, we define

(2.1) D p (θ) := 0 j 1 1 -e i(1-2j)θ p -1 -1 and E p (s) := 2 π π 0 D p (θ) s sin 2 θ dθ.
Then for all s ∈ C and y 2, we have

(2.2) E(s, y) = p y E p (s). Proof. Taking G p (M ♮ ) = det I -p -s ′ M ♮ -s if p y 1 otherwise in (1.
12), we get

E L(s ′ , g ♮ (ω); y) s = p y E L p (s ′ , g ♮ p (ω)) s = p y Ω det 1 -p -s ′ g ♮ p (ω) -s dµ(ω) = p y 2 π π 0 1 -2p -s ′ cos θ + p -2s ′ -s sin 2 θ dθ.
Taking s ′ = 1 and noticing (1.15) and (2.1), we get the desired result.

Lemma 2.2. For all p and σ > 0, we have

E ′′ p (σ)E p (σ) -E ′ p (σ) 2 > 0.
In particular for all σ > 0 and y 2, we have φ 2 (σ, y) > 0.

Proof. By using the definition (2.1) of E p (σ), it is easy to see that

E ′′ p (σ)E p (σ) -E ′ p (σ) 2 = 4 π 2 π 0 D p (θ) σ log 2 D p (θ) sin 2 θ dθ π 0 D p (θ) σ sin 2 θ dθ - 2 π π 0 D p (θ) σ log D p (θ) sin 2 θ dθ 2 = 4 π 2 π 0 π 0 D p (θ 1 ) σ D p (θ 2 ) σ log 2 D p (θ 1 ) -log D p (θ 1 ) log D p (θ 2 ) × × sin 2 θ 1 sin 2 θ 2 dθ 1 dθ 2 .
In view of the symmetry in θ 1 and θ 2 , the same formula holds if we exchange the roles of θ 1 and θ 2 . Thus it follows that

E ′′ p (σ)E p (σ) -E ′ p (σ) 2 = 2 π 2 π 0 π 0 D p (θ 1 ) σ D p (θ 2 ) σ log 2 D p (θ 1 ) D p (θ 2 ) sin 2 θ 1 sin 2 θ 2 dθ 1 dθ 2 .
This proves the first assertion and the second follows immediately.

Lemma 2.3.

There is an absolute constant c 2 such that for t 4 log c and y ce t , the equation φ 1 (σ, y) = 2(log t + γ) has a unique positive solution in σ. Denoting by κ(t, y) this solution, we have κ(t, y) ≍ e t uniformly for t 4 log c and y ce t .

Proof. According to Lemma 4.3 below with the choice of J = 1, we have

φ 1 (σ, y) = 2(log 2 σ + γ) + O(1/ log σ) for y σ 2. Thus φ(ce t , y) = 2 log(t + log c) + 2γ + O 1 t + log c > 2 log t + 2γ and φ(c -1 e t , y) = 2 log(t -log c) + 2γ + O 1 t -log c < 2 log t + 2γ,
provided that c is a large constant and t 4 log c. On the other hand, in view of Lemma 2.2, we know that for any y 2, φ 1 (σ, y) is an increasing function of σ in (0, ∞). Hence the equation φ 1 (σ, y) = 2(log t + γ) has a unique positive solution κ(t, y) and c -1 e t κ(t, y) ce t for t 4 log c and y ce t . This completes the proof. § 3. Preliminary lemmas This section is devoted to establish some preliminary lemmas, which will be useful later. Lemma 3.1. Let j 0 be a fixed real number. Then we have

(3.1) π 0 e 2u cos θ (1 -cos θ) j sin 2 θ dθ ≍ j e 2u u -(j+3/2)
(u 1).

The implied constant depends on j only.

Proof. First we write

π 0 e 2u cos θ (1 -cos θ) j sin 2 θ dθ = π/2 0 e 2u cos θ (1 -cos θ) j + e -2u cos θ (1 + cos θ) j sin 2 θ dθ = 1 0 e 2ut (1 -t) j + e -2ut (1 + t) j (1 -t 2 ) 1/2 dt ≍ 1 0 e 2ut (1 -t) j+1/2 dt + 1 0 e -2ut (1 -t) 1/2 dt.
By the change of variables u(1

-t) = v, it follows that 1 0 e 2ut (1 -t) j+1/2 dt = e 2u u -(j+3/2) u 0 e -2v v j+1/2 dv ≍ e 2u u -(j+3/2) , 1 0 e -2ut (1 -t) 1/2 dt 1 0 e -2ut dt ≪ u -1 .
We obtain the desired result by insertion of these estimates into the preceeding relation.

Lemma 3.2. Let j 0 be an integer and

(3.2) E p,j (σ) := 2 π π 0 D p (θ) σ (1 -cos θ) j sin 2 θ dθ.
(In particular E p,0 (σ) = E p (σ).) Then we have

E p,j (σ) = 2 j+3 π 1 0 1 - 1 p 2 + 4u p -σ u j+1/2 (1 -u) 1/2 du
and the estimate

(3.3) E p,j (σ)/E p (σ) ≪ (p/σ) j
holds uniformly for all primes p and σ > 0. Further if p σ 0, we have

(3.4) E p (σ) ≍ 1.
The implied constant in (3.3) depends on j only and the one in (3.4) is absolute.

Proof. By the change of variables u = sin 2 (θ/2), a simple computation shows that the first assertion is true. Obviously (3.3) holds for j = 0. Now assume that it is true for j. An integration by parts leads to

E p (σ) ≫ j σ p j 1 0 1 - 1 p 2 + 4u p -σ u j+1/2 (1 -u) 1/2 du ≫ j σ p j 1 0 1 - 1 p 2 + 4u p -1 4σ p + 1 2(1 -u) × × 1 - 1 p 2 + 4u p -σ u j+1+1/2 (1 -u) 1/2 du.
On the other hand, we have

0 < u < 1 ⇒ 1 - 1 p 2 + 4u p -1 4σ p + 1 2(1 -u) 1 + 1 p -2 4σ p 16σ 9p .
Inserting it into the preceeding estimate, we see that and let h(u) be defined as in (1.22). Clearly we have

E p (σ) ≫ j σ p j+1 1 0 1 - 1 p 2 + 4u p -σ u j+1+1/2 (1 -u) 1/2 du ≍ j σ p j+1 E p,j+1 (σ)
h(u) = g(u) if 0 u < 1, g(u) -2u if u 1, (3.6) h ′ (u) = g ′ (u) if 0 u < 1, g ′ (u) -2 if u 1, (3.7) h ′′ (u) = g ′′ (u) (u 0, u = 1). (3.8) Lemma 3.3. We have h(u) ≍ u 2 if 0 u < 1, log(2u) if u 1, (3.9) h ′ (u) ≍ u if 0 u < 1, u -1 if u 1, (3.10) h ′′ (u) ≍ 1 if 0 u < 1, u -2 if u 1, (3.11) h ′′′ (u) ≍ u if 0 u < 1, u -3 if u 1. (3.12)
Proof. When 0 u < 1, we have

e 2u cos θ = ∞ n=0 (u cos θ) n n! .
From this we deduce that

h(u) = log 2 π ∞ n=0 u n n! π 0 (cos θ) n sin 2 θ dθ (3.13) = log 1 + ∞ ℓ=1 2 • (2ℓ -1)!! (2ℓ)!(2ℓ + 2)!! u 2ℓ ,
where we have used the following facts:

π 0 (cos θ) 2ℓ+1 sin 2 θ dθ = 0 and 2 π π 0 (cos θ) 2ℓ sin 2 θ dθ =    1 if ℓ = 0, 2 (2ℓ -1)!! (2ℓ + 2)!! if ℓ 1
and where n!! denotes the product of all positive integer from 1 to n having same parity than n. Now we easily deduce, from (3.13), the desired results (3.9)-(3.12) in the case of 0 u < 1. The estimates of (3.9)-(3.12) for u > 1 are simple consequences of (3.1), by noticing the following relations

h ′ (u) = -2 π 0 e 2u cos θ (1 -cos θ) sin 2 θ dθ π 0 e 2u cos θ sin 2 θ dθ , h ′′ (u) = 4 π 0 e 2u cos θ (1 -cos θ) 2 sin 2 θ dθ π 0 e 2u cos θ sin 2 θ dθ -4     π 0 e 2u cos θ (1 -cos θ) sin 2 θ dθ π 0 e 2u cos θ sin 2 θ dθ     2 .
This completes the proof. § 4. Estimates of φ n (σ, y)

The aim of this section is to prove some estimates of φ n (σ, y) for n = 0, 1, 2, 3, 4.

