A J Williams
email: amywilliams1@ufl.edu

J Eigenbrode

M Millan

R H Williams

A Buch

S Teinturier

D P Glavin

C Freissinet

C Szopa

O Mcintosh

S S Johnson

C Knudson

J M T Lewis

A Mcadam

R Navarro-González

V Fox

A B Bryk

R Summons

A Mojarro

A Steele

H Franz

W Brinckerhoff

C Malespin

P R Ma- Haffy

ORGANIC MOLECULES DETECTED WITH THE FIRST TMAH WET CHEMISTRY EXPERIMENT, GALE CRATER, MARS

Introduction:

The Curiosity rover is ascending Mt. Sharp, in Gale crater, exploring stratigraphic packages of rocks for evidence of habitability [1] and searching for organic compounds using the Sample Analysis at Mars (SAM) instrument suite. SAM performs sample analysis via pyrolysis and gas chromatography mass spectrometry (GC-MS). SAM additionally performs wet chemistry experiments, either by N-methyl-N-(tertbutyldimethylsilyl)-trifluoracetamide (MTBSTFA) derivatization or tetramethylammonium hydroxide (TMAH) thermochemolysis. Coupled with wet chemistry, the GC-MS can detect non-to semi-volatile organics that could derive from martian macromolecules, as occurs on Earth. The SAM TMAH experiment hydrolyzes molecules in drill samples, releasing them from their host macromolecules, and then through methylation renders them sufficiently volatile for detection by GC-MS [2]. This work reports on the results from the first in situ TMAH wet chemistry experiment conducted by SAM on Mars. Ongoing analyses indicate that the TMAH experiment was successful, and that a variety of aromatic molecules were generated. Some of these molecules may be indigenous to Mars whereas others are internal to SAM. Laboratory experiments are underway to confirm identification by retention time comparison using SAM flight spare GC columns.

Background: Two of the nine wet chemistry cups on SAM contain the TMAH reagent; the other seven contain MTBSTFA [2]. Each TMAH cup contains an outer reservoir filled with ~ 500 μL of TMAH in methanol (1:3v) with pyrene and 1-fluoronaphthalene in solution used as internal standards. Inside is a second sealed reservoir filled with nonanoic acid that will react with TMAH when the reservoir is punctured and serves as an additional internal calibration standard.

The first TMAH wet chemistry experiment on Mars was performed in September 2020 at the Mary Anning (MA) drill site in the Glen Torridon region. The drill site was selected based on anticipated mineralogic similarity to the Glen Etive drill site, which would prioritize abundant clay minerals and minimal iron oxide phases. A location that was substantially stratigraphically below the diagenetically altered material near the contact with the Greenheugh pediment was also desired to limit the impact of alteration on the sample site.

Methods: Six sample portions were delivered to a punctured SAM TMAH cup. The sample was saturated with TMAH in the cup and raised into the pyrolysis oven. A split of the volatiles released from pyrolysis was monitored by the quadrupole mass spectrometer (QMS) for the entire temperature ramp for evolved gas analysis (EGA). For GC-MS, the volatiles released during pyrolysis were diverted through the SAM hydrocarbon trap under He flow as cuts at selected temperature intervals. The cut from >560°C was sent only to the QMS for EGA. Volatiles trapped on the SAM hydrocarbon trap were released by heating the trap to ~300°C and then split between GC1 (MXT-20) and GC2 (MXT-5) chromatographic channels for their separation.

Results:

A Successful TMAH Experiment. Engineering data indicate the TMAH cup was punctured and powdered sample from the MA drill hole delivered. Trimethylamine byproduct masses from the TMAH reaction were detected. The 1-fluoronaphthalene recovery standard was also identified, but not the nonanoic acid and pyrene standards.

High molecular weight molecules. 'Bands' of masses grouped together and having mass-to-charge (m/z) 190 to 485, represent high molecular weight molecules detected by the SAM QMS. These data may indicate that large, complex molecules were present.

Aromatic organics. A variety of methylated, oxygen-, sulfur-, or nitrogen-bearing aromatic organics were detected in GC-MS and/or EGA data. The origin of these molecules is discussed below.

Discussion: The successful first in situ TMAH experiment demonstrates that a variety of organic molecules can be liberated by the SAM TMAH experiment.

A) Organics detected in EGA and GC-MS. Some of the molecules that were identified in both the EGA and GC-MS data may be indigenous to the sample, as opposed to being SAM-internal byproducts. This interpretation is still assessed. Examples of molecules identified in both EGA and GC-MS include benzene, toluene, trimethyl-and tetramethyl-benzene, naphthalene, and methylnaphthalene (Fig. 1). TMAH has a basic pH and is a methylating agent that enhances the cleavage of polar bonds and methylates pyrolysis products in situ. The presence of methylated single and double ring aromatics suggests that these organics might derive from a macromolecular source that was cleaved and methylated by TMAH thermochemolysis. We note that with limited TMAH reagent, no TMAH blank was conducted, so we cannot rule out carry over from previous analyses.

B) Organics detected with GC-MS only.

Molecules detected only with GC-MS may be either indigenous to the sample or be formed from reactions with TMAH within the SAM gas processing system (e.g., reactions with Tenax in the hydrocarbon trap or with the column stationary phase). Examples of the organics detected in GC-MS only include pentamethyl-benzene, benzoic acid methyl ester, dimethyl-, trimethyl-, and tetramethyl-benzenamine, dihydronaphthalene, 2-butyl-thiophene, and benzothiophene. Pentamethylbenzene may be part of a multi-methylated benzene suite. Benzoic acid has been detected with SAM from both Mars-indigenous [3][4][5] and SAM-internal [5,6] sources. The benzoic acid methyl ester therefore reflects the reaction of TMAH methylating benzoic acid of indeterminate source. The multi-methylated benzenamine suite is also currently of indeterminate source. The benzenamine may be Mars-indigenous or a byproduct of benzene reacting with the N in TMAH. Ongoing benchtop experiments will elucidate these sources. Regardless, the methylation of the benzenamine reflects an expected reaction with TMAH.

The thiophene-bearing molecules have been previously detected on Mars with SAM [6] and are likely Mars-indigenous. These molecules were not methylated but may have been liberated from a macromolecular source by the TMAH reaction.

C) SAM-internal organics. Several known SAMinternal molecules were identified in the EGA and GC data, including mono-and bisilylated water from reactions with water and MTBSTFA, and derivatized boric acid from the reaction of MTBSTFA with the hydrocarbon trap. Aromatic byproducts detected here and associated with the reaction of MTBSTFA with the hydrocarbon trap Tenax include benzene, toluene, 1-ethynyl-3-methyl-benzene, benzoic acid, naphthalene, and diphenylmethane [6]. Detection of benzene, toluene, and naphthalene here as SAM-internal byproducts does not confirm the source of these molecules in the EGA data.

Sources of organics. Meteoritic input is one possible source of organics on the martian surface. Several organics identified with the TMAH experiment are also liberated from the Murchison meteorite with SAM-like TMAH thermochemolysis benchtop experiments, including toluene, trimethylbenzene, methylnaphthalene, 2-butyl-thiophene, and benzothiophene. During TMAH thermochemolysis, the Murchison insoluble organic matter is degraded into its constituent parts, some of which are detectable with GC-MS [7]. Molecules detected with neat pyrolysis of Murchison include benzene and naphthalene [8].

Amines and amides are not prevalent in Murchison pyrolysate [9] and benzenamines are also not generated during TMAH thermochemolysis of Murchison. However, UV irradiation experiments of benzene and naphthalene in the presence of water and NH3-containing ices can generate benzenamine and dihydronaphthalene [10]. It is therefore possible that UV irradiation has contributed to radiation processing of meteoritic organics.

TMAH methylates carboxylic acids with high efficiency. Carboxylic acid methyl esters (CAMEs) have been generated from TMAH thermochemolysis of Murchison [11]. The lack of CAMEs in these data suggest that 1) CAMEs are not present in amounts detectable with SAM as run or 2) flight operating conditions require optimization before they can be detected clearly.

Conclusions: Multiple aromatic molecules were detected with the first TMAH experiment on SAM and may be a mix of SAM-internal and radiation processed macromolecular organic matter of either exogenous or martian origin.

Figure 1 .

 1 Figure 1. Single ion EGA and chromatograms from the SAM TMAH experiment. Single-ring benzene and toluene are at top, and double-ring (methyl)naphthalene are at bottom.


Frédéric Bayart

Lucas Oger

SPECTRUM OF HYPERBOLIC AND PARABOLIC WEIGHTED COMPOSITION OPERATORS

We compute the spectrum of a weighted composition operator induced by a hyperbolic or a parabolic symbol when it acts on the Fréchet space of holomorphic functions on the unit disc. We discuss situations where these results extend and do not extend to the unit ball.

FRÉDÉRIC BAYART, LUCAS OGER

In this paper, we focus on the Fréchet space Hol(Ω) endowed with the compact-open topology. When Ω = D, the spectrum of W m,φ has already been thoroughly studied in [3, 2, 1]. Our aim in this paper is to complete the picture by filling the cases left open in these papers and by studying how the results can be extended to the ball.

The results obtained so far depend on the nature of the Denjoy-Wolff point of the symbol and on the behaviour of the multiplier near this attractive fixed point. Recall that for a holomorphic self-map φ of D, one of the following two properties hold.

r The map φ has a fixed point ξ inside D. In that case, φ is said to be elliptic. If φ is not an automorphism, then the sequence of iterates (φ [n] ) of φ converges to ξ in Hol(D). r The map φ has no fixed point in D. In that case, there exists ξ ∈ T such that (φ [n] ) converges to ξ in Hol(D). Moreover, φ admits an angular derivative φ ′ (ξ) ∈ (0, 1] at ξ. The symbol φ is called parabolic if φ ′ (ξ) = 1 and hyperbolic if φ ′ (ξ) < 1. Elliptic automorphisms behave very differently and we refer to [1] for results concerning them. Regarding the other cases, assuming some regularity of m and φ at the Denjoy-Wolff point ξ if it belongs to T, the following table summarizes the results of [2] on the point spectrum of W m,φ .

Elliptic, φ ′ (ξ) ̸ = 0 Elliptic, φ ′ (ξ) = 0, φ ̸ = ξ Hyperbolic Parabolic

Introduction

This paper is a contribution to a series of works which intend to compute the spectrum of weighted composition operators acting on spaces of holomorphic functions. Let Ω be a domain in C d , and Hol(Ω) the set of all complex-valued holomorphic functions on Ω.

For φ : Ω → Ω holomorphic and m ∈ Hol(Ω), the weighted composition operator with symbol φ and multiplier m is defined as

W m,φ (f ) = m(f • φ), f ∈ Hol(Ω).
We shall only consider the cases where Ω = D is the unit disc or Ω = B d is the (euclidean) unit ball in C d . There is a huge literature on this subject (see e.g. [START_REF] Cowen | Composition operators on spaces of analytic functions[END_REF][START_REF] Shapiro | Composition operators and classical function theory[END_REF] for a general study, or the recent articles [START_REF] Bayart | Hyperbolic composition operators on the ball[END_REF][START_REF] Bourdon | Spectra of some composition operators and associated weighted composition operators[END_REF]), in particular when they are considered as acting on Banach spaces of holomorphic functions like Hardy spaces or Bergman spaces.

m = 1 {φ ′ (ξ) n : n ∈ N} {1} C * C * m(ξ) ̸ = 0 {m(ξ)φ ′ (ξ) n : n ∈ N} {m(ξ)} C * ? m(ξ) = 0 ∅ ∅ ? ?
Table 1. Results in the disc Moreover, if φ is not an automorphism, then σ(W m,φ ) = σ p (W m,φ ) ∪ {0}. Otherwise, σ(W m,φ ) = σ p (W m,φ ).

The cases in grey are left open in [START_REF]Denjoy-Wolff theory and spectral properties of weighted composition operators on Hol(D)[END_REF], even if some interesting examples are given. Our first main result will be that, for these three cases, under natural regularity assumptions on φ and m (precise statements will be given later on), σ p (W m,φ ) = C * . This property keeps true if we assume that ξ is a pole for m, again assuming some regularity assumptions. As in [START_REF]Denjoy-Wolff theory and spectral properties of weighted composition operators on Hol(D)[END_REF], the proof will be based on a careful study on how φ [n] (z) goes to ξ and on a renormalization of m(φ [n] (z)). Nevertheless, we will have to change the normalization for parabolic maps and we will need much more precise estimates, leading to replace the results of Valiron [START_REF] Valiron | Sur l'itération des fonctions holomorphes dans un demi-plan[END_REF], Pommerenke [START_REF] Pommerenke | On the iteration of analytic functions in a halfplane. I[END_REF] and Baker-Pommerenke [START_REF] Baker | On the iteration of analytic functions in a half-plane II[END_REF] by those of Bourdon-Shapiro [START_REF] Bourdon | Cyclic phenomena for composition operators[END_REF].

Regarding the several variables situation, we will mainly concentrate on the twodimensional case. A similar classification occurs and we will be able, when m(ξ) ̸ = 0, to prove similar statements, using now [START_REF] Bayart | The linear fractional model on the ball[END_REF] and [START_REF] Bracci | Solving the Schröder equation at the boundary in several variables[END_REF]. When m(ξ) = 0, the situation will become much more involved. Indeed, a new difficulty arises : in dimension two or more, we cannot easily factorize a holomorphic function vanishing at some point.

We will obtain some partial results, and also exhibit a very simple example of a weight m and symbol φ, regular at the Denjoy-Wolff point, with m(ξ) = 0, such that σ p (W m,φ ) = ∅. Hence the results dramatically break down on the ball.

We end up this introduction with the following lemma, that we will use repeatedly.

