The chaos expansion of a random variable with uniform distribution is given. This decomposition is applied to analyze the behavior of each chaos component of the random variable logζ on the so-called critical line, where ζ is the Riemann zeta function. This analysis gives a better understanding of a famous theorem by Selberg.

Riemann zeta functionMalliavin calculusmultiple Wiener–Itô integralsSelberg theorem60F0560H051M06LabexANR-11-LABX-0007-01C. Tudor was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01) and MATHAMSUD project SARC (19-MATH-06). Introduction

Let H be a real and separable Hilbert space and let (W(h),h∈H) be an isonormal Gaussian process on a standard probability space Ω,F,P. It is well-known that any square integrable random variable, measurable with respect to the sigma-algebra generated by W can be decomposed in chaos, i.e. it can be written as an orthogonal sum F=∑n≥0Fn where for every n≥0, the random variable Fn is an element of the nth Wiener chaos. The knowledge of the concrete chaos expansion of F (i.e. the knowledge the exact form of Fn for every n≥0) gives an important information on the random variable F. If F is a random variable with a given common probability distribution, it is in general hard to get its exact chaos expansion, except in some particular case (Gaussian distribution, Gamma distribution, etc). In this work, our purpose is to find the chaos expansion of a random variable with uniform distribution and to apply this result to a well-known problem in number theory. We will consider the random variable F given by
F=e−12W(f)2+W(g)2
with f,g orthonormal elements of the Hilbert space H. Then F follows the uniform distribution over the unit interval [0,1]. By using the tools of Malliavin calculus, Stroock formula and the properties of the Hermite polynomials, we derive the chaos decomposition of the random variable defined by (1). Then we will apply our result to the Selberg theorem, which concerns the behavior of the Riemann zeta fumction on the critical line. Let us recall the context. The Riemann zeta function ζ is defined, for ℜs>11$]]>, by
ζ(s)=∑n=1∞1ns
and when ℜs≤1, the function ζ is defined as an analytic continuation of (2). The distribution of the zeta zeros is one of the outstanding problems in mathematics. It is known that ζ(−2n)=0 for every n≥1. The points −2n are called the trivial zeros of the Riemann zeta function. The most mysterious facts about the Riemann zeta function concerns the distribution of its nontrivial zeros. The Riemann hypothesis claims that all the nontrivial zeros of the Riemann zeta function lie on the critical lineℜs=12. Therefore, the behavior of the function ζ on the critical line and close to this critical line has been intensively studied.

A famous result by Atle Selberg says that, if T>00$]]> and t is a random variable uniformly distributed over the interval [T,2T], then the sequence
logζ12+it12loglogT→X1+iX2in distribution asT→∞
where X1+iX2 is a complex-valued standard normal random variable, i.e. X1,X2∼N(0,1) are independent random variables. There are several versions of this theorem. In particular, the result (3) holds if t∼U[0,T] or, more generally, if t∼U[aT,bT] with b>a≥0a\ge 0$]]>. We will work with t∼U[0,T] and we will assume throughout in the sequel t=TU with U∼U[0,1]. As in the literature (e.g., [7, 16]), we will still call Selberg’s theorem the result concerning the convergence of (3) with t=TU.

Recall that for any z∈C, logz=log|z|+iargz. Then the convergence (3) is actually equivalent to (4) and (5) below (see [11–13])
logζ12+it12loglogT→T→∞(d)X1∼N(0,1)
and
argζ12+it12loglogT→T→∞(d)X2∼N(0,1)
where ”→(d)” stands for the convergence in distribution.

The idea behind the proof of the limit theorems (4) and (5) is (see, among others, [11–15]) to approximate logζ12+it by the (renormalized) Dirichlet series
112loglogT∑p≤Tε1p12+it=112loglogT∑p≤Tεcos(tlogp)p+i∑p≤Tεsin(tlogp)p.
with t=TU, U∼U[0,1] and ε small enough. We will work throughout with ε=1. This approximation of logζ12+it by the Dirichlet series (6) is in L2 sense, since (see [13], see also [7] for a detailed proof)
Elogζ12+it−∑p≤T1p12+it2≤C
where C is a strictly positive constant not depending on T. This implies that the sequence
112loglogTlogζ12+it−∑p≤T1p12+itT>0converges to zero inL2(Ω)asT→∞.0}}\\ {} & & \displaystyle \hspace{2.5pt}\text{converges to zero in}\hspace{2.5pt}{L^{2}}(\Omega )\hspace{2.5pt}\text{as}\hspace{2.5pt}T\to \infty .\end{array}\]]]>