Lemma 4.1. For any fixed integer J 1, we have

(4.1) φ 0 (σ, y) = σ 2 log 2 σ + 2γ + J j=1 b j,0 (log σ) j + O J R J (σ, y)
uniformly for y σ 3, where R J (σ, y) is defined as in (1.20) and

(4.2) b j,0 := ∞ 0 h(u) u 2 (log u) j-1 du.
Proof. By the definition (2.1) of D p (θ) and the one of E p (σ), it is easy to see that for p σ 1/2 , we have

D p (θ) σ = e 2(σ/p) cos θ 1 + O σ p 2 , (4.3) E p (σ) = 1 + O σ p 2 2 π π 0 e 2(σ/p) cos θ sin 2 θ dθ. (4.4)
From these, we deduce that (4.5)

σ 1/2 <p y log E p (σ) = σ 1/2 <p y g(σ/p) + O(σ 1/2 / log σ)
where g(u) is defined as in (3.5).

In order to treat the sum over p σ, we write

E p (σ) = (1 -1/p) -2σ E * p (σ),
where

E * p (σ) := 2 π π 0 1 + 2(1 -cos θ) p 1 - 1 p -2 -σ sin 2 θ dθ.
By using the change of variables u = sin 2 (θ/2), we have

E * p (σ) = 8 π π 0 1 + 4 p 1 - 1 p -2 sin 2 (θ/2) -σ sin 2 (θ/2) cos 2 (θ/2) dθ 8 π p/2σ 0 1 + 4 p 1 - 1 p -2 u -σ u(1 -u) du 8 π 1 + 8 σ -σ p/2σ 0 u(1 -u) du C p σ 3/2
, where C > 0 is a constant. On the other hand, we have trivially E * p (σ) 1 for all p and σ > 0. Thus | log E * p (σ)| ≪ log(σ/p) for p σ 1/2 and (4.6)

p σ 1/2 log E * p (σ) ≪ p σ 1/2 log(σ/p) ≪ σ 1/2 .
Combining (4.5) and (4.6), we can write

p y log E p (σ) = 2σ p σ 1/2 log(1 -1/p) -1 + σ 1/2 <p y g(σ/p) + O(σ 1/2 ).
In view of (3.6) and the following estimate

σ 1/2 <p σ 2σ log(1 -1/p) -1 -2σ/p ≪ σ 1/2 <p σ σ/p 2 ≪ σ 1/2 / log σ,
the preceeding estimate can be written as (4.7)

p y log E p (σ) = 2σ p σ log(1 -1/p) -1 + σ 1/2 <p y h(σ/p) + O(σ 1/2 ).
By using the prime number theorem in the form

(4.8) π(t) := p t 1 = t 2 dv log v + O te -8 √ log t ,
it follows that (4.9)

σ 1/2 <p y h σ p = y σ 1/2 h(σ/t) log t dt + O(R 0 ),
where

R 0 := h σ y ye -8 √ log y + h σ 1/2 σ 1/2 e -4 √ log σ + y σ 1/2 (σ/t)|h ′ (σ/t)|e -8 √ log t dt ≪ σ 2 y e -8 √ log y + σ 1/2 e -2 √ log σ + σ σ 1/2 e -2 √ log t dt + σ 2 y σ e -8 √ log t t 2 dt ≪ σe - √ log σ
by use of Lemma 3.3.

In order to evaluate the integral of (4.9), we use the change of variables u = σ/t to write

y σ 1/2 h(σ/t) log t dt = σ σ 1/2 σ/y h(u) u 2 log(σ/u) du = σ σ 1/2 σ -1/2 h(u) u 2 log(σ/u) du + O R ′ 0 where R ′ 0 := σ σ/y 0 |h(u)| u 2 log(σ/u) du + σ σ -1/2 0 |h(u)| u 2 log(σ/u) du ≪ σ 2 y log y + σ 1/2 log σ .
On the other hand, we have

σ 1/2 σ -1/2 h(u) u 2 log(σ/u) du = 1 log σ σ 1/2 σ -1/2 h(u) u 2 (1 -(log u)/ log σ) du = J j=1 1 (log σ) j σ 1/2 σ -1/2 h(u) u 2 (log u) j-1 du + O 1 (log σ) J+1 .
Extending the interval of integration [σ -1/2 , σ 1/2 ] to (0, ∞) and bounding the contributions of (0, σ -1/2 ] and [σ 1/2 , ∞) by using (3.9) of Lemma 3.3, we have

σ 1/2 σ -1/2 h(u) u 2 (log u) j-1 du = b j,0 + O (log σ) j σ 1/2 .
Combining these estimates, we find that (4.10)

σ 1/2 <p y h σ p = σ J j=1 b j,0 (log σ) j + O J R J (σ, y) .
Now the desired result follows from (4.7), (4.10) and the prime number theorem in the form (4.11)

p σ log(1 -1/p) -1 = log 2 σ + γ + O e -2 √ log σ .
This completes the proof.

Remark 3. In view of (1.3), we can write (4.1) as

φ 0 (σ, y) = σ log(B + 1 log σ) A + 1 + J j=1 b j,0 (log σ) j + O J R J (σ, y)
uniformly for y σ 3. In the case σ < 0, a similar asymptotic formula (with A - 1 , B - 1 and corresponding b - j,0 in place of A + 1 , B + 1 and b j,0 ) can be established uniformly for y -σ 3. As indicated in the introduction, Lemma 4.1 can be easily generalised to the general case m 1. Thus we give an improvement and generalisation of Corollaries A and C of [START_REF] Royer | Interprétation combinatoire des moments négatifs des valeurs de fonctions L au bord de la bande critique[END_REF], of Theorem B of [START_REF] Habsieger | L-functions of automorphic forms and combinatorics : Dyck paths[END_REF], and an improvement of Theorem 1.12 of [START_REF] Cogdell | On the complex moments of symmetric power L-functions at s = 1[END_REF]. It is worthy to indicate that our method seems to be simpler and more natural. Lemma 4.2. We have

(4.12) E ′ p (σ) E p (σ) =            log D p (0) + O 1 σ for all p and σ > 0, 1 2 g ′ σ p log D p (0) + O 1 p 2 + σ p 3 if p σ 1/2 ,
where g(u) is defined as in (3.5).

Proof. First we write

E ′ p (σ) = 2 π π 0 D p (θ) σ log D p (θ) sin 2 θ dθ (4.13) = E p (σ) log D p (0) + R ′ , where (4.14) R ′ := 2 π π 0 D p (θ) σ log D p (θ) D p (0) sin 2 θ dθ. Since log D p (θ) D p (0) = -log 1 + 2p(1 -cos θ) (p -1) 2 2p(1 -cos θ) (p -1) 2 8(1 -cos θ) p , it follows from (3.3) of Lemma 3.2 with j = 1 that R ′ E p (σ) ≪ E p,1 (σ) pE p (σ) ≪ 1 σ
for all p and σ > 0. This implies, via (4.13), the first estimate of (4.12).

We have log

D p (θ) = (cos θ)(2/p) + O 1/p 2 = (cos θ) log D p (0) + O 1/p 2 .
Inserting it and (4.3) into the first relation of (4.13) and in view of (4.4), we can write, for p σ 1/2 ,

E ′ p (σ) = 1 + O σ p 2 2 π π 0 e 2(σ/p) cos θ (cos θ) log D p (0) + O 1 p 2 sin 2 θ dθ = 1 + O σ p 2 2 π π 0 e 2(σ/p) cos θ (cos θ) sin 2 θ dθ log D p (0) + O E p (σ) p 2 .
From this and (4.4), we deduce

E ′ p (σ) E p (σ) = 1 + O σ p 2 1 2 g ′ σ p log D p (0) + O 1 p 2 ,
which implies the second estimate of (4.12). This completes the proof.

Lemma 4.3. Let J 1 be a fixed integer. Then we have

φ 1 (σ, y) = 2 log 2 σ + 2γ + J j=1 b j,1 (log σ) j + O J R J (σ, y)
uniformly for y σ 3, where the constant b j,1 is given by

(4.15) b j,1 := ∞ 0 h ′ (u) u (log u) j-1 du
and R J (σ, y) is defined as in (1.20).

Proof. We have

φ 1 (σ, y) = p y E ′ p (σ)/E p (σ).
Using the first relation of (4.12) for p σ 2/3 and the second for σ 2/3 < p y, we obtain

φ 1 (σ, y) = p σ 2/3 log D p (0) + 1 2 σ 2/3 <p y g ′ σ p log D p (0) + O 1 σ 1/3 .
In view of (3.7), the preceeding formula can be written as

(4.16) φ 1 (σ, y) = p σ log D p (0) + σ 2/3 <p y h ′ σ p log 1 - 1 p -1 + O 1 σ 1/3 .
Similarly to (4.10), we can prove that (4.17)

σ 2/3 <p y h ′ σ p log 1 - 1 p -1 = J j=1 b j,1 (log σ) j + O J R J (σ,
E ′′ p (σ)E p (σ) -E ′ p (σ) 2 E p (σ) 2 =          O 1 σ 2 if p σ 1/2 , 1 p 2 g ′′ σ p + O min 1 σ 2 p , 1 σp 2 if p > σ 1/2 ,
where g(u) is defined as in (3.5).