Lemma 1.1. Let m 1 , m 2 ∈ Hol(Ω) and φ : Ω → Ω holomorphic. Then

σ p (W m 1 ,φ ) • σ p (W m 2 ,φ ) ⊂ σ p (W m 1 m 2 ,φ ).
Proof. Let λ 1 ∈ σ p (W m 1 ,φ ) and λ 2 ∈ σ p (W m 2 ,φ ). By definition, there exist two nonzero maps f 1 , f 2 of Hol(Ω) such that

m 1 (f 1 • φ) = λ 1 f 1 , m 2 (f 2 • φ) = λ 2 f 2 .
Hence, considering f = f 1 f 2 and m = m 1 m 2 , we get

m(f • φ) = m 1 m 2 (f 1 • φ)(f 2 • φ) = m 1 (f 1 • φ)m 2 (f 2 • φ) = λ 1 f 1 λ 2 f 2 = λ 1 λ 2 f.
Finally, we obtain λ 1 λ 2 ∈ σ p (W m,φ ), the desired result. □ Notation. Throughout the paper, if f, g : X → R are two functions defined on the same set X, the notation f ≲ g will mean that there exists some constant C > 0 such that f ≤ Cg on X. The notation f ≍ g will simply mean f ≲ g and g ≲ f .

On the disc

We begin this section by recalling some definitions.

Definition 2.1. The Stolz angle with vertex ξ ∈ T and radius R > 0 is

S(ξ, R) := z ∈ D : |z -ξ| 1 -|z| < R .
The horodisc of vertex ξ ∈ T and radius α > 0 is

H(ξ, α) := {z ∈ D : |z -ξ| 2 < α(1 -|z| 2 )} = D ξ 1 + α , α 1 + α where D(w, r) is the (euclidean) disc of center w and radius r > 0. The right half-plane of C is H := {z ∈ C : ℜ(z) > 0}.
In what follows, as it is often the case, we will transfer the problem on the right half-plane H through the Cayley transform

τ (z) = 1 + z 1 -z .
For φ : D → D holomorphic with Denjoy-Wolff point at 1, we will set Φ = τ • φ • τ -1 , which is a self-map of H, with ∞ as attractive fixed point.

Setting w = τ (z), we will repeatedly use

1 -z = 2 1 + w and 1 -φ(z) = 2 1 + Φ(w)
.

The analogues of the Stolz angles (with vertex 1) on the right half-plane are the angular sectors {w ∈ H : arg(w) < α}, for α ∈ (0, π/2). We will still call them Stolz angles at ∞.

As mentioned in the introduction, we will sometimes need that φ has some regularity properties at its Denjoy-Wolff point. Here are the relevant definitions. For n ∈ N and 0 ≤ ε < 1, we say that φ ∈ C n+ε (ξ) if φ has a n th derivative at ξ, and if

φ(z) = n k=0 φ (k) (ξ) k! (z -ξ) k + o(|z -ξ| n+ε ), z ∈ D. Definition 2.3. A holomorphic map φ : D → D is regular if φ is continuous on D, and if φ(D) ⊂ D ∪ {ξ},
where ξ is the Denjoy-Wolff point of φ.

We assume now, and for the rest of the paper, that ξ = 1 and φ(1) = 1.

If we translate Definition 2.2 onto the map Φ, it means that we may write

Φ(w) = a 0 w + a 1 + n-2 j=2 a j (w + 1) j + o 1 |w + 1| n-2+ε , with a 0 , • • • , a n-2 ∈ C.
∈ N 0 , φ [n] (K) ⊂ S(ξ, R).
The computation of σ p (W m,φ ) when φ is hyperbolic and m does not vanish at the Denjoy-Wolff point ξ of φ has already been done in [START_REF]Denjoy-Wolff theory and spectral properties of weighted composition operators on Hol(D)[END_REF], where the following theorem is obtained. Recall that ℓ is the angular limit of f ∈ Hol(D) at ξ ∈ T if there exists a Stolz angle S such that f (z) → ℓ when z → ξ and z ∈ S. We shall improve it by slightly relaxing the assumptions. This will be important to handle the case m(ξ) = 0. Theorem 2.6. Let φ : D → D be a hyperbolic holomorphic map, with Denjoy-Wolff point ξ, and m ∈ Hol(D) with non-zero angular limit m(ξ) at ξ. Assume that

(1) m(z) = m(ξ) + Λ(z),
and that for all Stolz angles S with vertex ξ, there exist β ∈ (0, 1) and a nondecreasing map g : R → R + such that ∀x ∈ (0, 1),

n≥0 g(x n ) < +∞ and z ∈ S, |z| ≥ 1 -β =⇒ |Λ(z)| ≤ g(|z -ξ|). Then, σ p (W m,φ ) = C * .
The assumptions of Theorem 2.5 mean that (1) is true with Λ(z) = m ′ (ξ)(z -ξ) + o(z -ξ). Therefore, we may choose g(x) = M x for a suitable M > 0. It is nevertheless natural (think at Definition 2.2) to be able to use g(x) = M x ε for some ε > 0, or even

g(x) = M |log(x)| 1+η , η > 0.
Proof. Since φ and m are non-constant maps, if m(f • φ) = 0, then f = 0, so we have 0 ̸ ∈ σ p (W m,φ ). For n ∈ N and z ∈ D, we define

f n (z) = 1 m(ξ) n n-1 k=0 m(φ [k] (z)). Hence, W m,φ (f n )(z) = m(z) 1 m(ξ) n n-1 k=0 m(φ [k+1] (z)) = m(ξ) 1 m(ξ) n+1 n k=0 m(φ [k] (z)) = m(ξ)f n+1 (z).
We show that (f n ) n≥0 converges uniformly on all compact subsets of D. Let K be such a compact. By Lemma 2.4, there exists a Stolz angle S with vertex ξ such that for all n ∈ N 0 , φ [n]

(K) ⊂ S. For all z ∈ K, m(φ [n] (z)) m(ξ) -1 = m(φ [n] (z)) -m(ξ) |m(ξ)| ≤ 1 |m(ξ)| g φ [n] (z) -ξ .
Let us now consider α > 0 such that K ⊂ H(ξ, α). Using Julia's lemma, and Julia-Wolff-Caratheodory theorem ( [START_REF] Bracci | Continuous semigroups of holomorphic selfmaps of the unit disc[END_REF], [START_REF] Shapiro | Composition operators and classical function theory[END_REF]), for all n ∈ N,

φ [n] (K) ⊂ H(ξ, αφ ′ (ξ) n ).
Therefore, for all n ∈ N and z ∈ K,

φ [n] (z) -ξ 2 < 2αφ ′ (ξ) n .
Hence, if n is sufficiently large and x ∈ (φ ′ (ξ) 1/2 , 1),

g( φ [n] (z) -ξ ) ≤ g(x n ).
Finally, (f n ) n≥0 converges uniformly on K to a nonzero function f , which satisfies

W m,φ (f ) = m(ξ)f , so that m(ξ) ∈ σ p (W m,φ ). Since σ p (C φ ) = C * (see [2, Theorem 6.1]), by Lemma 1.1, we obtain C * ⊂ σ p (W m,φ ). □ 2.
2. Hyperbolic symbol with vanishing weight at the Denjoy-Wolff point. In this section, we intend to prove the following result.

Theorem 2.7. Let φ : D → D be a hyperbolic regular holomorphic map, with Denjoy-Wolff point ξ, and m ∈ Hol(D). Assume that there exist ε ∈ (0, 1), a ∈ C * , α ∈ R * and Λ ∈ Hol(D) such that φ ∈ C 1+ε (ξ),

m(z) = a(z -ξ) α (1 + Λ(z)),
and for all Stolz angles S with vertex ξ, there exist β ∈ (0, 1) and a nondecreasing map g : R → R + such that ∀x ∈ (0, 1),

n≥0 g(x n ) < +∞ and z ∈ S, |z| ≥ 1 -β =⇒ |Λ(z)| ≤ g(|z -ξ|).
Then σ p (W m,φ ) = C * .

In particular, we may apply this theorem when m is meromorphic at ξ.

Proof. We shall prove the theorem on H, by assuming that ξ = 1. Let Φ : H → H be holomorphic, which extends to a continuous function on H ∪ {∞} and maps Φ(H) into H. We shall also assume that Φ may be written

Φ(w) = λw + Θ(w),
with λ > 1 and |Θ(w)| ≲ |w + 1| 1-ε . Let also m : H → C be such that m(w) = a (w + 1) α 1 + Λ(w) .

In this context, for all Stolz angles S with vertex ∞, there exist C > 0 and a nondecreasing map g : R → R + such that ∀x ∈ (0, 1),

n≥0 g(x n ) < +∞ and w ∈ S, |w| ≥ C =⇒ |Λ(w)| ≤ g(1/|w|).
By the linear fractional model (see [START_REF] Bourdon | Cyclic phenomena for composition operators[END_REF]Theorem 4.9]), there exist a holomorphic selfmap

H : H → H and c > 1 such that H • Φ = cH and H(w) = (w + 1)(1 + G(w)) with |G(w)| ≲ |w + 1| -ε for |w| large enough.
Our strategy for the proof is the following. We will first consider the weighted composition operator W H -α ,Φ and show that it admits a nonzero eigenvalue by exhibiting an eigenvector. Next, we will set m 1 = mH α and we will apply Theorem 2.6 to W m 1 ,Φ . We will finally conclude by Lemma 1.1.

Step 1 : We show that σ p (W H -α ,Φ ) ∩ C * ̸ = ∅. Set λ = α/2 log(c), and

F = exp[λ log 2 (H)].
Then,

W H -α ,Φ (F ) = exp[λ log 2 (H • Φ)] H α = exp[λ log 2 (cH)] H α = exp[λ log 2 (c) + 2λ log(c) log(H) + λ log 2 (H)] H α = e λ log 2 (c) H 2λ log(c)-α F = e λ log 2 (c) F.
Since e λ log 2 (c) ̸ = 0, we obtain the desired result.

Step 2 : We show that σ p (W m 1 ,Φ ) = C * . Indeed, using the estimates on m and H, we know that m 1 = mH α can be written

m 1 (w) = a(1 + Λ(w))(1 + G(w)) α . In addition, (1 + G(w)) α = 1 + αG(w) + η(w), with |η(w)| ≲ |w + 1| -2ε . Finally, we may write m 1 (w) = a + Λ(w),
with Λ = a(Λ + αG + αGΛ + η + ηΛ). By sums and products, for all Stolz angles S with vertex ∞, there exist C > 0 and a nondecreasing map g such that for all x ∈ (0, 1), g(x n ) < +∞, and

w ∈ S, |w| ≥ C =⇒ | Λ(w)| ≤ g(1/|w|).
We obtain the result by applying Theorem 2.6.

Conclusion : Using Lemma 1.1 together with the results of Step 1 and

Step 2, we get σ p (W m,Φ ) = C * . □ 2.3. General discussion on parabolic symbols. We turn to the case where the symbol φ is a parabolic self-map of D, and we will always assume that 1 is its Denjoy-Wolff point. The strategy to prove that, under some regularity assumptions, σ p (W m,φ ) = C * is the same as in the hyperbolic case. We will first prove that this holds true if m does not vanish at 1 by renormalizing the sequence m(φ [n] (z)). Then, we will handle the case where m vanishes at 1 by exhibiting an eigenvector when m has a specific form related to the symbol, and we will conclude in the general case by factorizing the multiplier.

Nevertheless, every step becomes harder. The key point is that φ [n] (z) converges much more slowly to 1 when φ is parabolic, and the convergence of the sequences which come into play will be more difficult to prove.

The linear fractional model gives us now the existence of a map H :

H → C such that H • Φ = H + a.
As in the proof of Theorem 2.7, we will need estimates on the behaviour of H near infinity, and even more precise estimates due to the parabolic nature of the symbol.

Our starting point is again [START_REF] Bourdon | Cyclic phenomena for composition operators[END_REF], now Theorem 4.12. Bourdon and Shapiro give there a linear fractional model with estimates for φ a parabolic self-map of D belonging to C 3+ε (1), with φ ′′ (1) ̸ = 0 and φ regular. Indeed, in such case, we get ℜ(φ ′′ (1)) ≥ 0, and we can write

φ(z) = 1 + (z -1) + φ ′′ (1) (z -1) 2 2 + φ ′′′ (1) (z -1) 3 6 + γ(z), with γ(z) = o(|z -1| 3+ε ). Setting again Φ = τ • φ • τ -1 , then for all w ∈ H, Φ(w) = w + a + b w + 1 + Θ(w), Θ(w) = o 1 |w + 1| 1+ε , with a = φ ′′ (1) and b = φ ′′ (1) 2 -2φ ′′′ (1)/3. If ℜ(φ ′′ (1)
) = 0, we may and shall assume that ℑ(φ ′′ (1)) > 0. Bourdon-Shapiro's theorem reads as follows.

Theorem 2.8 ([9], Theorem 4.12).