Our purpose is to bring a new contribution to understanding the limit theorems (4) and (5). More exactly, we will analyze the asymptotic behavior of each chaos component of 112loglogTlog|ζ12+it| with t∼U[0,T]. That is, if Jn(T) denotes the projection of 112loglogTlogζ12+it on the nth Wiener chaos, we want to study the limit behavior in distribution of Jn(T) as T→∞ for each fixed n≥0. This will allow to understand which chaos projection Jn(T) is dominant with respect to the others and determines the limit behavior of the logζ.

Many other related works in the old and recent literature treated the distribution of the zeros of the Riemann zeta function. Some related results, among many others, are [1–4, 6].

Our analysis will be based on the study of the Dirichlet approximation (6). Using the chaos expansion of the uniformly distributed random variable U, we will find the chaos expansion of the random variable cos(TUlogp) with U∼U[0,1] via Malliavin calculus and we will study the limit in distribution of each chaos component. We will see that every chaos converges to zero, but their sum tends in distribution to the Gaussian law. All the chaoses contribute to the limit and there is no term that is bigger than the others and gives the limit behavior.

Our work has the following structure. Section 2 contains some preliminaries on Wiener chaos and Malliavin calculus needed throughout the work. Section 3 is devoted to the chaos expansion of a uniformly distributed random variable U. In Section 4, we obtain the chaos decomposition of the Dirichlet series that approximates logζ12+iTU and we then analyze the asymptotic behavior, as T→∞, of each chaos component.

Preliminaries: the multiple stochastic integral and the Malliavin derivative

We also present the elements from the Malliavin calculus that will be used in the paper. We refer to [9] for a more complete exposition. Consider H as a real separable Hilbert space and (B(φ),φ∈H) an isonormal Gaussian process on a probability space (Ω,A,P), that is, a centered Gaussian family of random variables such that EB(φ)B(ψ)=⟨φ,ψ⟩H.

We denote by D the Malliavin derivative operator that acts on smooth functions of the form F=g(B(φ1),…,B(φn)) (g is a smooth function with compact support and φi∈H, i=1,…,n)
DF=∑i=1n∂g∂xi(B(φ1),…,B(φn))φi.

It can be checked that the operator D is closable from S (the space of smooth functionals as above) into L2(Ω;H) and it can be extended to the space D1,p which is the closure of S with respect to the norm
‖F‖1,pp=EFp+E‖DF‖Hp.

We denote by Dk,∞:=∩p≥Dk,p for every k≥1. In this paper, H will be the standard Hilbert space L2([0,T]).

We will make use of the chain rule for the Malliavin derivative (see Proposition 1.2.4 in [9]). That is, if φ:R→R is a continuously differentiable function and F∈D1,2, then φ(F)∈D1,2 and
Dφ(F)=φ′(F)DF.

Denote by In the multiple stochastic integral with respect to B (see [9]). This In is actually an isometry between the Hilbert space H⊙n (symmetric tensor product) equipped with the scaled norm n!‖·‖H⊗n and the Wiener chaos of order n which is defined as the closed linear span of the random variables Hn(B(φ)) where φ∈H, ‖φ‖H=1 and Hn is the Hermite polynomial of degree n≥1Hn(x)=(−1)nexpx22dndxnexp−x22,x∈R.
The isometry of multiple integrals can be written as: for m,n positive integers,
EIn(f)Im(g)=n!⟨f˜,g˜⟩H⊗nifm=n,EIn(f)Im(g)=0ifm≠n.
The Malliavin derivative D acts on the Wiener chaos as an annilihilation operator: if F=In(f) with symmetric f∈L2([0,T]n), then DtF=nIn−1(f(·,t)) where “·” stands for n−1 variables in [0,T].