Proof. First we write

E ′′ p (σ) = 2 π π 0 D p (θ) σ log 2 D p (θ) sin 2 θ dθ (4.19) = E p (σ) log 2 D p (0) + R ′′ , where R ′′ := 2 π π 0 D p (θ) σ log 2 D p (θ) -log 2 D p (0)
E ′′ p (σ)E p (σ) -E ′ p (σ) 2 E p (σ) 2 = R ′′ -2R ′ log D p (0) E p (σ) - R ′ E p (σ) 2 ,
where R ′ is defined as in (4.14).

From the definitions of R ′ and R ′′ , a simple calculation shows that

R ′′ -2R ′ log D p (0) = 2 π π 0 D p (θ) σ log 2 D p (θ) D p (0) sin 2 θ dθ. Since log 2 D p (θ) D p (0) = log 2 1 + 2p(1 -cos θ) (p -1) 2 = 4(1 -cos θ) 2 p 2 + O (1 -cos θ) 2 p 3 , we have R ′′ -2R ′ log D p (0) = 4 p 2 E p,2 (σ) + O E p,2 (σ) p 3 ,
where E p,j (σ) is defined as in (3.2). By using (3.3) with the choice of j = 2 and the trivial estimate E p,2 (σ) 4E p (σ), we deduce

(4.21) R ′′ -2R ′ log D p (0) E p (σ) = 4 p 2 E p,2 (σ) E p (σ) + O min 1 σ 2 p , 1 p 3 .
Similarly we have

log D p (θ) D p (0) = -log 1 + 2p(1 -cos θ) (p -1) 2 = - 2(1 -cos θ) p + O (1 -cos θ) p 2 ,
and therefore

R ′ = - 2 p E p,1 (σ) + O E p,1 (σ) p 2 . Now (3.
3) with j = 1 and the trivial estimate E p,1 (σ) 2E p (σ) imply

R ′ E p (σ) 2 = 4 p 2 E p,1 (σ) E p (σ) 2 + O E p,1 (σ) 2 p 3 E p (σ) 2 (4.22) = 4 p 2 E p,1 (σ) E p (σ) 2 + O min 1 σ 2 p , 1 p 3 .
Inserting (4.21) and (4.22) into (4.20) and in view of (4.14), we deduce (4.23)

E ′′ p (σ)E p (σ) -E ′ p (σ) 2 E p (σ) 2 = 4 p 2 h p (σ) + O min 1 σ 2 p , 1 p 3
for all p and σ > 0, where

h p (σ) := E p,2 (σ) E p (σ) - E p,1 (σ) E p (σ) 2 .
When p σ

(σ) = g ′′ σ p 1 + O σ p 2 = g ′′ σ p + O min σ p 2 , 1 σ .
Inserting it into (4.23) and in view of Lemma 3.1, we get, for p σ 1/2 ,

E ′′ p (σ)E p (σ) -E ′ p (σ) 2 E p (σ) 2 = 1 p 2 g ′′ σ p + O min 1 σ 2 p , 1 σp 2 .
This completes the proof.

Lemma 4.5. Let J 1 be a fixed integer. Then we have

φ 2 (σ, y) = 1 σ J j=1 b j,2 (log σ) j + O J R J (σ, y) uniformly for y σ 2, where b j,2 := ∞ 0 h ′′ (u)(log u) j-1 du. In particular b 1,2 = 2.
Proof. From Lemma 4.4 and (3.8), we deduce easily that

φ 2 (σ, y) = p y E ′′ p (σ)E p (σ) -E ′ p (σ) 2 E p (σ) 2 = σ 1/2 <p y g ′′ (σ/p) p 2 + O 1 σ 3/2 log σ = σ 1/2 <p y h ′′ (σ/p) p 2 + O 1 σ 3/2 log σ .
Similarly to (4.10), we can prove that

σ 1/2 <p y h ′′ (σ/p) p 2 = 1 σ J j=1 b j,2 (log σ) j + O J R J (σ, y) ,
by using (3.11), (3.12) and (4.8). Now the desired result follows from the preceeding two estimates.

Finally

b 1,2 = 1 0 h ′′ (u) du + ∞ 1 h ′′ (u) du = h ′ (1-) -h ′ (1+) = h ′ (1-) -h ′ (1-) -2 = 2.
This completes the proof.

Similarly (even more easily, since we only need an upper bound instead of an asymptotic formula), we can prove the following result.

     1 if |τ | c 1 σ 1/2 log σ or |τ | y 1/δ , e -c2τ 2 /[σ(log σ) 2 ] if c 1 σ 1/2 log σ |τ | σ, e -c3|τ | δ if σ |τ | y 1/δ .
Proof. First we write

E p (s) = 2 π π 0 D p (θ) -1 -s sin 2 θ dθ = 2 π π 0 sin 2 θ (1 -s)(D p (θ) -1 ) ′ d D p (θ) -1 1-s .
Since (D p (θ) -1 ) ′ = 2p -1 sin θ, after a simplification and an integration by parts it follows that

E p (s) = p π(s -1) π 0 D p (θ) s-1 cos θ dθ = p π(s -1) π/2 0 D p (θ) s-1 -D p (π -θ) s-1 cos θ dθ. This implies that (5.2) E p (s) E p (σ) = σ -1 s -1 E * p (s) E * p (σ) with E * p (s) := π/2 0 D p (θ) s-1 -D p (π -θ) s-1 cos θ dθ. 1 • Case of σ 1/δ < |τ | y 1/δ Write E * p (s) = π/2 0 D p (θ) s-1 1 -∆ p (θ) s-1 cos θ dθ with ∆ p (θ) := 1 -2p -1 cos θ + p -2 1 + 2p -1 cos θ + p -2 .
It is clear that for all p, the function θ → ∆ p (θ) is increasing on [0, π/2]. It follows that

E * p (σ) π/4 0 D p (θ) σ-1 1 -∆ p (θ) σ-1 cos θ dθ 1 -∆ p (π/4) σ-1 π/4 0 D p (θ) σ-1 cos θ dθ
for all p and σ 1. This implies that

(5.3) 1 E * p (σ) π/4 0 D p (θ) σ-1 cos θ dθ 1 1 -∆ p (π/4) σ-1 .
Similarly since the function θ → D p (θ) σ-1 cos θ is decreasing on [0, π/2] for all p and σ 2, we can deduce, via (5.3), that

(5.4) 1 E * p (σ) π/2 π/4 D p (θ) σ-1 cos θ dθ 1 1 -∆ p (π/4) σ-1 .
From (5.3) and (5.4), we deduce that

E * p (s) E * p (σ) 2 1 -∆ p (π/4) σ-1 .
It is easy to verify that for all p σ 2, we have

∆ p π 4 σ-1 1 - √ 2 p + 1 p 2 σ-1 1 - σ -1 4p .
Combining these estimates with (5.2), we obtain

E p (s) E p (σ) 8p |s -1| p 4 |τ | (p σ).
By multiplying this inequality for σ < p |τ | δ ( y) and the trivial inequality |E p (s)| |E p (σ)| for the others p, we deduce, via the prime number theorem, that

E(s, y) E(σ, y) exp - σ<p |τ | δ log |τ | + 4 σ<p |τ | δ log p e -{1/δ-4+o(1)}|τ | δ . 2 • Case of c 1 σ 1/2 log σ |τ | σ 1/δ
For p σ 1/2 2, we can write

|E * p (s)| π/2 0 D p (θ) σ-1 + D p (π -θ) σ-1 cos θ dθ = 1 + O σ p 2 π/2 0
e 2[(σ-1)/p] cos θ + e -2[(σ-1)/p] cos θ cos θ dθ and

|E * p (σ)| = π/2 0 D p (θ) σ-1 -D p (π -θ) σ-1 cos θ dθ = 1 + O σ p 2 π/2 0 e 2[(σ-1)/p] cos θ -e -2[(σ-1)/p] cos θ cos θ dθ.
From these, we deduce that (5.5)

E * p (s) E * p (σ) 1 + O σ p 2 + 1 e σ/p
(2 σ 1/2 p σ)

where we have used the following facts π/2 0 e 2[(σ-1)/p] cos θ cos θ dθ ≫ e σ/p and π/2 0 e -2[(σ-1)/p] cos θ cos θ dθ ≪ 1.