Let φ : D → D be a parabolic holomorphic map, with Denjoy-Wolff point 1. Assume that φ is regular, C 3+ε (1), and that φ ′′ (1) ̸ = 0. Then, there exists a function H :

H → C such that H • Φ = H + a, and (2)
H(w) = w - b a log(1 + w) + B(w), w ∈ Ω,
with B holomorphic, bounded on Ω and continuous on Ω, where

r if ℜ(a) > 0, then Ω = H, r if ℜ(a) = 0 and ℑ(a) > 0, then Ω = H + = {z ∈ C : ℜ(z) > 0, ℑ(z) > 0}.
It turns out that we will need the following enhancement of this theorem, giving a slightly more precise information on the behaviour of H. Proposition 2.9. Under the assumptions of Theorem 2.8, there exist R > 0, a domain Ω ⊂ H and a map H :

H → C such that H • Φ = H + a and (3) H(w) = w + c - b a log(1 + w) + θ(w), w ∈ Ω, with c ∈ C, θ : H → C holomorphic, satisfying |θ(w)| ≲ |w| -ε/2 for |w| sufficiently large and r if ℜ(a) > 0, then Ω = H, r if ℜ(a) = 0 and ℑ(a) > 0, then Ω = H R = {z ∈ C : ℜ(z) > 0, ℑ(z) > R}.
Proof. We follow the method of [START_REF] Bourdon | Cyclic phenomena for composition operators[END_REF] but we have to go a step further in the expansion. We start from the last formula p.74 and [9, Proposition 4.19] : there exists some R > 0 such that H is the uniform limit on compact subsets of Ω of the maps H n defined by

H n (w) = w -w 0 -bw n-1 j=0 1 w(j)w 0 (j) + bS 2,n (w) + S 3,n (w), with w 0 ∈ Ω, w(n) = Φ [n] (w) + 1, and
S 2,n (w) = n-1 j=0 w 0 -(H j (w) -H 0 (w)) w(j)w 0 (j) , S 3,n (w) = n-1 j=0 [Θ(w(j)) -Θ(w 0 (j))].
To achieve the proof, we need to study all the terms of H n . First, [9, Proposition 4.18] gives us the very useful inequality

|w(n)| ≳ |w| + n, w ∈ Ω, n ≥ 0.
r Let us start with S 3,n . For j ≥ 1 and w ∈ Ω, if we set γ = max(|w| , j), then

|w| ε/2 |Θ(w(j))| ≲ |w| ε/2 |w(j)| 1+ε ≲ |w| ε/2 (|w| + j) 1+ε ≤ γ ε/2 γ 1+ε ≤ 1 γ 1+ε/2 ≤ 1 j 1+ε/2 .
Moreover, the sum Θ(w 0 (j)) converges, so

S 3 (w) = lim n→∞ S 3,n (w) = j≥0 Θ(w(j)) - j≥0 Θ(w 0 (j)) = θ 3 (w) + c 3 , with |θ 3 (w)| ≲ |w| -ε/2 for |w| large enough.
r We now focus on S 2,n . For j ≥ 1 and w ∈ Ω,

|w| ε |w(j)w 0 (j)| ≲ |w| ε (|w| + j)(|w 0 | + j) ≤ γ ε γj ≤ 1 γ 1-ε j ≤ 1 j 2-ε . Since |H j (w) -H 0 (w)| ≲ log(j + 1) (cf. [9, p. 76]), we also get |w| ε H j (w) -H 0 (w) w(j)w 0 (j) ≲ |w| ε log(j + 1) (|w| + j)(|w 0 | + j) ≤ log(j + 1) γ 1-ε j ≤ log(j + 1) j 2-ε .
Therefore, we have shown that

S 2 (w) = lim n→∞ S 2,n (w) = θ 2 (w) + c 2 ,
with |θ 2 (w)| ≲ |w| -ε for |w| large enough. For the moment, setting

d 1 = c 2 + c 3 -w 0 and ∆ 1 = θ 2 + θ 3 , we get H(w) = w -bw j≥0 1 w(j)w 0 (j) + d 1 + ∆ 1 (w), |∆ 1 (w)| ≲ |w| -ε/2 .
Now, note that using Lemma 4.20 of [START_REF] Bourdon | Cyclic phenomena for composition operators[END_REF], for each w ∈ Ω and each j ≥ 0,

|w + ja| ≳ (|w| + j), |w(j) -w -ja| ≲ log(j + 1).
Hence, let us write

bw j≥0 1 w(j)w 0 (j) = I 1 (w) + I 2 (w) + I 3 (w) + b j≥1 w (w + ja)(w 0 + ja) ,
with I 1 (w) = bw/(w + 1)(w 0 + 1), and

I 2 (w) = b j≥1 w w + ja 1 w 0 (j) - 1 w 0 + ja , I 3 (w) = b j≥1 w w 0 (j) 1 w(j) - 1 w + ja .
Then,

I 1 (w) = b w 0 + 1 - b (w 0 + 1)(w + 1) = c 4 -θ 4 (w),
with |θ 4 (w)| ≲ |w| -1 for |w| large. In addition,

I 2 (w) = b j≥1 1 w 0 (j) - 1 w 0 + ja =c 5 -b j≥1 ja w + ja 1 w 0 (j) - 1 w 0 + ja =θ 5 (w) , with ja w + ja 1 w 0 (j) - 1 w 0 + ja ≲ j log(j + 1) (|w| + j)(|w 0 | + j) 2 ≤ log(j + 1) |w| 1/2 j 3/2 , so |θ 5 (w)| ≲ |w| -1/2
. Finally, to study I 3 (w), we only have to see that

w w 0 (j) 1 w(j) - 1 w + ja ≲ |w| log(j + 1) (|w 0 | + j)(|w| + j) 2 ≤ log(j + 1) |w| 1/2 j 3/2
.

Thus, |I 3 (w)| ≲ |w| -1/2 . Setting d 2 = d 1 -c 4 -c 5 and ∆ 2 = ∆ 1 + θ 4 + θ 5 -I 3 , we get H(w) = w -b j≥1 w (w + ja)(w 0 + ja) + d 2 + ∆ 2 (w), |∆ 2 (w)| ≲ |w| -ε/2 .
Next, note that

j≥1 w (w + ja)(w 0 + ja) = j≥1 1 w 0 + ja - 1 w + ja =J 1 (w) + w 0 j≥1 1 (w + ja)(w 0 + ja) =J 2 (w)
.

For the second term, we have

|J 2 (w)| ≤ |w 0 | j≥1 1 (w + ja)(w 0 + ja) ≤ |w 0 | |w| 1/2 j≥1 1 j 3/2 ≲ 1 |w| 1/2 ,
and for the first one,

J 1 (w) - ∞ 1 1 w 0 + ta - 1 w + ta dt = j≥1 j+1 j a(t -j) (w 0 + ja)(w 0 + ta) dt =c 6 - j≥1 j+1 j a(t -j) (w + ja)(w + ta) dt =θ 6 (w) , with, for j ≤ t ≤ j + 1, a(t -j) (w 0 + ja)(w 0 + ta) ≲ 1 (|w 0 | + j) 2 ≤ 1 j 2 and a(t -j) (w + ja)(w + ta) ≲ 1 (|w| + j) 2 ≤ 1 |w| 1/2 j 3/2 .
Finally,

J 1 (w) - ∞ 1 1 w 0 + ta - 1 w + ta dt = c 6 + θ 6 (w), |θ 6 (w)| ≲ 1 |w| 1/2 . Therefore, setting d 3 = d 2 -c 6 and ∆ 3 = ∆ 2 + θ 6 -J 2 , we obtain H(w) = w -b ∞ 1 1 w 0 + ta - 1 w + ta dt + d 3 + ∆ 3 (w), |∆ 3 (w)| ≲ |w| -ε/2 .
But it is well-known that

∞ 1 1 w 0 + ta - 1 w + ta dt = 1 a log w + a w 0 + a , so
H(w) = w - b a log w + a w 0 + a + d 3 + ∆ 3 (w), |∆ 3 (w)| ≲ |w| -ε/2 .
We conclude, using the identity

log w + a w 0 + a -log(w + 1) = log 1 + a -1 w + 1 =θ 7 (w)
-log(w 0 + a) Regarding the behaviour of the sequences (φ [n] (z)), we will need the following lemma.

Lemma 2.10 ([6, Lemma 5.8]). Under the assumptions of Theorem 2.8, for all w ∈ H,

Φ [n] (w) = an + b a log(n) + B n (w),
with B n bounded (independently of n) on all compact subsets of H. Assume also that there exist a neighborhood U of 1, r > 0, δ > 0 and Λ : D → C such that setting D r = τ -1 (H r ) and

V (r) = U ∩ D if ℜ(φ ′′ (1)) > 0, U ∩ D r if ℜ(φ ′′ (1)) = 0 and ℑ(φ ′′ (1)) > 0, the map m satisfies ∀z ∈ D, m(z) = m(1) + Λ(z),
with m(1) ̸ = 0, and

∀z, z ′ ∈ V (r), |Λ(z) -Λ(z ′ )| ≲ |z -1| 1+δ + |z ′ -1| 1+δ + |z -z ′ | 1 2 +δ Then, σ p (W m,φ ) = C * .
In particular, our assumptions imply that m admits a nonzero limit at 1 along D r . They are satisfied if m ∈ C 1+δ (1) and m(1) ̸ = 0.

Proof. We may assume that r ≥ R, where R is given by Proposition 2.9. We move on H and consider m ∈ Hol(H) which can be written m(w) = m(∞) + Λ(w), with m(∞) ̸ = 0 and for all w ∈ Ω,

(4) |Λ(w) -Λ(w ′ )| ≲ 1 |w + 1| 1+δ + 1 |w ′ + 1| 1+δ + w -w ′ (w + 1)(w ′ + 1) 1 2 +δ , where Ω = H ∩ {|w| > M } if ℜ(φ ′′ (1)) > 0, and Ω = H r ∩ {|w| > M } otherwise, with M > 0 sufficiently large.
Arguing as in the proof of Theorem 2.6, it is sufficient to show that m(∞) is an eigenvalue of W m,Φ . Let K be a compact subset of H containing 1. We claim that Φ [n] (K) ⊂ Ω provided n is large enough. This is a consequence of the Denjoy-Wolff theorem if ℜ(φ ′′ (1)) > 0. If ℜ(φ ′′ (1)) = 0, we set w 0 = 1 + i(r + 1) and consider ρ > 0 so that K ⊂ ∆(w 0 , ρ), where ∆(ζ, r) is the pseudo-hyperbolic disc with center ζ and radius r. Using Schwarz-Pick's lemma, for all n ∈ N,

Φ [n] (K) ⊂ ∆(Φ [n] (w 0 ), ρ) = ℜ(Φ [n] (w 0 ))∆(1, ρ) + ℑ(Φ [n] (w 0 )).
However, the sequence (ℜ(Φ [n] (w 0 ))) n≥0 is bounded (see [START_REF] Bourdon | Cyclic phenomena for composition operators[END_REF]Proposition 4.25]), and ℑ(Φ [n] (w 0 )) ---→ n→∞ +∞. Thus, for n large, Φ [n] (K) ⊂ Ω.

In particular, there exists n 0 ∈ N such that for n ≥ n 0 , m(Φ [k] (1)) ̸ = 0. Hence, for all w ∈ H and n ≥ n 0 , we may set

F n (w) = n k=n 0 m(Φ [k] (w)) m(Φ [k] (1))
.

We intend to show that (F n ) converges uniformly on K. For w ∈ K and n large,

m(Φ [n] (w)) m(Φ [n] (1)) -1 ≲ m(Φ [n] (w)) -m(Φ [n] (1)) ≤ 1 Φ [n] (w) + 1 1+δ + 1 Φ [n] (1) + 1 1+δ + B n (w) -B n (1) (Φ [n] (w) + 1)(Φ [n] (1) + 1) 1 2 +δ .
Since |Φ [n] (w) + 1|, |Φ [n] (1) + 1| ≥ C(K)n, an application of Lemma 2.10 yields the desired convergence. We conclude since

W m,Φ (F )(w) = m(w)F (Φ(w)) = m(w) lim n→+∞ n k=n 0 m(Φ [k+1] (w)) m(Φ [k] (1)) = lim n→+∞ m(Φ [n+1] (1)) n+1 k=n 0 m(Φ [k] (w)) m(Φ [k] (1)) = m(∞)F (w).□
When φ is a parabolic automorphism (hence, excluding the identity), the problem is much simpler to solve. Indeed, in such case, there exists a ∈ R * such that for all w ∈ H, Φ(w) = w + ia. Hence, for all compact subsets K of H containing 1, and for all w ∈ K,

|w| ≤ C =⇒ Φ [n] (w) = |w + ina| ≥ n |a| -|w| ≥ n |a| -C.
For n 0 ≥ (M + C)/ |a| (M > 0 being in the definition of Ω) and n ≥ n 0 , we obtain m(Φ [n] (w)) ̸ = 0, and

m(Φ [n] (w)) m(Φ [n] (1)) -1 ≲ m(Φ [n] (w)) -m(Φ [n] (1)) .
Under the same hypotheses as in Theorem 2.11, (F n ) converges uniformly on K, and we get the result. Theorem 2.12. Let φ : D → D be a parabolic holomorphic map, with Denjoy-Wolff point 1. Assume that φ is regular, C 3+ε (1), and that φ ′′ (1) ̸ = 0. Let m ∈ Hol(D). Assume also that there exist a neighborhood U of 1, r > 0, δ > 0 and Λ : D → C such that setting D r = τ -1 (H r ) and

V (r) = U ∩ D if ℜ(φ ′′ (1)) > 0, U ∩ D r if ℜ(φ ′′ (1)) = 0 and ℑ(φ ′′ (1)) > 0, the map m satisfies ∀z ∈ D, m(z) = (z -1) α (β + Λ(z)), with α ∈ R * , β ∈ C * , and ∀z, z ′ ∈ V (r), |Λ(z) -Λ(z ′ )| ≲ |z -1| 1+δ + |z ′ -1| 1+δ + |z -z ′ | 1 2 +δ . Then, σ p (W m,φ ) = C * .
Proof. Again, we shall prove the theorem on H and we follow a strategy similar to that of Theorem 2.7. Therefore, we fix m : H → C, which may be written

m(w) = (w + 1) -α (β + Λ(w)),
where Λ satisfies (4). By Theorem 2.8 and Proposition 2.9, we shall find a map H :

H → C such that H • Φ = H + a, and (5)
H(H) ∩ {ya : y ≤ 0} ⊂ {ya : -s ≤ y ≤ 0}.
We conclude by considering H + sa instead of H.

Step 1 : We show that σ p (W H -α ,Φ ) ∩ C * ̸ = ∅. Indeed, set F (w) = Γ α (H(w)/a), where Γ is the gamma function. Then, F is well-defined since H(w)/a ̸ ∈ R -, and

H -α (w)(F • Φ)(w) = H -α (w)Γ α H(w) a + 1 = H -α (w) H α (w) a α F (w) = a -α F (w).
Therefore, a -α ∈ σ p (W H -α ,Φ ).