Chaos expansion of uniformly distributed random variables

The uniformly distributed random variable U will be chosen of a particular form that allows to use the techniques of the Malliavin calculus. Actually, in the sequel we will assume
U=e−12(W(f)2+W(g)2)
with the following conditions fixed throughout our work: f,g∈H, ‖f‖=‖g‖=1 and ⟨f,g⟩=0 (all the scalar products and norms in the paper will be considered in H if no further specification is made). In (10), (W(h),h∈H=L2([0,1])) stands for a Gaussian isonormal process as described in Section 2. In particular, this implies that W(f) and W(g) are independent standard normal random variables. The fact that the random variable (10) is uniformly distributed over [0,1] follows from the simple computations: with F:R→R an arbitrary function such that EF(U)<∞,
EF(U)=12π∫R2Fe−x2+y22e−x2+y22dxdy=∫0∞dρFe−ρ22e−ρ22ρ=∫01F(u)du
where we used the change of variables with polar coordinates.

We analyze the chaos expansion of
U2k=e−k(W(f)2+W(g)2)
for every k>00$]]>. This will be done by using the techniques of the Malliavin calculus. We will also need some properties of the Hermite polynomials. Let Hn denote the nth Hermite polynomial, see (8). Recall that, if Y∼N(0,σ2), then
EH2m(Y)=(σ2−1)m(2m)!2mm!
and EHn(Y)=0 if n is odd.

We will use the following two auxiliary lemmas that concern the Hermite polynomials.

For everys≥0and for everyk>00$]]>,Ee−kW(f)2H2s(2kW(f))=(−1)s(2s)!2ss!(2k+1)−s−12.

With the notation σ2=2k2k+1, we have
Ee−kW(f)2H2s(2kW(f))=12π∫Re−x2(2k+1)2H2s(2kx)dx=12k+112πσ2∫Re−x22σ2H2s(x)dx=12k+1EH2s(Y)
where Y∼N(0,σ2). The conclusion comes from (12). □

Recall that D(s)(s≥1) denotes the sth iterated Malliavin derivative. If s=0 then by convention f⊗s=1.

For anyk>00$]]>, consider the random variableF=e−kW(f)2. Then for everys≥0,D(s)F=(2k)s2(−1)se−kW(f)2Hs2kW(f)f⊗s.Moreover, we haveED(s)F=0ifsis oddandED(2s)F=(−1)s(2s)!kss!(2k+1)−s−12f⊗2s,for everys≥0.

Denote by g(x)=e−x2 for x∈R. By the chain rule of the Malliavin derivative (7), we have, for every s≥0,
D(s)F=g(s)kW(f)ks2f⊗s.
We will use the following relation between the derivatives of the function g and the Hermite polynomials:
g(s)(x)=(−1)se−x22s2Hs(2x),for everys≥0,x∈R.
It is easy to see that if s is odd, then the of (14) expectation vanishes. If s is even, by plugging (18) into (17), we obtain (14). Consequently,
ED(s)F=(2k)s2s!f⊗sEe−kW(f)2Hs2kW(f)
and by using (13), we obtain (15) and (16). □

The next step is to get the chaos expansion of U2k with U a random variable uniformly distributed over [0,1]. If f,g are two functions, by f⊗˜g we denote the symmetrization of their tensor product.

Let U be given by (10). Then for everyk>00$]]>the random variableU2kadmits the following Wiener chaos expansion:U2k=∑n≥0kn(2k+1)n+1I2n(h2n),withh2n=(−1)n∑s=0n1s!(n−s)!f⊗2s⊗˜g⊗(2n−2s).

We will first show that for every k>00$]]>ED(n)U2k=0ifnis odd
and for every n≥1ED(2n)U2k=kn(2k+1)n+1(2n)!h2n
with hn given by (20). By using the Leibniz rule for the Malliavin derivarive (see, e.g., [9], Exercise 1.2. 13), we can write, for n,k≥1,
D(n)U2k=D(n)(e−kW(f)2e−kW(g)2)=∑s=0nCnsD(s)e−kW(f)2⊗˜D(n−s)e−kW(g)2,
so
ED(n)U2k=∑s=0nCnsED(s)e−kW(f)2⊗˜ED(n−s)e−kW(g)2
where we used the independence of W(f) and W(g). By Lemma 2, we immediately get that ED(n)U2k=0 if n is odd and
ED(2n)U2k=(−1)nkn(2k+1)n+1∑s=0nC2n2s(2s)!(2n−2s)!s!(n−s)!f⊗2s⊗˜g⊗(2n−2s)=(−1)nkn(2k+1)n+1∑s=0n(2n)!s!(n−s)!f⊗2s⊗˜g⊗(2n−2s)=kn(2k+1)n+1(2n)!h2n
with h2n given by (20).