Inserting (5.5) into (5.2), for 2 σ 1/2 p σ we obtain

E p (s) E p (σ) exp -log s -1 σ -1 + C σ p 2 + 1 e σ/p e -τ 2 /(2σ 2 )+Cσ/p 2 +Ce -σ/p if 3 |τ | σ, e -1 2 log(1+τ 2 /σ 2 )+Cσ/p 2 +Ce -σ/p if σ |τ | σ 1/δ ,
where C > 0 is an absolute constant. Now by multiplying these inequalities for σ/(4 log σ) p σ/(2 log σ) and the trivial inequality |E p (s)| E p (σ) for the other p, we get

E(s, y) E(σ, y) exp - σ/(4 log σ) p σ/(2 log σ) τ 2 2σ 2 - Cσ p 2 - C e σ/p exp - τ 2 16σ(log σ) 2 -10C - 10C σ log σ exp - c 2 τ 2 σ(log σ) 2 if c 1 σ 1/2 log σ |τ | σ, and
E(s, y) E(σ, y) exp - σ/(4 log σ) p σ/(2 log σ) 1 2 log 1 + τ 2 σ 2 - Cσ p 2 - C e σ/p (5.6) exp - σ 8 log σ log 1 + τ 2 σ 2 -10C - 10C σ log σ exp -c 3 |τ | δ if σ |τ | σ 1/δ
. This completes the proof. § 6. Proof of Theorem 3

We follow the argument of Granville & Soundararajan [START_REF] Granville | The distribution of values of L(1, χ d )[END_REF] to prove Theorem 3. We shall divide the proof in several steps which are embodied in the following lemmas.

The first one is a classic integration formula (see [START_REF] Granville | The distribution of values of L(1, χ d )[END_REF], page 1019).

Lemma 6.1. Let c > 0, λ > 0 and N ∈ N. Then we have

(6.1) 1 2πi c+i∞ c-i∞
y s e λs -1 λs

N ds s =      0 if 0 < y < e -λN , ∈ [0, 1] if e -λN y < 1, 1 if y 1.
The second one is an analogue for (3.6) and (3.7) of [START_REF] Granville | The distribution of values of L(1, χ d )[END_REF] (see also Lemma 3.1 of [START_REF] Wu | Note on a paper by A. Granville[END_REF]).

Lemma 6.2. Let t 1, y 2e t and 0 < λ e -t . Then we have

(6.2) Φ(t, y) 1 2πi κ+i∞ κ-i∞ E(s, y) (e γ t) 2s e λs -1 λs ds s Φ(te -λ , y), (6.3)
Φ(te -λ , y) -Φ(t, y) 1 2πi κ+i∞ κ-i∞ E(s, y) (e γ t) 2s
e λs -1 λs e 2λs -e -2λs ds s .

Proof. Denote by 1 X (ω) the characteristic function of the set X ⊂ Ω. Then by Lemma 6.1 with N = 1 and c = κ, we have

1 {ω∈Ω:L(1,g ♮ (ω);y)>(e γ t) 2 } (ω) 1 2πi κ+i∞ κ-i∞ L(1, g ♮ (ω); y) (e γ t) 2 s e λs -1 λs 2 ds.
Integrating over Ω and interchanging the order of integrations yield Φ(t, y)

Ω 1 2πi κ+i∞ κ-i∞ L(1, g ♮ (ω); y) (e γ t) 2 s e λs -1 λs 2 ds dµ(ω) = 1 2πi κ+i∞ κ-i∞ E(s, y) (e γ t) 2s e λs -1 λs 2 ds.
This proves the first inequality of (6.2). The second can be treated by noticing that

1 {ω∈Ω:L(1,g ♮ (ω);y)>(e γ-λ t) 2 } (ω) = 1 {ω∈Ω:L(1,g ♮ (ω);y)>(e γ t) 2 } (ω) + 1 {ω∈Ω:(e γ t) 2 L(1,g ♮ (ω);y)>(e γ-λ t) 2 } (ω) 1 2πi κ+i∞ κ-i∞ L(1, g ♮ (ω); y) (e γ t) 2 s e λs -1 λs 2 ds.
From (6.2), we can deduce

Φ(te -λ , y) -Φ(t, y) 1 2πi κ+i∞ κ-i∞ E(s, y) (e γ-λ t) 2s e λs -1 λs 2 ds - 1 2πi κ+i∞ κ-i∞ E(s, y) (e γ+λ t) 2s e λs -1 λs 2 ds = 1 2πi κ+i∞ κ-i∞ E(s, y) (e γ t) 2s e λs -1
λs 2 e 2λs -e -2λs ds.

This completes the proof. Lemma 6.3. Let t 1, y 2e t and 0 < κλ 1. Then we have

1 2πi κ+iκ κ-iκ E(s, y) (e γ t) 2s e λs -1 λs 2 ds = E(κ, y) κ √ 2πσ 2 (e γ t) 2κ 1 + O κλ + log κ κ .
Proof. First in view of (4.24) we write, for s = κ + iτ and |τ | κ,

E(s, y) = exp σ 0 + iσ 1 τ - σ 2 2 τ 2 -i σ 3 6 τ 3 + O σ 4 τ 4
and e λs -1

λs 2 = 1 κ 1 - i κ τ + O κλ + τ 2 κ 2 .
Since σ 1 = log t + γ, we have

E(s, y) (e γ t) 2s e λs -1 λs 2 = E(κ, y) κ(e γ t) 2κ e -(σ2/2)τ 2 1 - i κ τ -i σ 3 6 τ 3 + O R(τ ) with R(τ ) := κλ + κ -2 τ 2 + σ 4 τ 4 + σ 2 3 τ 6 .
Now we integrate the last expression over |τ | κ to obtain (

t) 2s e λs -1

λs 2 ds = E(κ, y) 2πκ(e γ t) 2κ κ -κ e -(σ2/2)τ 2 1 + O R(τ ) dτ,
where we have used the fact that the integrals involving (i/κ)τ and (iσ 3 /6)τ 3 vanish.

On the other hand, using lemmas 4.5 and 4.6 we have

κ -κ e -(σ2/2)τ 2 dτ = 2π σ 2 1 + O exp - 1 2 κ 2 σ 2 , κ+iκ κ-iκ e -(σ2/2)τ 2 R(τ ) dτ ≪ 1 √ σ 2 κλ + 1 κ 2 σ 2 + σ 2 3 σ 3 2 + σ 4 σ 2 2 ≪ 1 √ σ 2 κλ + log κ κ .
Inserting these into (6.4), we obtain the desired result. uniformly for t 1, y 2e t , κ 2 and 0 < λκ 1, where

R 1 := λ -1 e -c3κ δ + λ -1 (κ/ log κ) 1/2 y -1/δ , R 2 := λκ(log κ) 1/2 + e -(c3/2)κ δ + λ -1 (κ/ log κ) 1/2 y -1/δ .
Proof. We split the integral in (6.5) into two parts according to κ |τ | y 1/δ or |τ | y 1/δ . Using Lemma 5.1 with σ = κ and the inequality (e λs -1)/s 2 ≪ 1/τ 2 , the integral in (6.5) is ≪ E(κ, y) (e γ t) 2κ λ e -c3κ δ κ + 1 y 1/δ , which implies (6.5), in view of Lemma 4.5 with J = 1.

Similarly we split the integral in (6.6) into four parts according to uniformly for t 1, y 2e t and 0 < λ e -t . Obviously the estimates (6.9) and (6.10) imply the desired result. This completes the proof of Theorem 3. § 7. Proof of Theorem 4

|τ | c 1 κ 1/2 log κ, c 1 κ 1/2 log κ < |τ | κ, κ < |τ | y 1/
Using Lemmas 4.1 and 4.5, we can write

E(κ, y) κ √ 2πσ 2 (e γ t) 2κ = exp φ(κ, y) -2κ(γ + log t) + O(log κ) = exp κ 2 log 2 κ -2 log t + J j=1 b j,0 (log κ) j + O J R J (κ, y)
.

On the other hand, Lemma 4.3 and (1.17) imply that

2 log 2 κ + 2γ + J j=1 b j,1 (log κ) j + O J R J (κ, y) = 2(log t + γ).
Combining these estimates, we can obtain

E(κ, y) κ √ 2πσ 2 (e γ t) 2κ = exp -κ J j=1 b j,1 -b j,0 (log κ) j + O J R J (κ, y) .
In view of (1.21), (4.2) and (4.15), we have b j,1 -b j,0 = a j . This completes the proof. § 8. Proof of Corollary 5

We first prove an asymptotic developpment of κ(t, y) in t.