Step 2 : We show that σ p (W m 1 ,Φ ) = C * , where m 1 = mH α . This will be a consequence of Theorem 2.11. Indeed, we may write, for w ∈ Ω,

m 1 (w) = (w + 1) -α β + Λ(w) w + 1 - b a log(1 + w) + (c -1) + θ(w) α = β + Λ(w) 1 - b a log(w + 1) w + 1 + c -1 w + 1 + θ(w) w + 1 α = β + Λ(w) 1 - αb a log(w + 1) w + 1 + α(c -1) w + 1 + αθ(w) w + 1 + Λ 2 (w) , with |θ(w)| ≲ |w| -ε/2 and |Λ 2 (w)| ≲ |w| -3/2 , for |w| large.
We expand the product and verify that we can write m 1 (w) = m 1 (∞) + Λ(w), with m 1 (∞) = β ̸ = 0 and Λ(w) satisfying ( 4). The last point follows from the easy fact that the product of two functions which are bounded on Ω and satisfy (4) still verifies (4). Thus we just have to check that each term satisfies this condition.

r The map p 1 : w → log(w+1) w+1 satisfies, for w, w ′ ∈ Ω,

|p 1 (w) -p 1 (w ′ )| ≲ w -w ′ (w + 1)(w ′ + 1) 3/4 .
r The map p 2 : w → (w + 1) -1 satisfies, for w, w ′ ∈ Ω,

|p 2 (w) -p 2 (w ′ )| ≤ w -w ′ (w + 1)(w ′ + 1)
.

r The map p 3 : w → θ(w) w+1 satisfies, for w, w ′ ∈ Ω,

|p 3 (w) -p 3 (w ′ )| ≲ 1 w + 1 1+ε + 1 w ′ + 1 1+ε . r The map Λ 2 satisfies, for w, w ′ ∈ Ω, |Λ 2 (w) -Λ 2 (w ′ )| ≲ 1 w + 1 3/2 + 1 w ′ + 1 3/2 .
We conclude by applying Lemma 1.1. □

Multivariable case

We now study the spectrum of weighted composition operators defined on the euclidean unit ball B d of C d . We also denote by S d-1 the unit sphere.

As proved by MacCluer in [START_REF] Maccluer | Iterates of Holomorphic self-maps of the unit ball in C n[END_REF], the Denjoy-Wolff theorem keeps true for holomorphic self-maps of B d . Precisely, let φ : B d → B d be such a map without any fixed point in B d . Then there exists ξ ∈ S d-1 such that φ [n] converges uniformly on all compact subsets of B d to ξ. Moreover, the boundary dilation coefficient λ(φ) of φ at ξ, defined by

λ(φ) = lim inf z→ξ 1 -∥φ(z)∥ 1 -∥z∥ ,
belongs to (0, 1]. We will say that φ is parabolic if λ(φ) = 1 and hyperbolic otherwise.

When m(ξ) ̸ = 0 and φ is hyperbolic, the one-dimensional proof can be easily modified to cover B d . The parabolic case will be substantially more difficult and we will use the results of [START_REF] Bayart | The linear fractional model on the ball[END_REF]. We shall restrict ourselves to the case d = 2. Since the point spectrum is stable under conjugation by automorphisms, we are allowed to reduce the matrix of dφ(ξ) and we will distinguish two cases.

r the parabolic diagonalizable case, with dφ(ξ) = 1 0 0 µ , r the parabolic nondiagonalizable case, with dφ(ξ) = 1 0 α 1 and α > 0,

(see [START_REF] Bayart | The linear fractional model on the ball[END_REF] for details). The orbits (φ [n] (0)) behave very differently for these two kinds of maps and we will need to handle them separately.

3.1. Hyperbolic symbols with nonvanishing weight at the Denjoy-Wolff point.

We begin this paragraph by recalling some definitions.

Definition 3.1. The horodisc with vertex ξ ∈ S d-1 and radius r > 0 is defined by

H(ξ, r) = z ∈ B d : |1 -⟨z, ξ⟩| 2 1 -∥z∥ 2 < r .
We obtain the following theorem, very close to the one-variable one.

m(z) = m(ξ) + Λ(z),
and that for all horodisc H with vertex ξ, there exist δ ∈ (0, 1) and a nondecreasing map g : R → R + such that ∀x ∈ (0, 1),

n≥0 g(x n ) < +∞ and z ∈ H, ∥z∥ ≥ 1 -δ =⇒ |Λ(z)| ≤ g(∥z -ξ∥). Then, σ p (W m,φ ) = C * .
Proof. By [16, Theorem 8.5.3] (see also [START_REF] Bayart | The linear fractional model on the ball[END_REF]Theorem 2.3]), setting λ ∈ (0, 1) the boundary dilation coefficient of φ, for all r > 0, φ(H(ξ, r)) ⊂ H(ξ, λr). By induction, we obtain φ

[n] (H(ξ, r)) ⊂ H(ξ, λ n r), n ∈ N.
Consider once again the sequence (f n ) defined by

f n (z) = 1 m(ξ) n n-1 k=0 m(φ [k] (z)) = n-1 k=0 m(φ [k] (z)) m(ξ) .
Then, as in the proof of Theorem 2.6,

m(φ [n] (z)) m(ξ) -1 ≤ 1 |m(ξ)| g φ [n] (z) -ξ . Let K be a compact subset of B d . Consider R > 0 such that K ⊂ H(ξ, R). Hence, for all z ∈ K, φ [n] (z) -ξ 2 = φ [n] (z) 2 + 1 -2ℜ φ [n] (z), ξ ≤ 2[1 -ℜ φ [n] (z), ξ ] + φ [n] (z) 2 -1 ≤ 2ℜ(1 -φ [n] (z), ξ ) ≤ 2 1 -φ [n] (z), ξ ≤ 2 √ Rλ n/2 1 -∥φ [n] (z)∥ 2 ≤ 2 √ Rλ n/2 .
Finally, for n large enough, if x ∈ (λ 1/4 , 1), we get φ

[n] (z) -ξ ≤ x n for n large enough. Thus, m(φ [n] (z)) m(ξ) -1 ≤ 1 |m(ξ)| g (x n ) ,
so that (f n ) converges uniformly on K. We conclude as in Theorem 2.6, using the fact that σ p (C φ ) = σ p (W 1,φ ) = C * (see [START_REF] Oger | Study of composition operators on the unit ball of n[END_REF]Proposition 3.3]). □ Lemma 3.4. Let ε ∈ (0, 1) and assume that φ ∈ D 5+ε (e 1 ) is parabolic non-diagonalizable.

Let A, M > 0. Then, for all compact sets K ⊂ H 2 , there exists n 0 ∈ N such that Φ [n] (K) ⊂ K M,A for n ≥ n 0 . Morever there exists a > 0 such that, for all η ∈ (0, 1), for all (w, z) ∈ K,

w(n) -n(n -1)a 2 ≲ n 1+η , |z(n) -na| ≲ n η , z(n) w(n) - 1 an ≲ 1 n 2-η .
Proof. The first assertion is [START_REF] Bayart | The linear fractional model on the ball[END_REF]Lemma 3.3]. The estimates are direct consequences of [5, Proposition 3.8 and Lemma 3.10]. □

We deduce the following result.

Theorem 3.5. Let ε > 0. Assume that φ ∈ D 5+ε (e 1 ) is parabolic nondiagonalizable. Let m ∈ Hol(D). Assume also that there exist M, A > 0 and δ ∈ (0, 1) such that, setting

V M,A = τ -1 (K M,A ), the map m satisfies m(z) = m(e 1 ) + Λ(z),
with m(e 1 ) ̸ = 0 and for all (w, z), (w

′ , z ′ ) ∈ V M,A , |Λ(w, z) -Λ(w ′ , z ′ )| ≲ ∥(w, z) -e 1 ∥ 1+δ + ∥(w ′ , z ′ ) -e 1 ∥ 1+δ + ∥(w, z) -(w ′ , z ′ )∥ 1 2 +δ . Then, σ p (W m,φ ) = C * .
Proof. We move on H 2 and consider m ∈ Hol(H 2 ), which can be written

m(w, z) = m(∞) + Λ(w, z),
with m(∞) ̸ = 0 and for all (w, z), (w

′ , z ′ ) ∈ K M,A , |Λ(w, z) -Λ(w ′ , z ′ )| 2 ≲ 1 + |z| 2 |w + 1| 2 1+δ + 1 + |z ′ | 2 |w ′ + 1| 2 1+δ + |w -w ′ | |w + 1| • |w ′ + 1| 1+2δ + z w + 1 - z ′ w ′ + 1 1+2δ .
As in the one-dimensional case we consider

F n (w, z) = n k=n 0 m(Φ [k] (w, z)) m(Φ [k] (1, 0)) ,
and it is sufficient to prove that (F n ) converges uniformly on all compact subsets of H 2 to conclude that m(∞) ∈ σ p (W m,Φ ). Setting (w 0 , z 0 ) = (1, 0), we get m(Φ [n] (w, z)) -m(Φ [n] (1, 0))

2 ≤ |Λ(w(n), z(n)) -Λ(w 0 (n), z 0 (n))| 2 ≤ 1 + |z(n)| 2 |w(n) + 1| 2 1+δ + 1 + |z 0 (n)| 2 |w 0 (n) + 1| 2 1+δ + |w(n) -w 0 (n)| |w(n) + 1| • |w 0 (n) + 1| 1+2δ + z(n) w(n) + 1 - z 0 (n) w 0 (n) + 1 1+2δ
. Now Lemma 3.4 allows us to conclude that m(Φ [n] (w, z)) -m(Φ [n] (1, 0)) ≲ 1 n β , for some β > 1. Indeed,

1 + |z(n)| 2 |w(n) + 1| 2 1+δ ≲ n 2 n 4 1+δ ≤ 1 n 2+2δ , whereas |w(n) -w 0 (n)| |w(n) + 1| • |w 0 (n) + 1| 1+2δ ≲ n 1+η n 4 1+2δ ≲ 1 n 2+2δ , provided η > 0 is small enough. Finally, z(n) 1 + w(n) - z 0 (n) 1 + w 0 (n) ≲ z(n) -z 0 (n) w(n)w 0 (n) + z(n) w(n) - z 0 (n) w 0 (n) ≲ 1 n 4-η + 1 n 2-η . By choosing η > 0 sufficiently small, z(n) 1 + w(n) - z 0 (n) 1 + w 0 (n) 1 2 +δ ≲ 1 n 1+γ for some γ > 0.
Thus, m(∞) ∈ σ p (W m,Φ ). To conclude, we need to observe that σ p (C Φ ) = C * . This follows from [5, Theorem 3.1], which gives the existence of G ∈ Hol(H 2 ) and a ∈ C * such that G•Φ = G+a. For all λ ∈ C, the function H = exp(λG) is then an eigenvector of C Φ corresponding to the eigenvalue exp(λa). □ 3.3. Parabolic diagonalizable symbol. We firstly come back to general considerations for φ a holomorphic self-map of B 2 with e 1 as Denjoy-Wolff point. Assume that φ ∈ D 2 (e 1 ) and (by reduction)

dφ(e 1 ) = λ 1 0 α 1 µ 1 .

Then, φ may be written

φ(w, z) = (1 + λ 1 (w -1) + γ 1 z 2 + o(|w -1|), α 1 (w -1) + µ 1 z + κ 1 z 2 + o(|w -1|))
To ensure that φ maps B 2 into itself, it is mandatory that ( 6)

λ 1 ≥ |µ 1 | 2 + 2|γ 1 |
(see [5, Equation (2.1)]) and the γ 1 z 2 term is the source of many troubles. If we translate this on Φ, we find that

Φ 1 (w, z) = λ ∞ w 1 -γ ∞ z 2 w+1 + o(w), Φ 2 (w, z) = µ ∞ z 1 -γ ∞ z 2 w+1 - α ∞ 1 -γ ∞ z 2 w+1 + F ∞ z 2 w + 1 + o z 2 w + 1 , with λ ∞ = 1/λ 1 , µ ∞ = µ 1 /λ 1 , γ ∞ = 2γ 1 /λ 1 , α ∞ = α 1 /λ 1 and F ∞ holomorphic in D.
In particular, (6) becomes

λ ∞ ≥ |µ ∞ | 2 + λ ∞ |γ ∞ |.
As for the one-dimensional case, the study of the orbits of φ done in [START_REF] Bayart | The linear fractional model on the ball[END_REF] require some regularity assumptions on φ. Recall that Wolff's lemma says that, for any (w, z) ∈ H 2 ,

ℜ(Φ 1 (w, z)) -|Φ 2 (w, z)| 2 ≥ ℜ(w) -|z| 2 .
Definition 3.6. We say that φ satisfies the strong Wolff 's condition if

λ ∞ > |µ ∞ | 2 + λ ∞ |γ ∞ |
and there exists δ > 0 such that for any (w, z) ∈ H 2 ,

ℜ(Φ 1 (w, z)) -|Φ 2 (w, z)| 2 ≥ ℜ(w) -|z| 2 + δ.
To study the spectrum of W m,Φ , we will need the following estimates on (w(n), z(n)). Lemma 3.7. Let ε ∈ (0, 1), and assume that φ ∈ D 6+ε (e 1 ) is parabolic diagonalizable and that φ satisfies the strong Wolff condition. There exist a ∈ C * and a sequence of complex numbers (a n ) such that for any compact K ⊂ H 2 , there exist C(K) satisfying, for all (w, z) ∈ K, for all n ≥ 1,

|w(n) -na| ≤ C(K) log(n + 1) |z(n)| ≤ C(K) n |z(n) -a n | ≤ C(K) log(n + 1) n 2 .
Proof. The two first inequalities already appear in [START_REF] Bayart | The linear fractional model on the ball[END_REF] (see Lemma 5.4 and the discussion after Lemma 5.3). For the third one, we shall need that our assumptions on φ ensure that on H 2 , we can write [5, p. 811]). If we iterate this, then we find

Φ 2 (w, z) = µ ∞ z + b w + 1 + p+q=2 q>0 z p (w + 1) q F p,q (w, z) + δ(w, z), with b, b p,q ∈ C, |F p,q (w, z)| ≲ 1, |δ(w, z)| ≲ 1 |w| 2+ε and |µ ∞ | < 1 (see
z(n) = µ n ∞    z + n-1 k=0 1 µ k+1 ∞    b w(k) + 1 + p+q=2 q>0 z p (k) (w(k) + 1) q F p,q (w(k), z(k)) + δ(w(k), z(k))       .
Now the known estimate on w(k) implies that

1 w(k) + 1 - 1 ka + 1 ≤ C(K) log(k + 1) k 2 .
Taking into account the estimate on |z(k)| we get for p + q = 2,

z p (k) (w(k) + 1) q ≤ C(K) k 2 .
Setting

a n = bµ n ∞ n-1 k=0 1 µ k+1 ∞ (ka + 1)
, we infer the result from the estimate

n-1 k=0 1 |µ ∞ | k+1 (k + 1) 2 ≲ 1 |µ ∞ | n n 2 .
□

We are now ready to state the theorem regarding the spectrum of W m,Φ .