In order to obtain the Wiener chaos decomposition of U2k, we will use the Stroock formula (see, e.g., [9]) to write
U2k=∑n≥01n!InED(n)U2k,
and using the formulas (21) and (22) for ED(n)Uk with n≥1 and k≥1, we get (19). □

It is worth to point out that the (trivial) formulas EU2k=12k+1 and EU4k=14k+1 can also be checked through the chaos expansion (26). Indeed, for n=0 in the right-hand side of (26) we get EU2k=12k+1 and, since ‖h2n‖2=∑s=0n(2s)!(2n−2s)!(2n)!1(s!(n−s)!)2 we get, via the isometry (9)
EU4k=E(U2k)2=1(2k+1)2∑n=0∞k2n(2k+1)2n(2n)!‖h2n‖2=1(2k+1)2∑n=0∞k2n(2k+1)2n∑s=0n(2s)!(2n−2s)!(s!(n−s)!)2=1(2k+1)2∑s=0∞(2s)!s!2∑n=s∞k2n(2k+1)2n(2n−2s)!(n−s)!)2=1(2k+1)2∑n=0∞k2n(2k+1)2n(2n)!n!22=14k+1,
since, with Z∼N(0,1),
∑n=0∞k2n(2k+1)2n(2n)!n!2=∑n=0∞k2n(2k+1)2n2nEZ2n=E∑n=0∞k2n(2k+1)2n(2n)!n!2=∑n=0∞k2n(2k+1)2n2nZ2n=Ee2k2(2k+1)2Z2=2k+11+4k.

Chaos analysis in Selberg’ s theorem

We use the results in the previous section in order to get the chaos expansion of the Dirichlet series (6) and to study the behavior, as T→∞, of each chaos component of logζ12+iTU with U given by (10).

Chaos expansion of the Dirichlet series

We first obtain the Wiener–Itô chaos expansion of the Dirichlet series (6) that approximates logζ on the critical line (in the sense of (35)). We will actually focus on the analysis of the real part of (6) but we stress that a similar study can be done for the imaginary part. Consider the family (XT)T>00}}$]]> given by
XT=∑p≤Tcos(TUlogp)p−Ecos(TUlogp)p,
where the sum is taken over the primes p and U is U[0,1] distributed, of the form (10).

ForT>00$]]>, letXTbe given by (25). DenoteXT:=112loglogTXT.ThenXT=∑n≥1c2n(T)I2n(h2n)where for everyn≥1,hnare defined by (20) andc2n(T)=112loglogT∑p≤T1p∑k≥0(−1)k(2k+1)!(Tlogp)2kk2k+1n.

From (19) and the series expansion of the cosinus function cosx=∑k≥0(−1)k(2k)!x2k, we obtain the chaos decomposition of the random variable cos(TlogpU) where T>00$]]>, p is a prime number and U is defined in (10). We actually have
cos(TUlogp)=∑k≥0(−1)k(2k+1)!(Tlogp)2k∑n≥0I2n(h2n)k2k+1n.
Thus 112loglogT∑p≤T1pcos(TlogpU)=∑n≥0J2n(T) with
J2n(T)=I2nhh∑p≤T1p∑k≥0(−1)k(2k+1)!(Tlogp)2kk2k+1n
and hn given by (20). Note that the chaos of order zero can be explicitly computed. In fact,
J0(T)=112loglogT∑p≤T1p∑k≥0(−1)k(2k+1)!(Tlogp)2k=112loglogT∑p≤T1p1Tlogpsin(Tlogp)=112loglogTEXT
and therefore
XT=112loglogTXT=∑n=1∞J2n(T)=∑n=1∞c2n(T)I2n(h2n).
□

Note that only Wiener chaoses of even order appear in the decomposition of XT.

Asymptotic behavior for each chaos

We now analyze the limit behavior of the projection of ℜlogζ12+iTU on each Wiener chaos. The first step is to do this study for the Dirichlet series (6).