Lemma 8.1. For each integer J 1, there are computable constants γ 0 , γ 1 , . . . , γ J such that the asymptotic formula (8.1) κ(t, y) = e t-γ0 1 + J j=1

γ j t j + O J R * J (t, y)
holds uniformly for t 1 and y 2e t , where

R * N (t, y) := 1 t N +1 + e t t y log y .
Further γ 0 is given by (1.24) and

γ 1 = -1 8 b 2 1,1 -1 4 b 2,1 .
Proof. By Lemma 4.3 and (1.17), we have

(8.2) 2 log t = 2 log 2 κ + J+1 j=1 b j,1 (log κ) j + O J R J+1 (κ, y) ,
where R J (κ, y) is defined as in (1.20). From (8.2), we easily deduce that

t = (log κ) J+1 j=1 exp b j,1 2(log κ) j exp O J R J+1 (κ, y) = (log κ) J+1 j=1 J+1 mj =0 1 m j ! b j,1 2(log κ) j mj + O J R J+1 (κ, y) .
Developping the product, we get

t = (log κ) J+1 j=0 b ′ j (log κ) j + O J R J+1 (κ, y) , where b ′ j := m1 0,...,mJ+1 0 m1+2m2+•••+(J+1)mJ+1=j b m1 1,1 • • • b mJ+1 J+1,1 (2m 1 )!! • • • (2m J+1 )!! = m1 0,...,mj 0 m1+2m2+•••+jmj =j b m1 1,1 • • • b mj j,1 (2m 1 )!! • • • (2m j )!! . Since b ′ 0 = 1 and b ′ 1 = b 1,1 /2 =
γ 0 , the preceeding asymptotic formula can be written as

(8.3) t = log κ + γ 0 + J j=1 b ′ j+1 (log κ) j + O J R * J (t, y) ,
where we have used the fact that κ(t, y) ≍ e t (see Lemma 2.3) and (log k)R J+1 (κ, y) ≍ R * J (t, y). With the help of (8.3), a simple recurrence argument shows that there are constants γ ′ n such that (8.4)

t = log κ + J j=0 γ ′ j t j + O J R * J (t, y) .
In fact taking J = 0 in (8.3), we see that (8.4) holds for J = 0. Suppose that it holds for 0, . . . , J -1, i.e.

t = log κ + J-j-1 i=0 γ ′ i t i + O R * J-j-1 (t, y) (j = 0, . . . , J -1), which is equivalent to (8.5) log κ = t 1 - J-j i=1 γ ′ i-1 t i + O R * J-j-1 (t, y) t (j = 0, . . . , J -1).
This holds also for j = J if we use the convention:

-1 i=0 = 0 and R * -1 (t, y) := 1, since log κ = t + O(1). Inserting it into (8.3), we easily see that (8.4) holds also for J. In particular we have

γ ′ 1 = b ′ 2 = 1 8 b 2 1,1 + 1 4 b 2,1 .
Now (8.1) is an immediate consequence of (8.4) with

γ j := m1 0,...,mJ 0 m1+2m2+•••+JmJ =j (-1) m1+•••+mJ γ ′ 1 m1 • • • γ ′ J mJ m 1 ! • • • m J ! .
This completes the proof. Now we are ready to prove Corollary 5.

Using (8.5), we have

J j=1 a j (log κ) j = J j=1 a j t j 1 - J-j i=1 γ ′ i-1 t i + O N R * J-j-1 (t, y) t -j (8.6) = J j=1 ρ j t j + O J R * J-2 (t, y) t 2 ,
where the ρ n are constants. In particular we have ρ 1 = a 1 = 1 and ρ 2 = γ 0 + a 2 . Now Theorem 4, (8.1) and (8.6) imply the result of Corollary with

a * 1 = ρ 1 = 1, a * j = ρ j + j-1 i=1 γ i ρ j-i (j 2).
This completes the proof of Corollary 5. § 9. Proof of Theorem 2

For each η ∈ (0, 1 2 ), define

H + k (1; η) := f ∈ H * k (1) : L(s, f ) = 0, s ∈ S ,
where S := {s :

= σ + iτ : σ 1 -η, |τ | 100k η } ∪ {s := σ + iτ : σ 1, τ ∈ R}, and
H - k (1; η) := H * k (1) H + k (1; η).
Then we have (see [START_REF] Lau | A density theorem on automorphic L-functions and some applications[END_REF], (1.11))

(9.1) H - k (1; η) ≪ η k 31η .
Our starting point in the proof of Theorem 2 is the evaluation of the moments of L(1, f ). For this, we recall a particular case of Proposition 6.1 of [START_REF] Lau | A density theorem on automorphic L-functions and some applications[END_REF]. Lemma 9.1. Let η ∈ (0, 1 31 ) be fixed. There are two positive constants c i = c i (η) (i = 4, 5) such that (9.2)

f ∈H + k (1;η) ω f L(1, f ) s = E(s) + O η e -c4 log k/ log 2 k
uniformly for

(9.3) k 16, 2 | k and |s| 2T k with T k := c 5 log k/(log 2 k log 3 k).
Here E(s) is defined by (1.15).

Let κ(t, y) be the saddle-point determined by (1.17) and κ t := κ(t, ∞). For k 16, 2 | k, λ > 0, N ∈ N and t > 0, introduce the two integrals

I 1 (k, t; λ, N ) := 1 2πi κt+i∞ κt-i∞ f ∈H + k (1;η) ω f L(1, f ) (e γ t)
k (t) + O η k -5/6 I 1 (k, t; λ, N ) F k (te -λN ) + O η k -5/6 , (9.4) Φ(t) I 2 (k, t; λ, N ) Φ(te -λN ) (9.5)
uniformly for k 16, 2 | k, λ > 0, N ∈ N and t > 0. The implied constants depend on η only.

Proof. By exchanging the order of sommation and by using Lemma 6.1 with c = κ t , we obtain

I 1 (k, t; λ, N ) = f ∈H + k (1;η) ω f 2πi κt+i∞ κt-i∞ L(1, f ) (e γ t) 2 s e λs -1 λs 2N ds s , f ∈H + k (1;η), L(1,f ) (e γ t) 2 ω f .
In view of the second estimate of (1.7) and of (9.1), we reintroduce the missing forms

I 1 (k, t; λ, N ) f ∈H * k (1), L(1,f ) (e γ t) 2 ω f + O f ∈H * k H + k (1;η) ω f f ∈H * k (1), L(1,f ) (e γ t) 2 ω f + O k -1+31η log k .
Clearly this implies the first inequality of (9.4), thanks to (1.6) and (1.7). Similarly, using Lemma 6.1 with c = κ t , we find I 1 (k, t; λ, N )

f ∈H + k (1;η) L(1,f ) (e γ t) 2 ω f + f ∈H + k (1;η) (e γ te -λN ) 2 L(1,f )<(e γ t) 2 ω f = f ∈H + k (1;η) L(1,f ) (e γ te -λN ) 2 ω f .
As before, we can easily show that the last sum is F k (te -λN ) + O k -5/6 . The estimates (9.5) can be proved in the same way as (6.2).

Lemma 9.3. Let η ∈ (0, 1 + e λκt λT k

2N

.

Combining (9.7) and (9.8) yields to the required estimate.

End of the proof of Theorem 2

For simplicity of notation, we write On the other hand, by using Theorem 3 and (1.25), it is easy to see that there is a positive constant c such that Φ(te λN ) ≍ Φ(t) ∼ E(κ t ) κ t √ 2πσ 2 (e γ t) 2κt ≫ e -c8e t /t ≫ e -c9(log k)/[(log 2 k) 7/2 log 3 k] for t T (k). Thanks to Lemma 4.5, the previous estimate can be written as Inserting (9.12) and (9.16) into (9.9) and (9.13) and (9.15) into (9.10), we obtain F k (t) Φ(t) 1 + e t-T (k)-C (t/T (k)) 1/2 + O e -c6e δt + (log k) -A and F k (t) Φ(t) 1 -e t-T (k)-C (t/T (k)) 1/2 + O e -c6e δt + (log k) -A .

This implies the first asymptotic formula of (1.13) by taking η = 1 200 and δ = 1 5 . The second can be established similarly. This completes the proof of Theorem 2. § 10. Proof of Theorem 1

The formula (1.9) is an immediate consequence of Theorem 2 and (1.25). Taking t = T (k) in (1.9), we find that

(1. 15 )

 15 E(s, y) := E L(1, g ♮ (ω); y) s and E(s) := E(s, ∞), where E(•) denotes the expected value. We define also (1.16) φ(s, y) := log E(s, y), φ n (s, y) := ∂ n φ ∂s n (s, y) (n 0).

π π 0 e

 0 .Thus(3.3) holds also for j + 1.Since (1+1/p) -2 D p (θ) (1-1/p) -2 for all primes p and any θ ∈ R, we have D p (θ) σ ≍ 1 uniformly for p σ 0 and θ ∈ R. This implies(3.4).Introduce the function(3.5) g(u) := log 2 2u cos θ sin 2 θ dθ (u 0)

  y) , using (3.10),(3.11) and (4.11) instead of (3.9), (3.10) and (4.8). Now the desired result follows from (4.16), (4.10) and (4.17).

Lemma 4 . 4 .

 44 We have(4.18)

Lemma 4 . 6 .Lemma 5 . 1 .

 4651 We have (4.24) φ n (σ, y) ≪ 1/(σ n-1 log σ) (n = 3, 4) uniformly for y σ 3. § 5. Estimate of |E(κ + iτ, y)| For any δ ∈ (0, 1 4 ), there are two absolute positive constants c 1 , c 2 and a positive constant c 3 = c 3 (δ) such that for all y σ 3 we have (5.1) E(σ + iτ, y) E(σ, y)

IF 6 = I 2 +

 62 j := I j (k, t; λ, N ) andI + j := I j (k, te λN ; λ, N ) k (t) I 1 + O k -5/O |I 1 -I 2 | + k -5/6 Φ(te -λN ) + O |I 1 -I 2 | + k -5/6 Φ(t) + |Φ(te -λN ) -Φ(t)| + O |I 1 -I 2 | + k -5/6

  k) A .Similarly we can prove (even more easily)(9.15) |I 1 -I 2 | ≪ Φ(t)/(log k) A .