Theorem 3.8. Let ε ∈ (0, 1), and assume that φ ∈ D 6+ε (e 1 ) is parabolic diagonalizable, and that φ satisfies the strong Wolff condition. Let m ∈ Hol(B 2 ) be such that m(w, z) = m(e 1 ) + Λ(w, z), with m(e 1 ) ̸ = 0 and, for any (w, z), (w

′ , z ′ ) ∈ U ∩ B 2 , |Λ(w, z) -Λ(w ′ , z ′ )| ≲ ∥(w, z) -e 1 ∥ 1+δ + ∥(w ′ , z ′ ) -e 1 ∥ 1+δ + ∥(w, z) -(w ′ , z ′ )∥ 1 2 +δ , with δ > 0 and U a neighbourhood of e 1 . Then σ p (W m,Φ ) = C * .
Proof. We follow eactly the proof of Theorem 3.5, using now the estimates coming from Lemma 3.4 which yield

1 + |z(n)| 2 |w(n) + 1| 2 ≲ 1 n 4 , |w(n) -w 0 (n)| |w(n) + 1| • |w 0 (n) + 1| ≲ log(n) n 2 , z(n) w(n) + 1 - z 0 (n) w 0 (n) + 1 ≲ log(n) n 2 .
We conclude the proof thanks to Lemma 1.1 and [5, Theorem 5.1] which asserts that σ p (C Φ ) = C * . □ 3.4. Hyperbolic symbols and vanishing weights. We now consider the case where the weight vanishes at the Denjoy-Wolff point. It turns out that the situation becomes much more complicated than in the disc and that, even for very simple symbols and multipliers, the point spectrum can be empty. We provide an example on H 2 .

Example 3.9. Let Φ(w, z) = (2w, z) and m(w, z) = z w+1 . Then σ p (W m,Φ ) = ∅. Proof. Suppose that f is a nonzero eigenvector of W m,Φ corresponding to λ ̸ = 0. For a fixed w with ℜ(w) > 0, z → f (w, z) is analytic in |z| 2 < ℜ(w). Therefore we can write it f (w, z) = n≥0 a n (w)z n .

Evaluating mf • Φ = λf at (w, 0), we find f (w, 0) = 0 thus a 0 (w) = 0. Now the relation mf • Φ = λf also implies that for all n ≥ 0,

λa n+1 (w) = a n (2w) w + 1 ,
which in turn yields a n (w) = 0 for all n ∈ N and all w ∈ C with ℜ(w) > 0. Therefore f = 0, a contradiction. □

Observe that the previous weight has a very simple expression on B 2 : m(w, z) = z. This is one of the easiest example of a weight vanishing at e 1 ! Nevertheless, we will be able to get positive results. We did not look for the most general statement, and we will only handle the case where m is a polynomial vanishing at e 1 with no linear term in z, and with a nonzero linear term in w -1.

Theorem 3.10. Let φ ∈ D 3 (e 1 ) be a hyperbolic map with a Denjoy-Wolff point at e 1 . Let m : B 2 → C be a polynomial writing m(w, z) = w -1 + p+q≥2 a p,q (w -1) p z q .

Then, σ p (W m,φ ) = C * .

Proof. We move on the Siegel half-space H 2 and show that, for m(w, z) = 1 w + 1 + p+q≥2 a p,q z q (w + 1) p+q , then σ p (W m,Φ ) = C * . First, we focus on the case λ ∞ > |µ ∞ | 2 . We will use some results of [START_REF] Bayart | The linear fractional model on the ball[END_REF], in particular Lemma 4. There exist κ ∈ (0, 1) and M > 0 such that r L M is Φ-stable, r For each compact set K ⊂ H 2 , for each (w, z) ∈ K, That E maps H 2 into H is not written explicitly in [START_REF] Bayart | The linear fractional model on the ball[END_REF], but this follows from the definition of E as the limit of w(n)/λ n ∞ .

We set m 1 = mE, and first show that σ p (W m 1 ,Φ ) = C * . Arguing as in the proof of Theorem 3.2, it is enough to prove that |m 1 (w(n), z(n)) -1| converges uniformly on each compact set K ⊂ H 2 . We may even assume that K ⊂ L M . We write |m 1 (w(n), z(n)) -1| ≤ E(w(n), z(n)) w(n) + 1 -1 + p+q≥2 |a p,q | E(w(n), z(n))z(n) q (w(n) + 1) p+q .

On the one hand, using [START_REF] Bayart | Hyperbolic composition operators on the ball[END_REF], E(w(n), z(n))z(n) q (w(n) + 1) p+q ≲ κ nq |w(n) + 1| p+ q 2 -1 . For p + q ≥ 2, we deduce p + q 2 -1 ≥ 0 and this gives rise to a convergent series. On the other hand,

E(w(n), z(n)) w(n) + 1 -1 ≲ E(w(n), z(n)) -w(n) w(n) + 1 + 1 λ n ∞ ≲ |z 2 (n)| + |w(n) + 1| 1/2 |w(n) + 1| + 1 λ n ∞ ≲ κ 2n + 1 λ n/2 ∞
and we also obtain uniform convergence of the induced series. Finally, we show that σ p (W E -1 ,Φ ) ∩ C * ̸ = ∅. But as in the one-dimensional case, it is easy to check that f = exp(µ log 2 E) is an eigenvector of W E -1 ,Φ for µ = -1/2 log(λ ∞ ).

We turn to the case |µ ∞ | 2 = λ ∞ , which is simpler (since it implies that γ ∞ = 0). From [START_REF] Bracci | Solving the Schröder equation at the boundary in several variables[END_REF], in particular Proposition 2.1 and Lemma 2.4, we know that r There exists E : H 2 → H such that

E • Φ = λ ∞ E, |E(w, z) -w| ≲ |w| 1/2 for all w ∈ H 2 .
r For all compact sets K ⊂ H 2 , for all (w, z) ∈ K,

|z(n)| ≲ |µ ∞ | n = λ n/2
∞ and |w(n)| ≳ λ n ∞ . This comes from the fact that w(n)/λ n ∞ and z(n)/µ n ∞ converge uniformly on compact subsets of H 2 to a holomorphic function and that E, which is the limit of w(n)/λ n ∞ , does not vanish. Hence, if m 1 = mE, we obtain

Definition 2 . 2 .

 22 Let ξ ∈ T, and φ : D ∪ {ξ} → D ∪ {ξ} a holomorphic map on D.

2. 1 .

 1 Hyperbolic symbol with nonvanishing weight at the Denjoy-Wolff point. We begin by a well-known result about the iterates of hyperbolic selfmaps of D within a Stolz angle. Lemma 2.4 ([12, Lemma 2.66]). Let φ : D → D be a hyperbolic holomorphic map, with Denjoy-Wolff point ξ ∈ T. Then, for all compact subsets K of D, there exists R > 0 such that for all n

Theorem 2 . 5 ([ 2 ,

 252 Theorem 6.3]). Let φ : D → D be a hyperbolic holomorphic map, with Denjoy-Wolff point ξ, and m ∈ Hol(D) with a non-zero angular limit m(ξ), and a finite angular derivative at ξ. Then, σ p (W m,φ ) = C * .

=c 7 ,

 7 with |θ 7 (w)| ≲ |w| -1 for |w| large. Finally, if c = d 3 + bc 7 /a and θ = ∆ 3 -bθ 7 /a, then H(w) = w -b a log(w + 1) + c + θ(w), |θ(w)| ≲ |w| -ε/2 . □

2. 4 . 3 . 2 . 11 . 1 .

 432111 Parabolic symbols with nonvanishing weight at the Denjoy-Wolff point. The preliminary results of the previous section allow us to get the following theorem. Here again, we set a = φ ′′ (1), b = φ ′′ (1) 2 -2φ ′′′ (1)/Theorem Let φ : D → D be a parabolic holomorphic map, with Denjoy-Wolff point Assume that φ is regular, C 3+ε (1), and that φ ′′ (1) ̸ = 0. Let m ∈ Hol(D).

2. 5 .

 5 Parabolic symbols with vanishing weight at the Denjoy-Wolff point. If the weight m does vanish, or admits a pole at the Denjoy-Wolff point of the symbol, we get a similar result.

  H(w) = w -b a log(1 + w) + c + θ(w), w ∈ Ω, with Ω = H or H + , and |θ(w)| ≲ |w + 1| -ε . First, we show that we may always assume that H(H) ∩ {ya : y ≤ 0} = ∅. r If ℜ(a) = 0, then H(H) ⊂ H (cf. [9, p.64]), which gives the result. r Otherwise, let x ≥ 0 be such that if B(w) = c + θ(w), then |B(w)| ≤ x/2 on H. We first show that for |w| large enough, say |w| > ρ and y ≤ -x/ℜ(a), then one cannot have H(w) = ya. Otherwise z := w -b a log(1 + w) = ya -B(w) would belong to D(w, |w|/2) and to B(ya, |y|ℜ(a)/2), but these two discs have empty intersection. Moreover, H is bounded on the compact set H ∩ {|w| ≤ ρ}, so we get the existence of s > 0 such that

Theorem 3 . 2 .

 32 Let φ : B d → B d be a hyperbolic holomorphic map, with Denjoy-Wolff point ξ ∈ S d-1 , and m ∈ Hol(B d ) having a non-zero K-limit at ξ. Assume that

2 ,

 2 Lemma 4.4 and Proposition 4.5. Let us review this material. For M > 0, we setL M = (w, z) ∈ H 2 : |w| > M and |z| |w + 1| 1/2 < 1 log |w + 1|.

r

  For each compact set K ⊂ H 2 , there existsn 0 ∈ N such that Φ [n 0 ] (K) ⊂ L M , r For each compact set K ⊂ H 2 , for all (w, z) ∈ K, |w(n)| ≳ λ n ∞ ,as proved in [5, Proposition 4.5] or [11, Lemma 2.4]), r There exists E : H 2 → H holomorphic such that on L M , E • Φ = λ ∞ E and |E(w, z) -w| ≲ |z| 2 + |w + 1| 1/2 .

|m 1 1 λ

 11 (w(n), z(n)) -1| ≤ E(w(n), z(n)) w(n) + 1 -1 + p+q≥2 |a p,q | E(w(n), z(n))z(n) q (w(n) + 1) p+q ≲ -1+q/2) ∞, and we conclude as above. □

Acknowledgments : This research is partly supported by the Bézout Labex, funded by ANR, reference ANR-10-LABX-58.

3.2. Parabolic non-diagonalizable symbol. In the following subsections, we will focus on two-variables symbols, and we will always assume that φ is a holomorphic self-map of B 2 with Denjoy-Wolff point at e 1 = (1, 0).

An element of C 2 will be denoted (w, z). To apply the results of [START_REF] Bayart | The linear fractional model on the ball[END_REF], we need a notion of regularity at e 1 which emphasizes the dissymmetry between w -1 and z as (w, z) goes to e 1 . Indeed, one always has |z| 2 ≤ 1 -|w| 2 ≤ 2(1 -|w|). This leads up to the following definition.

As in the one-variable case, we will move from B 2 to another domain where the computations will be easier. The relevant domain here is the Siegel half-space, defined by

We will set Φ = τ • φ • τ -1 . Regarding distances between points, it is easy to check that if (w 0 , z 0 ) = τ (w, z) and

In this paragraph, we will assume that φ is parabolic and that dφ(e 1 ) = 1 0 α 1 , with α > 0. We call it a parabolic non-diagonalizable map.

As for the one variable case, the construction of eigenvectors will follow from a careful analysis of the behaviour of φ [n] (w, z). Moving to H 2 , for (w, z) ∈ H 2 , we will denote (w(n), z(n)) = Φ [n] (w, z). To study how (w(n), z(n)) goes to ∞, it is convenient to introduce the following sets. For A, M > 0, let

From the results of [START_REF] Bayart | The linear fractional model on the ball[END_REF], we will extract the following lemma.

Open question. What happens if the symbol φ is parabolic and the weight m does vanish at the Denjoy-Wolff point ?


Christine Coustau
email: christine.coustau@sophia.inra.fr

Joachim Kurtz

Yannick Moret

A Novel Mechanism of Immune Memory Unveiled at the Invertebrate-Parasite Interface

Keywords: trained immunity, immune priming, host-parasite interaction, Schistosoma mansoni, Biomphalaria glabrata

HAL is a

Pinaud et al. recently provided the first global investigation of the molecular processes

underlying innate immune memory in an invertebrate species. They showed that the memory response of the snail Biomphalaria glabrata to Schistosoma mansoni infection is associated with a shift from cellular to humoral mechanisms. Specificity and memory of immune systems are important traits with strong implications for the evolution of host-parasite interactions, the dynamics of infectious diseases, and potential strategies for the prevention and control of infections of economically valuable species. The classical view that immune memory is restricted to the adaptive immune systems of vertebrates is challenged by an increasing number of studies reporting immune memory traits from both vertebrate and invertebrate innate immune systems [START_REF] Kurtz | Memory in the innate and the adaptive immune systems[END_REF][START_REF] Netea | Trained immunity: a memory for innate host defense[END_REF]. However, the existence of immune memory processes in invertebrates (also referred to as immune priming) remains controversial because it is still largely unexplained at the cellular and molecular levels (but see [START_REF] Rodrigues | Hemocyte differentiation mediates innate immune memory in Anopheles gambiae mosquitoes[END_REF]).