Chaoses of the Dirichlet series

From Proposition 1, we notice that the (renormalized) Dirichlet series (25) can be expanded into an infinite sum of random variables in chaoses of even orders. The projection on the 2nth Wiener chaos is given by
J2n(T)=c2n(T)I2n(h2n)
with c2n(T), h2n from (27), (20) respectively. We will show that for every n≥1,
J2n(T)→T→∞0almost surely and inL2(Ω),
i.e. the projection of each Wiener chaos of XT converges to zero when T→∞. Note that in (29) only the coefficients c2n(T) depend on T and but the random parts I2n(h2n) do not. Therefore it suffices to study the behavior of c2n(T) as T→∞. This is done in the lemma below.

LetT>00$]]>,n≥1and letc2n(T)be defined by (27). Then for everyn≥1,c2n(T)→T→∞0.

Let us first give the proof for n=1. This will illustrate what happens in the general case. For n=1, we have, writting k2k+1=121−12k+1,
c2(T)=112loglogT∑p≤T1p∑k≥0(−1)k(2k+1)!(Tlogp)2kk2k+1=12112loglogT∑p≤T1p∑k≥0(−1)k(2k+1)!(Tlogp)2k−12112loglogT∑p≤T1p∑k≥0(−1)k(2k+1)!(Tlogp)2k12k+1.
The two sums above can be calculated. Since
∑k≥0(−1)k(2k+1)!(Tlogp)2k=1Tlogpsin(Tlogp)
and
∑k≥0(−1)k(2k+1)!(Tlogp)2k12k+1=∑k≥0(−1)k(2k+1)!(Tlogp)2k1(Tlogp)2k+1∫0Tlogpy2kdy=1Tlogp∫0Tlogpsinyydy,
we will have
c2(T)=12112loglogT∑p≤T1p1Tlogpsin(Tlogp)−∫0Tlogpsinyydy.
By (34) and the trivial inequality
∫0tsinyydy−π2≤Ct
we clearly get c2(T)≤CT−12→T→∞0. Concerning the general case, we use again the identity k2k+1=121−12k+1 and the Newton’s formula to get c2n(T)=112loglogT∑p≤T1p∑k≥0(−1)k(2k+1)!(Tlogp)2kk2k+1n=112loglogT∑p≤T1p∑k≥0(−1)k(2k+1)!(Tlogp)2k12−12(2k+1)n=(−1)n2n112loglogT∑p≤T1p∑k≥0(−1)k(2k+1)!(Tlogp)2k∑a=0nCna(2k+1)−a(−1)n−a:=(−1)n2n∑a=0nCna(−1)n−aAT(a) where, for every a=0,1,..,n, we used the notation
AT(a):=112loglogT∑p≤T1p∑k≥0(−1)k(2k+1)!(Tlogp)2k(2k+1)−a.
The only element depending on T in the decomposition of c2n(T) is the one denoted by AT(a). We will show that for every a=0,1,…,n,
AT(a)→T→∞0.
We already proved the results for a=0,1, so we assume a≥2. We write
AT(a)=112loglogT∑p≤T1p∑k≥0(−1)k(2k+1)!(Tlogp)2k(Tlogp)−a(2k+1)∫0Tlogpy2ka=112loglogT∑p≤T1p∑k≥0(−1)k(2k+1)!(Tlogp)2k−2ka−a∫0Tlogp...∫0Tlogpdy1…dya(y1y2...ya)2k=112loglogT∑p≤T1p∫0Tlogp...∫0Tlogpdy1…ya1Tlogp1y1..ya∑k≥0(−1)k(2k+1)!y1..ya(Tlogp)a−12k+1=112loglogT∑p≤T1p∫0Tlogp...∫0Tlogpdy1…ya1Tlogp1y1..yasin(y1..ya(Tlogp)1−a).
By the change of variables y˜i=yiTlogp for i=1,..,a, we obtain
AT(a)=112loglogT∑p≤T1p∫01...∫01dy1…ya1Tlogp1y1..ya×sin(y1..ya(Tlogp)).
By choosing δ∈(0,1) small enough and by writting
sin(y1..ya(Tlogp))y1..ya(Tlogp)=sin(y1..ya(Tlogp))y1..ya(Tlogp)δsin(y1..ya(Tlogp))y1..ya(Tlogp)1−δ,
so that
sin(y1..ya(Tlogp))y1..ya(Tlogp)≤1y1..ya(Tlogp)1−δ,
we can bound AT(a) as follows:
AT(a)≤C112loglogTTδ−1∑p≤T1p(logp)δ−1≤C112loglogTTδ−1∑p≤T1p≤C112loglogTTδ−1/2logT
and this converges to zero as T→∞ if δ<12. In the last inequality we used the following estimate (see, e.g., [16, 15]): for every s∈C with ℜs<1 we have
∑p≤xp−s∼x1−s(1−s)logx.
From (31) and (33) we deduce that for every n≥1 and for T large enough, |c2n(T)|≤C112loglogTTδ−1/2logT with δ∈(0,12) and the conclusion follows. □