(10. 1 ) e -c ′ 1 (

 11 log k)/{(log 2 k) 7/2 log 3 k} ≪ F k (T (k)) ≪ e -c ′ 2 (log k)/{(log 2 k) 7/2 log 3 k} ,where c ′ 1 and c ′ 2 are two positive constants. Clearly (10.1) and (1.8) imply(1.11). The related results on G k (t) and G k (T (k)) can be proved similarly. This completes the proof of Theorem 1.

  sin 2 θ dθ.

	Using (4.13) and (4.19), we can deduce
	(4.20)

  1/2 , the inequality (3.3) of Lemma 3.2 implies that h p (σ) ≪ (p/σ) 2 . From this and (4.23) we deduce the first estimate of (4.18).

	If p σ 1/2 , we can use (4.3), (3.11) and (3.8) to write
	4h p

  Lemma 6.4. Let δ and c 3 be two constants determined by Lemma 5.1. Then we have

	(6.5)			κ±i∞ κ±iκ	E(s, y) (e γ t) 2s	e λs -1 λs 2 ds ≪	κ	E(κ, y) √ σ 2 (e γ t) 2κ R 1 ,
	(6.6)	κ+i∞ κ-i∞	E(s, y) (e γ t) 2s	e λs -1 λs 2 e 2λs -e -2λs ds ≪	κ	E(κ, y) √ σ 2 (e γ t) 2κ R 2 ,

  δ , |τ | y 1/δ . (e γ t) 2κ λκ 1/2 log κ + e -c3κ δ + λ -1 y -1/δ ,

	By Lemma 5.1 with σ = κ and the inequalities
							(e λs -1)/λs ≪ min{1, 1/(λ|τ |)},
				(e 2λs -e -2λs )/s ≪ min{λ, 1/|τ |},
	the integral in (6.6) is, as before,
			≪ ε	E(κ, y)
	which implies (6.6), as before.
	Now we are ready to complete the proof of Theorem 3. Lemma 6.3 and (6.5) of Lemma 6.4
	give
	(6.7)	1 2πi	κ+i∞ κ-i∞	E(s, y) (e γ t) 2s	e λs -1 λs 2 ds =	κ	E(κ, y) 2πσ 2 (e γ t) 2κ 1 + O R ′ √
	where		R ′ :=	log κ κ	+ κλ +	e -c3κ δ + (κ/ log κ) 1/2 y -1/δ λ	.
	Taking λ = κ -2 and noticing y 2e t ≍ κ and 1/δ > 4, we deduce
	(6.8)							R ′ ≪ t/e t .
	Combining (6.7) and (6.8) with (6.2), we obtain
	(6.9)		Φ(t, y)		κ	E(κ, y) 2πσ 2 (e γ t) 2κ 1 + O √	t e t	Φ(te -λ , y)
	uniformly for t 1, y 2e t and 0 < λ e -t .
	On the other hand, (6.3) of Lemma 6.2 and (6.6) of Lemma 6.4 imply
		Φ(te -λ , y) -Φ(t, y) ≪	κ	E(κ, y) √ σ 2 (e γ t) 2κ λκ(log κ) 1/2 +	(κ/ log κ) 1/2 e c3κ δ	+	(κ/ log κ) 1/2 λy 1/δ
			≪	κ	E(κ, y) √ σ 2 (e γ t) 2κ λκ(log κ) 1/2 +	(κ/ log κ) 1/2 e c3κ δ
	when y -1/(2δ) κ -1/2 (log κ) -1 λ κ -1 . Since Φ(te -λ , y) -Φ(t, y) is a non-decreasing function of λ, we deduce
	(6.10)	Φ(te -λ , y) -Φ(t, y) ≪	κ	E(κ, y) √ σ 2 (e γ t) 2κ λκ(log κ) 1/2 +	(κ/ log κ) 1/2 e c3κ δ	+	κ(log κ) 1/2 y 1/(2δ)

  1 200 ] be fixed and c 4 be the positive constant given by Lemma 9.1. Then we have(9.6) |I 1 (k, t; λ, N ) -I 2 (k, t; λ, N )| ≪ e -c4(log k)/ log 2 k (1 + e λκt ) 2N log T k (e γ t) 2κt + E(κ t ) + e -c4(log k)/ log 2 k N (e γ t) 2κt1 + e λκt λT k2Nuniformly for λ > 0, N ∈ N, k 16, 2 | k and t T (k), where T (k) is given by (1.10). The implied constant depends on η only.Proof. By the definitions of I 1 and I 2 , we can writeI 1 (k, t; λ, N ) -I 2 (k, t; λ, N )In order to estimate the last integral, we split it into two parts according to |τ | T k or |τ | > T k .In view of (1.18), it is easy to see that κ t T k for t T (k). Thus we may apply (9.2) of Lemma 9.1 for s = κ t + iτ with |τ | T k . Note that |(e λs -1)/(λs)| 1 + e λκt for s = κ t + iτ , which is easily seen by looking at the cases |λs| 1 and|λs| > 1. The contribution of |τ | T k to |I 1 (k, t; λ, N ) -I 2 (k, t; λ, N )| is (9.7) ≪ e -c4(log k)/ log 2 k (1 + e λκt ) 2N log T k (e γ t) 2κt .Since κ t T k for t T (k), we can apply (9.2) of Lemma 9.1 to write, for s = κ t + iτ with τ ∈ R,

	=	1 2πi	κt+i∞ κt-i∞	f ∈H + k (1;η)	ω f L(1, f ) s -E(s)	e λs -1 λs	2N	ds s(e γ t) 2s .
		f ∈H + k (1;η)
						1 + e λκt	2N dτ
						|τ | T k	λ|τ |		|τ |
			≪	E(κ

ω f L(1, f ) s -E(s) f ∈H + k (1;η) ω f L(1, f ) κt + E(κ t ) 2E(κ t ) + O e -c4(log k)/ log 2 k . Thus the contribution of |τ | > T k to |I 1 (k, t; λ, N ) -I 2 (k, t; λ, N )| is (9.8) ≪ E(κ t ) + e -c4(log k)/ log 2 k (e γ t) 2κt t ) + e -c4(log k)/ log 2 k

N (e γ t) 2κt

and

(9.10)

In view of (6.10) and Theorem 3, we have

for λN e -t . Take Since T k = e T (k)+ 3 2 log 3 k+2C+log c5 , it is easy to see that

Inserting these estimates into the preceeding inequality, a simple calculation shows that (9.12)

provided the constant C is suitably large, where c 6 = c 6 (η, δ) is a positive constant.

Similarly by using (6.10) with te λN in place of t, we have Φ(t) -Φ(te λN ) ≪ Φ(te λN ) λN κ te λN (log κ te λN ) 1/2 + e -(c3/2)κ δ te λN

.

Since for t T (k) we have

the preceeding estimate can be writen as

from which we deduce that (9.13) Φ(t) -Φ(te λN ) Φ(t) e t-T (k)-C (t/T (k))


Suzanne Lesecq

Laurent Ouvry

Marc Correvon

Gabriela Dudnik

John Barrett

Alan Mcgibney

François Birot

Hugues De Chaumont

Richard Banach

Jean-Marc Van Gyseghem

J.-M Van Gyseghem

J Herveg

C Jackson

SensL S Buckley

A Di Matteo

C Ó'murchú

A Mathewson

INSPEX: Integrated Smart Spatial Exploration System

Keywords:

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction and objectives

A large amount of work has been recently done on obstacle avoidance systems for autonomous vehicles. To detect different types of obstacles across the full range of possible lighting and weather conditions, the obstacle detection systems usually combine multiple subsystems including LiDAR, radar, IR and optical. The data from these subsystems is merged with vehicle orientation and navigation subsystems. These systems are typically heavy, power hungry, and require high computational capabilities. INSPEX partners have asked themselves "what if we could miniaturise an obstacle detection system like this and reduce its power consumption so that it could be portable/wearable?" This is very challenging but it would open many new applications in, for example, assistive guidance for the visually impaired, human guidance in low visibility conditions (night, smoke, fog) obstacle detection for small or humanoid robots, and small drone obstacle detection for obstacle avoidance procedures. This encapsulates the objective of INSPEX: to combine several range sensors with an IMU, environmental sensing, signal and data processing, wireless communications, energy management and user interface, all in a miniature, low power, small size, light weight device (Fig. 1), with a flexible system integration methodology that will allow the INSPEX smart spatial exploration system to be applied to a range of different application areas.