In their recent work on the immune response of the snail Biomphalaria glabrata to infection by the human blood fluke Schistosoma mansoni, Pinaud et al. [START_REF] Pinaud | A shift from cellular to humoral responses contributes to innate immune memory in the vector snail Biomphalaria glabrata[END_REF] demonstrated that the observed heightened response to a secondary infection was associated with a shift from cellular to humoral processes. While a classical cellular encapsulation occurs during the first infection, secondarily infecting parasites are killed without encapsulation. Using experimental transfer of plasma, they inferred that the soluble fraction of the snail blood caused the parasite death. This result was unexpected for two reasons. First, previous studies on innate immune memory mostly showed that cellular rather than humoral processes play a role, especially when the immune memory involves specificity [START_REF] Pham | A specific primed immune response in Drosophila is dependent on phagocytes[END_REF][START_REF] Roth | Phagocytosis mediates specificity in the immune defence of an invertebrate, the woodlouse Porcellio scaber (Crustacea: Isopoda)[END_REF]. Moreover, the memory response of the mosquito Anopheles gambiae to Plasmodium infection is based on a sustained hemocyte differentiation protecting mosquitoes from secondary infections [START_REF] Rodrigues | Hemocyte differentiation mediates innate immune memory in Anopheles gambiae mosquitoes[END_REF]. Second, until now, invertebrate immune memory was believed to rely either on a sustained response, involving the long-lasting upregulation of the same immune effectors after the initial immune challenge (Figure 1A), or a recalled response, leading to a faster and stronger defense (Figure 1B). Both processes implicitly involve the same immune effector systems during the primary and the secondary immune responses. By contrast, the work of Pinaud et al. reveals that the primary and secondary immune responses, developed against the same parasite, involve different immune effector systems resulting in a shift from a cellular to a humoral response (Figure 1C).

A first insight into the molecular mechanisms associated with this immune shift was provided by a comparison of whole-snail transcriptome at various times after a first infection and after a secondary exposure to the parasite. A substantial set of differentially expressed transcripts was over-or under-represented exclusively following the secondary challenge. Among the overexpressed candidates were transcripts coding for various fibrinogen-related proteins (FREPs). FREPs are complex pattern recognition proteins undergoing somatic rearrangement [START_REF] Zhang | Diversification of Ig superfamily genes in an invertebrate[END_REF]. They are known to interact with the highly polymorphic mucins (SmPoMucs) from S. mansoni and to participate in Biomphalaria immune responses (for review, see [START_REF] Coustau | Advances in gastropod immunity from the study of the interaction between the snail Biomphalaria glabrata and its parasites: a review of research progress over the last decade[END_REF]).

Experimentally reduced expression of three of the major FREP members via RNA interference resulted in a partial loss of protection, demonstrating that FREPs are involved in the memory response. Other transcripts over-represented during the secondary response included cytotoxic and cytolytic molecules such as lipopolysaccharide-binding protein/bactericidal/permeabilityincreasing protein (LBP/BPI) and biomphalysin. Analysis of primed plasma proteomes revealed that a number of proteins, consistent with the transcriptomic approach, were specifically present in the plasma after a secondary challenge. Altogether, these results showed that the primary and secondary challenges result in the expression of different sets of genes and in the presence of different plasmatic proteins, further supporting the initial histological observation of a shift in immune responses.

The precise mechanisms determining this immune shift should be further characterized. In particular, it is important to distinguish the immune memory response from (i) the effects that are potentially mediated by the parasites themselves, and from (ii) immune responses resulting from a constant stimulation by parasites. It would further be interesting to see whether the form of memory observed here is primarily based on receptor, signaling, or effector molecules.

Intriguingly, FREPs could serve as both receptor and effector, just like vertebrate antibodies.

In this context, it is noteworthy that the vertebrate immune response entails a shift from cellular (innate, granulocyte-based responses) to humoral (antibody-based) and back to cellular (memory B and T cells) responses. While such details of the immune responses of the snail still need to be uncovered, the obvious complexity of the induced responses led the authors to support views that question the artificial dichotomy between innate and adaptive immunity [START_REF] Kurtz | Memory in the innate and the adaptive immune systems[END_REF].

Recently, elucidation of mechanisms determining memory processes from the vertebrate innate immune system led vertebrate immunologists to propose the concept of 'trained immunity' [START_REF] Netea | Trained immunity: a memory for innate host defense[END_REF].

It refers to an immune response that (i) is induced after a primary infection or vaccination; (ii) increases resistance to reinfection, in a more or less specific manner; and (iii) is independent of the vertebrate-acquired immune system. These criteria are met by the memory type responses of invertebrates and plants, emphasizing that this concept has rather large evolutionary implications. Furthermore, while the concept of trained immunity has been proposed for the development of innate immune memory within the individual, it may also include transgenerational effects in which the immunological experience of parents could be used to reprogram the immune system of the offspring in both animals and plants [START_REF] Moret | Entomology -immune defence in bumblebee offspring[END_REF].

In conclusion, a major achievement of this work is to provide molecular evidence for a so-far unreported process of memory type immune response in invertebrates, further highlighting the diversity of invertebrate immune processes. Indeed, it seems unlikely that FREPs, here shown to be involved in the snail immune memory response, also determine immune memory responses in phyla such as insects or sea urchins, where immune recognition relies on other pattern recognition proteins. More generally, while major immune signaling pathways (i.e., Imd, Toll, JAK/STAT) are well conserved across invertebrate phyla, non-self-recognition molecules such as FREPs or Down syndrome cell adhesion molecule (Dscam) are highly diversified [START_REF] Cerenius | Variable immune molecules in invertebrates[END_REF]. Invertebrate immune systems may be viewed as a mosaic of evolutionary conserved processes and independent immune innovations. Deciphering the cellular and molecular basis of invertebrate immune responses from an increasing variety of phylogenetically distant models will shed light on the nature and origin of these evolutionary innovations, including immune memory type novelties.

Figure 1 .

 1 Figure 1. Basic Models of Immune Memory Responses in Invertebrates. In response to a primary infection (black arrows), a cellular response (blue line) is mounted and may either (A) be sustained in a long-lasting upregulation of immune effectors protecting from a secondary infection (blue arrow); (B) be recalled in a faster and stronger manner after a secondary infection (blue arrow); or (C) result in a shift to a humoral response (broken red line) upon secondary infection (blue arrow).


ON UNIVERSALITY OF GENERAL DIRICHLET SERIES FR ÉD ÉRIC BAYART AND ATHANASIOS KOUROUPIS

In the present work we establish sufficient conditions for a Dirichlet series induced by general frequencies to be universal with respect to vertical translations. Applying our methodology we give examples of universal Dirichlet series such as the alternating prime zeta function n≥1 (-1) n p -s n .

Introduction

The study of the universal properties of Dirichet series goes back to 1975 with the seminal work of Voronin on the Riemann zeta function [START_REF] Voronin | A theorem on the "universality" of the Riemann zeta-function[END_REF]. Voronin's theorem says:

Let K be a compact subset of {1/2 < e(s) < 1} with connected complement, let f be a nonvanishing function continuous on K and holomorphic in the interior of K. Then dens τ ≥ 0 : sup s∈K |ζ(s + iτ ) -f (s)| < ε > 0 where dens(A) denotes the lower density of A ⊂ R + , that is dens(A) = lim inf

T →∞ T 0 1 A (t) dt.
Let us introduce the following definitions: let Ω 1 ⊂ Ω ⊂ C be two domains such that Ω 1 +iτ ⊂ Ω 1 for all τ > 0, and, for all compact sets K ⊂ Ω, there exists τ > 0 with K +iτ ⊂ Ω 1 . Let D : Ω 1 → C be holomorphic. We say that D is universal in Ω if, for all compact subsets K of Ω with connected complement, for all nonvanishing functions f : K → Ω, continuous on K and holomorphic in the interior of K,

dens τ ≥ 0 : sup s∈K |D(s + iτ ) -f (s)| < ε > 0.
We say that f is strongly universal if the restriction that it is non-vanishing can be eased. Since Voronin's work, the area of universality gained popularity. Many authors studied aspects of (strong) universality for various classes of Dirichlet series n a n e -λns , where (a n ) ⊂ C N and (λ n ) is an increasing sequence of nonnegative real numbers tending to +∞. The case (λ n ) = (log n) corresponds to ordinary Dirichlet series. The survey paper [START_REF] Matsumoto | A survey on the theory of universality for zeta and L-functions[END_REF] provides a thorough examination of the subject up to 2015.

The first author in [START_REF] Bayart | Universality of general Dirichlet series with respect to translations and rearrangements[END_REF] improving the work of [START_REF] Laurinčikas | The universality of general Dirichlet series[END_REF] on strong universality of general Dirichel series obtained the following result:

Let P ∈ R d [X] with d ≥ 1 and lim +∞ P = +∞, let Q ∈ R d-1 [X]
, let ω ∈ R\2πZ and let κ ∈ R. Assume moreover that the sequence (log(P (n)) n≥1 is Q-linearly independent. Then the Dirichlet series D(s) = n≥1 Q(n)(log n) κ e iωn (P (n)) -s is strongly universal in {(2d -1)/2d < e(s) < 1}. This generalizes the case of the Lerch zeta function (see [START_REF] Laurinčikas | The Lerch zeta-function[END_REF]) when Q(n) = 1, κ = 0 and P (n) = n + α with α transcendental.

One of the objectives of this article is to study potential cases of universal Dirichlet series for which the methods of [START_REF] Bayart | Universality of general Dirichlet series with respect to translations and rearrangements[END_REF] are not applicable, see for example [START_REF] Bayart | Universality of general Dirichlet series with respect to translations and rearrangements[END_REF]Question 6.8]. The first one concerns the frequencies (λ n ). The simplest example of a frequency (λ n ) such that (log(λ n )) is Q-linearly independent is probably the sequence (p n ) of prime numbers. However, this sequence is not regular enough to be handled by the methods of [START_REF] Bayart | Universality of general Dirichlet series with respect to translations and rearrangements[END_REF]. The main barrier for this problem is that the sequence of primes is not regular enough to estimate its partial sums using the classical techniques from harmonic analysis like the method of non-stationary phase/ Van der Corput type lemmas 2.5 or decoupling [START_REF] Bourgain | Decoupling, exponential sums and the Riemann zeta function[END_REF]. We are able to prove strong universality for Dirichlet series with type of frequencies and with similar (and even slightly relaxed) conditions on the coefficients. However, the universal property is only shown on a smaller strip.

Theorem 1.1. Let D(s) = n≥1 a n e -λns be a Dirichlet series and let d ∈ N. Assume that

• λ n = log P (p n ) where P ∈ R d [X] and lim +∞ P = +∞. • there exist C, κ > 0 such that for n ≥ 2, n d-1 C(log n) κ ≤ |a n | ≤ Cn d-1 (log(n)) κ . • σ c (D) = 0. • The sequence (λ n ) is Q-linearly independent. Then D is strongly universal in {(3d -1)/d < e(s) < 1}. Corollary 1.2. The Dirichlet series n (-1) n p -s n is strongly universal in the strip { 2 3 < e(s) < 1}.
Observe that for the examples coming from [START_REF] Bayart | Universality of general Dirichlet series with respect to translations and rearrangements[END_REF] or from Theorem 1.1, the Dirichlet series itself converges in its strip of universality. This does not cover the case of the Riemann zeta function or that of the Hurwitz zeta functions n (n + α) -s , α transcendental, which have a pole at 1 and are known to be universal in { 1 2 < e(s) < 1}. We extend those results to a large class of general Dirichlet series, even in the case 1 is a branching point and not a pole. In what follows we denote by C σ the half-plane { e(s) > σ} and by C + σ its restriction to the complex numbers of positive imaginary part, {s ∈ C : e(s) > σ, m(s) > 0}.

Theorem 1.3. Let P ∈ R d [X] with lim +∞ P = +∞, let Q ∈ R d-1 [X] and let κ ∈ R. Assume moreover that the sequence (log(P (n))) is Q-linearly independent. Then the Dirichlet series D(s) = n Q(n)(log(n)) κ [P (n)] -s admits a holomorphic continuation to C + 1-1 d ∪ C 1 and even to C 1-1 d \{1} if κ ∈ N 0 .
Moreover, it is strongly universal in the strip {(2d -1)/d < e(s) < 1} . Notation. Throughout the paper, if f, g : E → R are two functions defined on the same set E, the notation f g will mean that there exists some constant C > 0 such that f ≤ Cg on E. It is well-known that D = n a n e -λns converges in the half-plane C σc(D) and that it defines a holomorphic function there. (3) for all σ 1 > σ 0 , there exist t 0 , B > 0 such that, for all s = σ + it with σ ≥ σ 1 and

t ≥ t 0 , |D(σ + it)| ≤ t B ; (4) for all σ 2 > σ 1 > σ 0 , sup σ∈[σ1,σ2] sup T >0 1 T ˆT 1 |D(σ + it)| 2 dt < +∞; (5) the sequence (λ n ) is Q-linearly independent.
Definition 2.2. We say that a Dirichlet series D = n a n e -λns belongs to D dens provided for all α, β > 0, there exist C > 0 and x 0 ≥ 1 such that, for all x ≥ x 0 ,

λn∈[x,x+ α x 2 ] |a n | ≥ Ce (σa(D)-β)x .
The main interest of introducing these definitions is the following theorem (see [START_REF] Bayart | Universality of general Dirichlet series with respect to translations and rearrangements[END_REF]). It should be pointed out that Definition 2.1 in [START_REF] Bayart | Universality of general Dirichlet series with respect to translations and rearrangements[END_REF] mentions the whole half-plane C σ0 and not the quarter half-plane as here. However, this does not change anything for the proofs. The key points are that the half vertical lines σ + it, t > 0, σ > σ 0 , are contained in C + σ0 and that for any compact set K included in the strip {σ 0 < e(s) < σ a (D)}, there exists τ > 0 such that K + iτ ⊂ C + σ0 .

2.3.

Two lemmas to estimate exponential sums. We shall need two inequalities which have been widely used in this context. The first one deals with exponential sums and is due to Montgomery and Vaughan (see [START_REF] Montgomery | Hilbert's inequality[END_REF]).

Lemma 2.4. Let (a n ) be a sequence of complex numbers such that n |a n | 2 < +∞. Let (λ n ) be a sequence of real numbers and set

θ n := inf m =n |λ n -λ m | > 0 for every n. Then ˆT 0 n a n e iλnt 2 dt = T n |a n | 2 + O n |a n | 2 θ n
where the O-constant is absolute.

We also need the following classical inequality for exponential sums, which goes back to J. G. van der Corput (see [START_REF] Bayart | Dynamics of linear operators[END_REF]Lemma 11.5]).