On the chaos decomposition of <inline-formula id="j_vmsta172_ineq_186"><alternatives>
<mml:math><mml:mo movablelimits="false">log</mml:mo><mml:mi mathvariant="italic">ζ</mml:mi></mml:math>
<tex-math><![CDATA[$\log \zeta $]]></tex-math></alternatives></inline-formula> on the critical line

Let us finish by some remarks concerning the asymptotic behavior of the chaos projection of logζ12+iTU. This random variable is obviously square integrable. It is close to the Dirichlet series (6) in the sense that (see [13, 7])
Elog|ζ12+iTU|−∑p≤Tcos(TlogpU)p2≤C,
so
E1loglogTlog|ζ12+iTU|−1loglogT∑p≤Tcos(TlogpU)p2≤C1loglogT→T→∞0.

From the results in the previous paragraph, we can deduce the asymptotic behavior of the chaoses that compose logζ on the real line. We have the following result.

Let U be given by (10). Assume that for everyT>00$]]>the square integrable random variable112loglogTlogζ12+iTUadmits the chaos decomposition112loglogTℜlogζ12+iTU=∑n≥0Kn(T).Then for everyn≥1,Kn(T)→T→∞0inL2(Ω).

Since 112loglogT∑p1pcos(TlogpU)=∑n≥0J2n(T) with Jn from (28), we can write, by using (35),
E112loglogTℜlogζ12+iTU−112loglogT∑p≤T1pcos(TlogpU)2=∑n≥0EK2n(T)−J2n(T)2+∑n≥0E|K2n+1(T)|2≤C112loglogT.
This clearly implies that for every n≥1,
Kn(T)→T→∞0inL2(Ω).
□

Let us make some comments on the result stated in Proposition 2. We denoted by Kn(T) the nth chaos component of the random variable FT:=112loglogT×logζ12+iTU, i.e for every T>00$]]> we have FT=∑n≥0Kn(T). We showed that for every n≥0, Kn(T)→T→∞0 almost surely and in L2(Ω). Let us discuss the meaning of this result. Suppose that we have a random sequence (XT)T>00}}$]]> which converges in distribution, as T→∞, to the standard normal distribution. Assume that for each T>00$]]> the random variable XT admits a chaos decomposition
XT=∑n≥0In(fn(T)).
In many situations, for such a limit theorem, there exists a dominant chaos for XT (see, e.g., Theorem 1.7 in [10] for an example on the average of solutions to some stochastic partial differential equations, or [8] for an example related to stochastic geometry). That is, there exists N0≥1 such that IN0(fN0(T)) converges in law, as T→∞, to N(0,1) while the other chaos components are negligible, i.e. EIn(fn(T))2→T→∞0 for n≠N0. This situation would be very convenient because, in order to understand the behavior of XT, it suffices to look at the convergence of the chaos of order N0.

In other situations, each chaos converges to a Gaussian limit, i.e. In(fn(T))→T→∞N(0,σn2) in law for every n≥0 and ∑n≥0σn2=1. We refer to, e.g., the paper [5] for such a situation.

We actually showed that we are not in the situations described above. We showed that for every n≥0, the behavior of Kn(T) is very close to the behavior of the nth chaos component of the Dirichlet series (6) (denoted by Jn(T) in (29)) and we proved that Jn(T)2→T→∞0 almost surely and in L2. This follows from the fact that Jn(T)=0 if n is odd and J2n(T)=c2n(T)I2n(h2n) with cn(T) a deterministic sequences converging to zero as T→∞. This suggests the complexity of the Selberg theorem since the contribution to the limit comes from every chaos.

Acknowledgments

We will like to thank the referee for having carefully read our work.

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