INSPEX first demonstrator targets the Visually Impaired and Blind (VIB) community, and field tests will be conducted. Therefore, ethical issues must be considered. Moreover, the device being connected, privacy concerns must be addressed right at the design phase (privacy-by-design). The VIB use-case is seen highly demanding in terms of miniaturisation, integration challenges, power efficiency and needs for communication with the smart environment. Indeed, for this particular application, the INSPEX mobile detection device (Fig. 3) should not exceed 200gr in weight and 100cm 3 in volume. 10 hours of lifetime in continuous use are expected with an initial target for power consumption smaller than 500mW. Obstacle location will be provided via an extra-auricular headset with Augmented Reality 3D Audio interface.

First architecture attempt

The INSPEX system is split in 3 devices with a modular architecture that provides its versatility (Fig. 3). The mobile detection device will integrate several range sensor technologies (i.e. LiDAR on chip, MEMS ultrasound, UltraWide Band (UWB) Impulse Radar) brought by the partners and optimised in the course of the project in order to fulfil the overall power budget. The choice for these range sensing technologies is conducted by the capability of the final mobile detection device to detect a large variety of obstacles (in shape, size, material, and colour) in different environmental conditions (temperature, humidity, luminosity, visibility) and particular situations (holes, stairs). The INSPEX system will function under various weather conditions (e.g. rain, snow, sand) over a large temperature range (typ. -20°C to 40°C) but also in low visibility conditions (e.g. night, dust, smoke, fog). An environmental sensing module will help reconfigure the system depending on these environmental conditions. into a set of cells [START_REF] Moravec | High resolution maps from wide angle sonar[END_REF], each cell containing the probability to find an obstacle at the cell location. Classically, Bayesian fusion [START_REF] Thrun | Probabilistic Robotics[END_REF] or evidence combination [START_REF] Elfes | Occupancy grids: A probabilistic framework for robot perception and navigation[END_REF] are used for data fusion. As they require floating-point computation, their integration on highly constrained embedded platforms is not possible due to their associated large power consumption. INSPEX will instead make use of integer computation [START_REF] Rakotovao | Multi-Sensor Fusion of Occupancy Grids based on Integer Arithmetic[END_REF] to decrease the power consumption associated with data treatment.

Being part of the IoT, the INSPEX system must offer connectivity to smarter environments. Context-aware communication capabilities will provide the user with a collaborative fabric of smart objects, giving the ability to determine the context of moving objects and to dynamically blend user experiences. A power manager will help expand the system autonomy. The number of range sensors powered-on will be dynamically adapted as a function of the available energy or of the environmental conditions, leading to an adaptation at run-time of the obstacle detection algorithm. The power manager will be verified using formal methods to ensure its proper design and functioning.

Range sensors implemented in the mobile detection device

Four state-of-the-art range sensors are brought by the partners to the project, namely, a MEMS ultrasound sensor (STM), a short range large field-of-view LiDAR-on-chip (CSEM), a long range narrow field-of-view LiDAR (Tyndall, SensL), and an UWB RF radar (CEA). Their organisation will be application dependant, e.g. to ensure the full coverage of the person height to better alarm the user on potential dangers. MEMS ultrasonic transducers (MUT) are used e.g. in non-destructive testing, speed sensing, collision warning, automation, flow metering (Doppler) and medical imaging. Depending on their actuation, capacitive MUTs (CMUTs) and piezoelectric MUTs (PMUTs) can be chosen. PMUTs have a lower power consumption than CMUTs, which can require a polarization voltage of around 200V. PMUTs are based on either bulk piezoelectric ceramic (with poor acoustic coupling to air or liquids) or on piezoelectric thin films (PZT, AlN) allowing them to be integrated into MEMS technology. When arrays of the latter are formed, ultrasound transducers allow the technology limits of conventional bulk ones to be overcome. 2D array MEMS transducers can be miniaturised to produce real-time proximity signals. Moreover, by using pulse-echo Time-of-Flight techniques, ultrasonic MEMS can work over 1m range with sub-mm ranging accuracy. In INSPEX, a piezo membrane MEMS (PMUT) together with its driving ASIC [30] will be optimised and integrated in the Mobile Detection Device.

Thanks to the huge market of the smartphone and the embedded cameras, small size and low power LiDAR are, and will be in the very near future, the best candidates for the autofocus (AF). Their range and bandwidth are compatible with the VIB use-case that targets obstacle detection within a range from less than 1 meter (short range) to 3-5 meters (long range). Such a technology is compliant with low power and low size requirements. INSPEX will optimise two LiDARs brought to the project by partners. The first one will be a long range LiDAR with a field-of-view of 2.2°. Its prototype has been demonstrated with a range of up to 21m [START_REF] Gnecchi | SiPM Sensors for ToF Ranging Applications[END_REF]. This is a single channel laser and the objective is to reduce the size by developing a chip based solution for the electronics. In conjunction with the miniaturization of the single channel module in order to increase the field of view, a 64 channel array version will be developed using the same electronics platform as will be used for the single channel version. The second LiDAR is a short range one with a large field-of-view based on an off-the-shelf component. Currently this commercial on-chip sensor allows to look at a single point in the range of up to 2m and is designed for camera auto-focus solutions. A fully footprint compatible 2 nd generation will be developed and integrated in INSPEX. This sensor is looking at 9 independent measurement points with a field of view of 30° and a range of up to 3m. It will allow faster scanning of the near proximity with increased spatial resolution.

Integrated CW or FMCW RF radars for presence detection already exist in the field of for instance domestic security or obstacle detection in the automotive domain. In domestic security, the requirement being to detect a presence, a large scale variation of the electromagnetic echo strength is enough to get sufficient detection probability. Such limited bandwidth Doppler radars hardly provide precise range information useable in dynamic obstacle tracking systems. In the automotive field, obstacle detection radar, usually operating at 24 or 77GHz, are optimised for performance (range, angle), at the expense of higher cost, high directivity and power consumption of more than 500mW [7]. UWB RF radars have also been successfully applied in healthcare applications, especially for respiration rate estimation of static humans, where sensitivity to very small movements is the key specification, but the measurement range is not adapted to (moving) obstacle detection needs. Smart white cane solutions integrating an RF Radar do not exist today at product level. At R&D scale, one can only find early demonstrators [START_REF] Yamauchi | Fusing ultra-wideband radar and LiDAR for small UGV navigation in all-weather conditions[END_REF]. INSPEX objective is to integrate a UWB RF radar sensor in its detection device with performance of 20m sensitivity range. It will have up to 100Hz refresh rate for high performance tracking of moving targets, 15cm range resolution, 60° azimuth aperture, coarse angle of arrival estimation, and approx. 100mW of power consumption. The form factor will be divided by 4 for the sensor to be compatible with integration in the envisioned Mobile Detection Device.

Fig. 1 :

 1 Fig. 1: INSPEX objective -integrating automotive all-conditions obstacle detection system [1] in a portable/wearable device

Fig. 3 :

 3 Fig. 3: INSPEX modular system architecture A low-power microcontroller will fuse data from the range sensors and build the Occupancy Grid (OG) environment model which is a spatial partition of the external world

Acknowledgements

The INSPEX project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 730953.


Isaiah Norton

Ibn Essayed

Fan Zhang

Sonia Pujol

Alex Yarmarkovich

Alexandra J Golby

Gordon Kindlmann

Demian Wassermann

Raul San

José Estepar

Yogesh Rathi

Walid Ibn Essayed

Demian Wasserman

Jose Estepar

Steve Pieper

Ron Kikinis

Hans J Johnson

Carl-Fredrik Westin

Lauren J O'donnell
email: odonnell@bwh.harvard.edu

Open Source Diffusion MRI Software for Brain Cancer Research

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Diffusion MRI (dMRI) extends MRI beyond static contrast techniques (e.g. T1-or T2-weighted imaging) to measure the molecular motion (diffusion) of water molecules [START_REF] O'donnell | An introduction to diffusion tensor image analysis[END_REF]. dMRI is applicable to a number of physical questions requiring understanding of tissue structure and compartmentation. In the brain, the cellular membranes and myelin of axonal bundles in the white matter directionally impede water diffusion with aggregate effects measurable at MRI scale (mm). dMRI is thus the only non-invasive technique that can map the brain's white matter fiber tracts (brain connections). dMRI also enables modeling and quantification of tissue microstructure. In cancer research, dMRI has two main applications: tractography to map white matter fiber tracts and imaging of cellular microstructure. dMRI tractography techniques (Figure 1A-B Standard neuroscience imaging software that relies on common reference atlas spaces is often not well suited for individual patient analysis due to the effects of brain tumors. To meet singlepatient clinical research needs, SlicerDMRI is built upon and deeply integrated with 3D Slicer, an NIH-supported open-source platform for medical image computing. 3D Slicer provides critical tools including automated and semi-automated image segmentation [START_REF] Pinter | Performing radiation therapy research using the open-source SlicerRT toolkit[END_REF] to label tumor tissue, image registration to align images across timepoints or modalities [START_REF] Johnson | BRAINSFit: mutual information rigid registrations of whole-brain 3D images, using the insight toolkit[END_REF], and data interchange with clinical informatics systems [START_REF] Fedorov | 3D Slicer as an image computing platform for the Quantitative Imaging Network[END_REF]. 3D Slicer is downloaded over 75,000 times per year, with active users and contributing developers from around the world. SlicerDMRI builds on this foundation to provide a unique environment for end-to-end dMRI analysis in clinical oncology including imaging studies and intervention planning research. The integrated functionalities of SlicerDMRI, combined with its user base, enhance its value as a testbed for implementing and testing new methods of dMRI visualization and analysis, for both clinical and pre-clinical research.