Lemma 2.5. Let a < b and let f, g : [a, b] → R be two functions of class C 2 . Assume that • f is monotonic with |f | < 1/2;
• g is positive, non-increasing and convex. Then

b n=a g(n)e 2πif (n) = ˆb a g(u)e 2πif (u) du + O(g(a) + |g (a)|).
2.4. The incomplete Gamma function/ Prym's function. We will make a short presentation and we refer the interested reader to [START_REF]Handbook of mathematical functions, with formulas, graphs, and mathematical tables[END_REF][START_REF] Frank | Asymptotics and special functions[END_REF][START_REF] Temme | Special functions[END_REF]. For e(a) > 0 and e(z) > 0, we define the incomplete Gamma function Γ(a, z) by

Γ(a, z) = ˆ+∞ z t a-1 e -t dt.
For fixed z, as in the classical case it has a meromorphic extension in C with simple poles at the nonpositive integers. This can be easily obtained from the recurrence relation:

Γ(a + 1, z) = aΓ(a, z) + z a e -z .
For a fixed value of a, Γ admits a holomorphic extension (its principal branch) to C\R -and even to C when a is a positive integer. When a is not a nonpositive integer, this follows for instance from the relation

Γ(a, z) = Γ(a)(1 -z a-1 γ * (a, z)),
where the function γ * is entire in both a and z. When a is a nonpositive integer, this follows from the corresponding statement for a = 0 (in that case, the incomplete Gamma function is also called the exponential integral). For this principal branch (and for a fixed a), we have the estimation

(1) Γ(a, z) = e -z ˆ+∞ 0 e -u (z + u) a-1 du = O(z a-1 e -z ),
as |z| → +∞.

2.5.

A remark on [START_REF] Bayart | Universality of general Dirichlet series with respect to translations and rearrangements[END_REF]. In [4, Theorem 1.6], the theorem on rearrangement universality of Dirichlet series states that a Dirichlet series n a n e -λns is rearrangement universal if for any f ∈ H(Ω), where Ω is the strip σ c (D) < e(s) < σ a (D), there exists a permutation σ of N such that n a σ(n) e -λ σ(n) s converges to f in H(Ω)). This theorem is false. Indeed it would imply that n (-1) n n -s is rearrangement universal. This cannot hold: any rearrangement of n (-1) n n -s will take values in R for real values of the parameter s. The mistake which is made in [START_REF] Bayart | Universality of general Dirichlet series with respect to translations and rearrangements[END_REF] lies on the fact that a lemma due to Banaszczyk is only true for some real Fréchet spaces and was applied to the complex Fréchet space H(Ω).

Proof of Theorem 1.1

The main difficulty in order to apply Theorem 2.3 is to estimate the square moments of D on vertical lines. We follow a method introduced in [START_REF] Balazard | The mean square of the logarithm of the zeta-function[END_REF] where the authors estimate the square moments of the logarithm of the zeta function. Let σ 0 ∈ (α, 1) with σ 0 > σ 2 (D) and let us set A = B/(σ 0 -α). Then for all T ≥ 1, for all σ > σ 1 > σ 0 ,

ˆT 0 |D(σ + it)| 2 dt T + n |a n | 2 exp(-2λ n σ) exp -2 e λn T A min(λ n -λ n-1 , λ n+1 -λ n ) .
Proof. The inverse Mellin transform (see [START_REF] Montgomery | Multiplicative number theory. I. Classical theory[END_REF]Appendix 3]) applied to the Γ function says that, for all x > 0,

(2)

e -x = 1 2πi ˆ2-σ0+i∞ 2-σ0-i∞
x -w Γ(w)dw.

Let T ≥ 2 and set X = T A . We apply (2) for x = e λn X , yielding to exp -

e λn X = 1 2πi ˆ2-σ0+i∞ 2-σ0-i∞ exp(-λ n w)X w Γ(w)dw.
Therefore, for any σ > σ 0 and any t ∈ [0, T ], setting s = σ + it, for any n ≥ 1,

a n exp (-λ n s) exp - e λn X = 1 2πi ˆ2-σ0+i∞ 2-σ0-i∞ a n exp (-λ n (s + w)) X w Γ(w)dw.
Since e(s + w) > 1 provided e(w) = 2 -σ 0 , we can sum these equalities and interchange summation and integral to get

(3) n≥1 a n exp (-λ n s) exp e -λn X = 1 2πi ˆ2-σ0+i∞ 2-σ0-i∞ D(s + w)X w Γ(w) dw.
We set τ = σ 0 -α and we introduce the following contour C, defined as the union of five segments or half-lines.

x y log 2 (T ) -τ C 1 C 5 C 2 C 3 C 4 2 -σ 0 On C 1 ∪ C 5 , |D(s + w)| 1.
Moreover, writing w = u + iv with u = 2 -σ 0 , Stirling's formula for the Γ-function (see again [START_REF] Montgomery | Multiplicative number theory. I. Classical theory[END_REF]Appendix 3]) says that

|Γ(u + iv)| e -C|v| for some C > 0 (independent of w ∈ C 1 ∪ C 5 ). Hence, ˆC1∪C5 |D(s + w)X w Γ(w)|dw X 2-σ0 ˆ+∞ log 2 (T ) e -Cv dv 1. Pick now w = u + iv ∈ C 2 ∪ C 3 ∪ C 4 . Then | m(s + w)| ≤ log 2 (T ) + T T e(s + w) ≥ σ 0 -τ = α. Therefore, |D(s + w)| T B . This implies ˆC2∪C4 |D(s + w)X w Γ(w)|dw T B T A(2-σ0) e -C log 2 (T )
1.

Finally,

ˆC3 |D(s + w)X w Γ(w)|dw T -Aτ T B ˆR |Γ(-τ + it)|dt
by our choice of A and τ . Summing the estimates we obtain (4) ˆC |D(s + w)X w Γ(w)|dw 1.

Let now R be the rectangle

C 2 ∪ C 3 ∪ C 4 ∪ C 6 with C 6 = [2 -σ 0 -i log 2 T, 2 -σ 0 + i log 2 T ] so that (5) ˆC = ˆ2-σ0+i∞ 2-σ0-i∞ + ˆR .
The function w → D(s + w)X w Γ(w) has a single pole 0 inside R, with residue D(s). Hence, by

, ( 4) and ( 5),

|D(s)| 1 + n≥1 a n exp(-λ n s) exp -e λn X .
We set b n = a n exp(-λ n σ) exp -e λn

X

. Taking the square and integrating over [0, T ], we find

ˆT 0 |D(σ + it)| 2 dt T + ˆT 0 n≥1 b n exp(-iλ n t) 2 dt.
We apply the Montgomery-Vaughan inequality, yielding to the following estimate:

ˆT 0 |D(σ + it)| 2 dt T + T n≥1 |a n | 2 exp(-2λ n σ) + n≥1 |b n | 2 min(λ n -λ n-1 , λ n+1 -λ n ) .
Since σ 0 > σ 2 (D), we have proven Lemma 3.1.

To estimate the sum appearing in the last lemma, we shall use the following result.

Lemma 3.2. Let (λ n ) be a sequence of frequencies, let (a n ) be a sequence of complex numbers.

Assume that there exist a ∈ R, b, c, β > 0 such that

|a n | n a , λ n ≥ b log n, min(λ n -λ n-1 , λ n+1 -λ n ) ≥ c exp(-βλ n ).
Then for all X > 0 and all σ > β/2,

n≥1 |a n | 2 exp(-2λ n σ) exp -2 e λn X min(λ n -λ n-1 , λ n+1 -λ n ) max 1, X 2a+(β-2σ)b+1 b .
In particular, if e λn+1 -e λn ≥ 1 for all n, we may choose β = 1 in the previous lemma. Indeed, the inequality

λ n+1 -λ n ≥ exp(-λ n ) is clear if λ n+1 -λ n ≥ 1. Otherwise, λ n+1 -λ n e λn+1-λn -1 = (e λn+1 -e λn )e -λn .
Proof. Denote by S the sum. Then

S n n 2a exp ((β -2σ)λ n ) exp -2 e λn X n n 2a+(β-2σ)b exp -2 n b X .
We split the sum into two parts. We first sum up to X 1/b and we denote by S 1 this sum. Then

S 1 n≤X 1/b n 2a+(β-2σ)b max 1, X 2a+(β-2σ)b+1 b .
Regarding the second sum, say S 2 , we write

S 2 = n≥X 1/b n 2a+(β-2σ)b exp - 2n b X ˆ+∞ X 1/b t 2a+(β-2σ)b exp - 2t b X dt
We do the change of variables u = t b /X which yields

S 2 X 2a+(β-2σ)b+1 b ˆ+∞ 1 u 2a+(β-2σ)b+1-b b exp(-2u)du X 2a+(β-2σ)b+1 b .
Using what has been done above, we may complete the first part of the proof of Theorem 1.1, that is: • there exist C, κ > 0 such that for n ≥ 2,

Corollary 3.
n d-1 C(log n) κ ≤ |a n | ≤ Cn d-1 (log(n)) κ . • σ c (D) = 0. • the sequence (λ n ) is Q-linearly independent.
Then D belongs to D w.a. (σ 0 ) where σ 0 = 1 -1 3d . Proof. Under the above conditions, σ a (D) = 1, σ c (D) = 0 and σ 2 (D) = 1 -1 2d . Therefore, conditions (1) to (3) of Definition 2.1 are satisfied. Let us prove (4). Let σ 1 > σ 0 and let ε < σ 1 -σ 0 . We set α = ε, and observe that D has order 1 in C α (see [START_REF] Hardy | The general theory of Dirichlet's series[END_REF]Theorem 12]). We apply Lemma 3.1 and then Lemma 3.2 with a = (d -1) + ε, b = d -ε and β = 1 to obtain that, for σ ≥ σ 1 , for T ≥ 2,

ˆT 0 |D(σ + it)| 2 dt T + T B(ε) with B(ε) = 2 (d -1) + ε) + (1 -2σ 0 )(d -ε) + 1 (d -ε)(σ 1 -ε) . Now, lim ε→0 B(ε) = 3d -2dσ 0 -1 dσ 1 = σ 0 σ 1 < 1.
Hence, choosing ε > 0 small enough, we have shown that

´T 0 |D(σ + it)| 2 dt T.
Let us proceed with the second half. First let us state the following lemma proved in [4, Lemma 6.1]: Lemma 3.4. Let P (X) = d k=0 b k X k be a polynomial of degree d, with b d > 0. Then, there exist x 0 , y 0 > 0 such that P induces a bijection from [x 0 , +∞] to [y 0 , +∞], and • there exist C, κ > 0 such that for n ≥ 2,

P -1 (x) = x 1/d (b 1/d d ) - b d-1 b (d-1)/d d + o(1), as x → +∞.
n d-1 C(log n) κ ≤ |a n | ≤ Cn d-1 (log(n)) κ .
Then D belongs to D dens .

Proof. Let α, β > 0. Without loss of generality, we may assume that P is one-to-one on [log(p 1 ), +∞). Then

λ n ∈ x, x + α x 2
if and only if p n ∈ P -1 (e x ), P -1 (e x+ α x 2 ) .

Using Lemma 3.4, there exist c 1 > 0, c 2 ∈ R such that for small ε > 0 and for every x sufficiently large:

p n ∈ c 1 e x d + c 2 + ε, c 1 e x d + α dx 2 + c 2 -ε implies that λ n ∈ x, x + α x 2 .
By Hadamard -De la Vallée Poussin estimate, we also know that Π(u) := card{n :

p n ≤ u} = ˆu 2 dt log(t) + O ue -c √ log u . If p n e x d , then n e x d
x . Therefore

Proof of Theorem 1.3

As above the main difficulty is to estimate the square moments of D. The situation is not as clear as in the previous case since D now will be defined via an analytic continuation. We need to understand how to define this analytic continuation and how close it is to the partial sums of D.

Lemma 4.1. Let P ∈ R d [X] with lim +∞ P = +∞, let Q ∈ R d-1 [X] and let κ ∈ R. Then the Dirichlet series D(s) = n Q(n)(log n) κ (P (n)) -s admits a holomorphic continuation to C + 1-1 d ∪ C 1 and even to C 1-1 d \{1} provided κ ∈ N 0 . Moreover, let σ 1 > 1 -1 d and σ 2 > 1.
(a) There exist t 0 , B > 0 such that, for all s = σ + it with σ ≥ σ 1 and t ≥ t 0 ,

|D(s)| ≤ t B .
(b) There exist δ, ε > 0 such that, for all x > 0, for all s = σ + it with σ ∈ [σ 1 , σ 2 ] and

1 ≤ t ≤ δx, D(s) = x n=2 Q(n)(log n) κ (P (n)) -s + O(x -ε ) + O (log P (x)) κ (s -1)P (x) s-1
(here, the O-constants do not depend neither on s nor on x).

Proof. As in the classical case of the Riemann zeta function, see for example [START_REF] Bayart | Dynamics of linear operators[END_REF], our plan is to use the regularity and smoothness of the coefficients and the frequencies of our Dirichlet series D to estimate its order and how close the partial sums approximate D. We will rely again on the principle of non-stationary phase, that is Lemma 2.5. But first we need to deal with some technical difficulties that arise from the "unknown" polynomials P and Q. We start with s = σ + it, σ > 1 and let N ≥ 1. We write

D(s) = N -1 n=2 Q(n)(log n) κ (P (n)) -s + +∞ n=N Q(n)(log n) κ (P (n)) -s
and we apply Euler's summation formula (see [5, (11.3)]). Setting

φ(u) = Q(u)(log u) κ (P (u)) -s and ρ(u) = u -u - 1 2 , we get (6)
D(s) = N -1 n=2 Q(n)(log n) κ (P (n)) -s + ˆ+∞ N φ(u)du + ˆ+∞ N ρ(u)φ (u)du + 1 2 φ(N ).
These integrals are convergent when s ∈ C 1 . Moreover it is easy to check that there exists ε > 0 such that, provided

s = σ + it with σ ≥ σ 1 > 1 -1 d , for any u > 2, |φ(u)| u -ε and |φ (u)| |s|u -1-ε .
In particular, the last integral in (6) defines a holomorphic function in

C 1-1 d which is O(|s|N -ε ) in C σ1 .
Let us now see how to control the first integral. Up to multiply Q by some constant, we may write it

Q(u) = P (u) + Q 1 (u) with deg(Q 1 ) ≤ d -2. As above, the integral ´+∞ N Q 1 (u)(log u) κ (P (u)) -s du defines an analytic function in C 1-1 d which is O(N -ε ).
Therefore we have obtained so far that D may be written in C 1

D(s) = N -1 n=2 Q(n)(log n) κ (P (n)) -s + ˆ+∞ N P (u)(log u) κ (P (u)) s du + R N (s)
where

R N is analytic in C 1-1 d and |R N (s)| |s|N -ε uniformly for σ ≥ σ 1 .
We choose N sufficiently large such that P is one-to-one on [N, +∞). By change of variables we obtain:

ˆ+∞ N P (u)(log u) κ (P (u)) s du = ˆ+∞ P (N ) (log P -1 (u)) κ u s du.
By Lemma 3.4 we have the following formula:

P -1 (u) = a d u 1/d (1 + ε 1 (u)) with |ε 1 (u)| u -1/d ,
where a d > 0. Therefore,

(log P -1 (u)) κ = log κ (a d u 1/d ) + ε 2 (u) with |ε 2 (u)| u -1/d log κ-1 (u).
As before, the integral

´+∞ P (N ) ε 2 (u)u -s du defines an analytic function in C 1-1 d which is O(P (N ) -ε ) in C σ1 .
On the other hand, setting b d = a d d and restricting ourselves to s ∈ C 1 , we may write ˆ+∞

P (N ) log κ (a d u 1/d ) u s du = ˆ+∞ P (N ) 1 d κ log κ (b d u) u s du = b s-1 d d κ ˆ+∞ b d P (N ) log κ (v) v s dv (v = b d u) = b s-1 d d κ ˆ+∞ log(b d P (N ))
y κ e (1-s)y dy

(y = log v) = b s-1 d d κ (s -1) κ+1 Γ(κ + 1, (s -1) log(b d P (N ))).
Hence we have shown that for s ∈ C 1 , we may write

D(s) = N -1 n=2 Q(n)(log n) κ (P (n)) -s + R N (s)+ b s-1 d d κ (s -1) κ+1 Γ(κ + 1, (s -1) log(b d P (N )))
where R N (s) is holomorphic in C 1-1 d and is O(|s|N -ε )+O(P (N ) -ε ) in C σ1 . Since we know that Γ(κ+1, •) admits an analytic continuation to C\R -we can conclude to the analytic continuation of

D to C + 1-1 d ∪ C 1 . When κ ∈ N, the analytic continuation even holds on C 1-1 d \{1}.
The estimation (a) (which is trivial for σ ≥ σ 2 > 1) follows easily for σ ∈ [σ 1 , σ 2 ] by what we already know on R N and by [START_REF]Handbook of mathematical functions, with formulas, graphs, and mathematical tables[END_REF].

Let us turn to the proof of (b). Choosing N ≥ x, we may write

D(s) = x n=2 Q(n)(log n) κ (P (n)) -s + N n=x+1 Q(n)(log n) κ (P (n)) -s + b s-1 d d κ (s -1) κ+1 Γ(κ + 1, (s -1) log(b d P (N ))) + O(|s|N -ε ) + O(P (N ) -ε ).
We apply Lemma 2.5 to the second sum with

g(u) = Q(u) log κ (u)(P (u)) -σ , f (u) = -t log(P (u)) 2π .
Observe that, for σ ∈ [σ 1 , σ 2 ] and u ∈ [x, N ], provided t ≤ δx with δ small enough,

|g(u)| x -ε , |g (u)| σx -1-ε ≤ x -ε , |f (u)| ≤ 1 2 .
Hence,

D(s) = x n=2 Q(n)(log n) κ (P (n)) -s + ˆN x Q(u) log κ (u) (P (u)) s du+ b s-1 d d κ (s -1) κ+1 Γ(κ + 1, (s -1) log(b d P (N ))) + O(x -ε + |s|N -ε + P (N ) -ε ).
We let N → +∞, yielding to:

D(s) = x n=2 Q(n)(log n) κ (P (n)) -s + ˆ+∞ x Q(u) log κ (u) (P (u)) s du + O(x -ε ), s ∈ C 1 .
Now writing Q(u) = P (u) + Q 1 (u) and repeating the argument in the first part of this proof one can obtain the following identity T -ε .

D(s) = x n=2 Q(n)(log n) κ (P (n)) -s + b s-1 d d κ (s -1) κ+1 Γ(κ + 1, (s -1) log(b d P (x))) + R(s), where R(s) is holomorphic in C 1-1 d and is O(x -ε ) in C σ1 .
Hence, applying Lemma 2.4

ˆT T /2 |D(σ + it)| 2 dt ˆT T /2 T /δ n=2 |Q(n)(log n) κ (P (n)) -s 2 dt + T 1-2ε T T /δ n=2 |Q(n)| 2 (log n) 2κ |P (n)| -2σ + T /δ n=2 |Q(n)| 2 (log n) 2κ |P (n)| -2σ log(P (n + 1)) -log(P (n)) + T 1-2ε .
The first sum is dominated by some constant since σ ≥ σ 1 > σ 2 (D). Regarding the second sum, for n ∈ [2, T /δ],

|Q(n)| 2 (log n) 2κ |P (n)| -2σ log(P (n + 1)) -log(P (n)) T n 2(d-1)-2dσ (log n) 2κ T n 2d(1-σ1)-2 (log n) 2κ ,
and we get the estimate

T /δ n=2 |Q(n)| 2 (log n) 2κ |P (n)| -2σ log(P (n + 1)) -log(P (n)) T, since 2d(1 -σ 1 ) < 1. Hence, we have obtained ˆT T /2 |D(σ + it)| 2 dt T,
for all T ≥ 1 and all σ ∈ [σ 1 , σ 2 ], where the involved constant does not depend neither on σ nor on T. Taking T 2 -j instead of T in the latter formula and summing over j, we get the proposition.

The second half of the proof of Theorem 1.3 has been proven in [4, Proposition 6.2], for the sake of completeness we repeat the argument below.

Proposition 4.3. Let P ∈ R d [X] with lim +∞ P = +∞, let Q ∈ R d-1 [X] and let κ ∈ R. Then the Dirichlet series D(s) = n Q(n)(log n) κ (P (n)) -s belongs to D dens .
Proof. Let α, β > 0. There exists x 0 > 1 such that for every x ≥ x 0 the polynomial P is positive and increasing and Q behaves like its leading term. By Lemma 3.4 there exist constants

c 1 > 0, c 2 ∈ R such that n ∈ c 1 e x d -c 2 + ε, c 1 e x d + α dx 2 -c 2 -ε implies that λ n ∈ x, x + α x 2 . Thus λn∈[x,x+ α x 2 ] |Q(n)(log n) κ | e x d x 2 e d-1 d x(1-β 2 )
e (1-β)x .

Theorem 1.3 now follows from Proposition 4.2, Proposition 4.3 and Theorem 2.3.

Random models and further discussion

One of the motivations behind our work is to give concrete examples of convergent universal objects like the alternating prime zeta function P -(s) = n≥1 (-1) n p -s n . As we have already proved P -is strongly universal in { 2 3 < e < 1}. But, the universality of this object in the whole critical strip remains an open question. Question 5.1. Is P -strongly universal in { 1 2 < e ≤ 2 3 }? It is worth mentioning that Theorem 1.1 implies that every series of the form

P χ (s) = n≥1 χ n p -s n , |χ n | = 1,
with σ c (P χ ) ≤ 0, is strongly universal in { 2 3 < e < 1}. We expect such P χ to be strongly universal in the whole critical strip even when we only have σ c (P χ ) ≤ 1/2. To justify our claim let us randomize our series. Let {X n } be a sequence of unimodular independent identically distributed Steinhaus or Rademacher (coin tossing) random variables and P X (s) = n≥1 X n p -s n . Kolmogorov's three-series theorem [15, Chapter 5] implies that P X converges almost surely in C 1 2 . To obtain that such series are strongly universal almost surely, we need to obtain more information about their order in the critical strip. Proposition 5.2. Let P X (s) = n≥1 X n p -s n , where {X n } is as above. Then, P X is of sublogarithmic order in the critical strip and as a consequence is strongly universal, almost surely.

Proof. We consider the corresponding randomized zeta functions [START_REF] Carlson | Contributions à la théorie des séries de Dirichlet. III[END_REF] ζ

X (s) = n≥1 1 1 -X n p -s n .
It is easy to see that ζ X converges absolutely for es > 1 + ε, ε > 0. It is also known that ζ X and the reciprocal 1/ζ X converge in C 1 2 , almost surely. For Steinhaus random variables (X 1 , X 2 , . . . ) ∈ T × T × . . . this can be obtained from the work of Helson [START_REF] Helson | Compact groups and Dirichlet series[END_REF] or as an application of Menchoff's theorem [START_REF] Bayart | Hardy spaces of Dirichlet series and their composition operators[END_REF]. In the case of Rademacher random variables X n = r n (t) = sign(sin(2π2 n t)), 0 < t < 1 this has been done by Carlson and Wintner [7,[START_REF] Wintner | Random factorizations and Riemann's hypothesis[END_REF].

Starting from [START_REF] Carlson | Contributions à la théorie des séries de Dirichlet. III[END_REF] one can prove that there exists an absolutely convergent Dirichlet series F (s, X) in C 1 2 such that: log ζ X -P X = F, in C 1 2 . Using the Borel-Carathéodory theorem in a similar manner as in the proof of the implication the Riemann hypothesis implies the Lindelöf hypothesis, [18, Chapter XIV], we obtain that for all ε > 0

|P X (σ + it)| = O (log t) 2-2σ+ε , t → ∞,
uniformly for σ ≥ σ 0 > 1 2 . Fix such X 0 , the fact that P X0 ∈ D dens follows immediately from Proposition 3.5. It remains to show that P X0 ∈ D w.a. ( 12 ). We will work as in Lemma 3.1. Let T > 2, we set X = T ε , τ = δ and σ ≥ σ 0 > 1 2 + 2δ, where ε, δ > 0 are sufficiently small. We consider the contour C = ∪ 5 i=1 C i , where

  Funding. A. Kouroupis is partially supported by the Onassis Foundation -Scholarship ID: F ZT 037-1/2023-2024.

2 . Preliminaries 2 . 1 .

 221 Abscissas of convergence. To a Dirichlet series D = +∞ n=1 a n e -λns we will associate three abscissas, its abscissa of convergence, σ c (D) := inf e(s) : n a n e -λns converges , its abscissa of absolute convergence σ a (D) := inf σ ∈ R : n |a n |e -λnσ converges , and also σ 2 (D) := inf σ ∈ R : n |a n | 2 e -2λnσ converges .

Theorem 2 . 3 .

 23 Let D be a Dirichlet series and let σ 0 > σ 2 (D). Assume that D ∈ D w.a. (σ 0 ) ∩ D dens . Then D is strongly universal in the strip {σ 0 < e(s) < σ a (D)}

Lemma 3 . 1 .

 31 Let D = n≥1 a n e -λns be a Dirichlet series with σ a (D) ≤ 1. Assume that D extends continuously to C α , 0 ≤ α < 1, analytically in C α , and that D has order B in C α .

3 .

 3 Let D(s) = n≥1 a n e -λns be a Dirichlet series and let d ∈ N. Assume that • λ n = log P (p n ) where P ∈ R d [X] and lim +∞ P = +∞.

Proposition 3 . 5 .

 35 Let D(s) = n≥1 a n e -λns be a Dirichlet series and let d ∈ N. Assume that • λ n = log P (p n ) where P ∈ R d [X] and lim +∞ P = +∞.

λn∈[x,x+ α x 2 ] 1 . 5 .

 215 |a n | e x(d-1) d x d-1+κ card n : p n ∈ c 1 e x d + c 2 + ε, c 1 e It follows immediately from Theorem 2.3, Corollary 3.3 and Proposition 3.Remark. To prove Theorem 1.1, we use rather weak properties of prime numbers to prove Corollary 3.3. Hence we could easily replace (log(P (p n ))) by another sequence of frequencies satisfying some conditions in the statement of this corollary. Nevertheless, in Proposition 3.5, we use the much more profound Hadamard -De la Vallée Poussin estimate.

  Using one last time (1), we obtain (b) of Lemma 4.1.From this and Lemma 2.4, we may deduce the first half of Theorem 1.3.

Proposition 4 . 2 .

 42 Let P ∈ R d [X] with lim +∞ P = +∞, let Q ∈ R d-1 [X] and let κ ∈ R. Assume moreover that (log(P (n))) is Q-linearly independent. Then the Dirichlet series D(s) = n Q(n)(log n) κ (P (n)) -s belongs to D w.a. (σ 0 ) with σ 0 = 1 -1 2d . Proof.It is clear that σ 2 (D) = (2d -1)/2d and thus it just remains to prove (4). We fix T ≥ 1 and we first estimate ´T T /2 |D(σ + it)| 2 dt where 1 -1 2d < σ 1 ≤ σ ≤ σ 2 . We apply the estimate given by Lemma 4.1 with x = T /δ so that O(x -ε ) = O(T -ε ) and log κ (P (x)) (s -1)P (x) s-1 log κ T T T d(σ-1)

We observe that n X n p -s n e -pn X = 1 2πi

To prove that the integral over the line segment C 3 is bounded note that our function P X0 is of zero order uniformly in C 1 2 +δ , thus

Applying Cauchy's theorem and the Montgomery-Vaughan inequality as in Lemma 3.1, we obtain 1

Therefore [START_REF] Balazard | The mean square of the logarithm of the zeta-function[END_REF] ). Question 5.3. Is it true that if the series P X converges, then it will be strongly universal in { 1 2 < e < 1}?

In view of the proof of Proposition 5.2, this question is clearly linked to the order of the Dirichlet series in C 1/2 . Question 5.4. Let α > 0. What is the order of n≥1 (-1) n p -s n in C α ?