SlicerDMRI is currently in use at multiple international research centers. Recent applications of SlicerDMRI in neurosurgery and oncology include: visual deficits in pituitary macroadenoma [START_REF] Au | Optic Nerve Tractography Prediction of Visual Deficit in Pituitary Macroadenoma[END_REF],

preoperative facial nerve localization in vestibular schwannoma [START_REF] Song | In vivo visualization of the facial nerve in patients with acoustic neuroma using diffusion tensor imaging-based fiber tracking[END_REF], tractography in edematous corticospinal tracts [START_REF] Chen | Corticospinal tract modeling for neurosurgical planning by tracking through regions of peritumoral edema and crossing fibers using two-tensor unscented Kalman filter tractography[END_REF], and microstructure alterations in Chiari malformation Type I [START_REF] Krishna | Diffusion tensor imaging assessment of microstructural brainstem integrity in Chiari malformation Type I[END_REF]. We maintain a list of publications using 3D Slicer diffusion functionality such as tensor estimation, fiber tractography, fiber selection, and tractography visualization at dmri.slicer.org/citations.

Computer Resource: SlicerDMRI Functionality SlicerDMRI includes clinician-accessible tools for end-to-end diffusion MRI analysis, including computation of tissue microstructure models (e.g. diffusion tensors), tracing of fiber tracts (tractography), and quantitative analysis of diffusion-derived measures. See Video 1 for a quick tutorial covering the installation and basic usage of SlicerDMRI. Technical computing expertise is not required to use the graphical user interface of SlicerDMRI. For studies in which multiple datasets will be processed in the same way (batch analysis), SlicerDMRI supports scripting of most computations.

Core Functionality

Here we describe the main functionality of SlicerDMRI, along with brief examples of usage in clinical cancer research, as well as relevant software modules in italics.

Support for clinical and research file formats

Tracing of brain connections via fiber tractography

The process of computationally tracing white matter fiber tracts using dMRI is called tractography. Slicer supports both single-fiber (DTI) tractography and multi-fiber tractography.

Quantification of dMRI in cancer research

SlicerDMRI provides tools to measure diffusion-derived values, both in regions of interest (DiffusionScalarMaps and LabelStatistics) and in fiber tracts (TractographyMeasurements). An example application in cancer research is measurement of FA to study tissue microstructure.

White matter tract identification in patients

Interactive selection of tractography data can be difficult for a busy clinician. One approach to simplification of tractography is to cluster (group) tracts using measures of shape similarity. The SlicerDMRI WhiteMatterAnalysis package includes tools to create data-driven tractography cluster atlases, and we have recently demonstrated automatic identification of fiber tracts in patients with brain tumors [START_REF] O'donnell | Automated white matter fiber tract identification in patients with brain tumors[END_REF].

Integration into the operating room

Building on 3D Slicer's capabilities for integration with surgical navigation devices [START_REF] Ungi | Open-source platforms for navigated image-guided interventions[END_REF][START_REF] Tokuda | OpenIGTLink: an open network protocol for image-guided therapy environment[END_REF],

SlicerDMRI has been used in the operating room under research committee oversight to develop new surgical visualization methods [START_REF] Golby | Interactive diffusion tensor tractography visualization for neurosurgical planning[END_REF][START_REF] Elhawary | Intraoperative realtime querying of white matter tracts during frameless stereotactic neuronavigation[END_REF]. The advantage of this hybrid platform approach is that clinicians may rely on reliable commercial systems for core navigation guidance, while also gaining access to, and providing feedback for, the development of novel visualization and selection techniques in the operative environment.

Software access SlicerDMRI installation and user support

The SlicerDMRI extension is available from the 3D Slicer Extension Manager, an "app store" that enables developers to independently implement and distribute software extensions.

Conclusion

SlicerDMRI provides a platform for diffusion imaging research in a clinical oncology setting with tools for end-to-end diffusion image analysis as well as interoperation with clinical imaging systems.

  ) may strongly inform which interventions will maximize safety for an individual patient with an acute disease state such as epilepsy or brain cancer. SlicerDMRI is a suite of open-source software tools for dMRI research with a strong focus on the needs of clinical researchers. SlicerDMRI is used for both neuroscience research and cancer imaging research. In this article we focus on the unique needs of patient-specific oncological neuroimaging research, where each tumor can have a differing presentation.

  Support of data interchange is crucial for integration in clinical research workflows. SlicerDMRI is compatible with clinical Digital Imaging and Communications in Medicine (DICOM) diffusionweighted imaging formats as well as research imaging formats such as nrrd and NIfTI. SlicerDMRI has recently introduced support for the new DICOM tractography interchange format.Integrated DTI estimation and visualizationDiffusion tensor imaging (DTI) is the most widely-used method for analyzing diffusion MRI data. SlicerDMRI provides a graphical user interface to calculate diffusion tensor images from 4 diffusion-weighted image sequences, with several options for brain masking and tensor fitting.Options for visualization in both 2D and 3D include tractography, diffusion tensors (as ellipsoids), and derived quantitative measures such as FA (fractional anisotropy) and MD (mean diffusivity) images. An example application in cancer research is visualization of FA to give insight into the location and condition of the white matter in proximity to a tumor. Slicer 4

  DTI tractography: After tensor estimation, tractography may be seeded (initiated) in two clinically relevant ways: from a region of interest created with the Slicer editing tools, or from an interactively visualized 3D object that can be manipulated by the user. After seeding, individual tracts may be isolated via region selection tools. All settings and selection objects may be interactively adjusted, in both 2D slice and 3D view windows, with tractography results updated immediately to match. Interactivity allows for rapid isolation of tracts of specific interest. Slicer 4.6 modules in this workflow include TractographySeeding and TractographyDisplay.Multi-fiber UKF tractography: In comparison with the single-tensor DTI model, which cannot model anatomical crossing of fibers, modern multi-fiber tracking approaches are better able to trace important tracts such as the corticospinal tract and arcuate fasciculus. SlicerDMRI includes the UKFTractography module[START_REF] Malcolm | Filtered multitensor tractography[END_REF], providing an expanded range of multi-fiber models, including multi-tensor and multi-compartment models, along with a tractography tracing technique based on the unscented Kalman filter that uses information from prior steps to stabilize fiber tracking. See Video 2 for visualization of UKF multi-fiber diffusion MRI tractography in a neurosurgical case.

Figure 1 :

 1 Figure 1: Example visualization of a neurosurgical case using SlicerDMRI, with tracts from the

  SlicerDMRI installation instructions, tutorials, sample data, and introductory and clinical application videos can be found at dmri.slicer.org. SlicerDMRI is cross-platform and can be used on Windows, Mac, and Linux. Community support for SlicerDMRI users is available through the 3D Slicer forums (more information at www.slicer.org). The SlicerDMRI developers actively monitor and respond to diffusion-related topics, and participation in the larger Slicer community provides access to experts with a breadth of knowledge on many imaging topics.SlicerDMRI software informationOriginally developed starting in 2001 at the MIT AI Lab in collaboration with researchers at the Surgical Planning Laboratory at Harvard Medical School[START_REF] Talos | Diffusion Tensor and Functional MRI Fusion with Anatomical MRI for Image-Guided Neurosurgery[END_REF], the SlicerDMRI source code is now freely available with a BSD-like license that permits unrestricted use, with all code downloadable at www.github.com/SlicerDMRI. Interested developers may freely include or extend SlicerDMRI functionality in their own extensions for public distribution through Slicer or for private use. Information about Slicer extension development is available in the Slicer developer manual on slicer.org. We welcome potential code contributions to SlicerDMRI via the

	GitHub pull request system. Although end-users do not need any programming experience to
	install or use the software, interested developers and collaborators are welcome to request or
	submit	features	on	the	GitHub	issue	tracker	at
	https://github.com/SlicerDMRI/SlicerDMRI/issues. Software issues may also be reported on the
	issue tracker.

The authors declare no potential conflicts of interest.

The authors gratefully acknowledge the support of NIH NCI ITCR grant U01CA199459 (Open Source Diffusion MRI Technology For Brain Cancer Research), NIH P41EB015898 (National Center for Image Guided Therapy, NCIGT), and NIH P41EB015902 (Neuroimaging Analysis Center, NAC). We are also thankful for other grant support over the lifetime of 3D Slicer, including NIH R01MH074794, NIH R01MH097979, and NIH U54EB005149 (National Alliance for Medical Image Computing, NA-MIC).