Hochschild homology, trace map and ζ -cycles

. In this paper we consider two spectral realizations of the zeros of the Riemann zeta function. The ﬁrst one involves all non-trivial ( i


Introduction
In this paper we give a Hochschild homological interpretation of the zeros of the Riemann zeta function. The root of this result is in the recognition that the map pEf qpuq " u 1{2 ÿ ną0 f pnuq which is defined on a suitable subspace of the linear space of complex-valued even Schwartz functions on the real line, is a trace in Hochschild homology, if one brings in the construction the projection π : A Q Ñ QˆzA Q from the rational adèles to the adèle classes (see Section 3). In this paper, we shall consider two spectral realizations of the zeros of the Riemann zeta function. The first one involves all non-trivial (i.e. non-real) zeros and is expressed in terms of a Laplacian intimately related to the prolate wave operator (see Section 4). The second spectral realization is sharper inasmuch as it affects only the critical zeros. The main players are here the ζ-cycles introduced in [7], and the Scaling Site [6] as their parameter space, which encodes their stability by coverings. The ζ-cycles give the theoretical geometric explanation for the striking coincidence between the low lying spectrum of a perturbed spectral triple therein introduced (see [7]), and the low lying (critical) zeros of the Riemann zeta function. The definition of a ζ-cycle derives, as a byproduct, from scale-invariant Riemann sums for complex-valued functions on the real half-line r0, 8q with vanishing integral. For any µ P R ą1 , one implements the linear (composite) map Σ µ E : S ev 0 Ñ L 2 pC µ q from the Schwartz space S ev 0 of real valued even functions f on the real line, with f p0q " 0, and vanishing integral, to the Hilbert space L 2 pC µ q of square integrable functions on the circle C µ " R˚{µ Z of length L " log µ, where pΣ µ gqpuq :" ÿ kPZ gpµ k uq.
The map Σ µ commutes with the scaling action R˚Q λ Þ Ñ f pλ´1xq on functions, while E is invariant under a normalized scaling action on S ev 0 . In this set-up one has Definition. A ζ-cycle is a circle C of length L " log µ whose Hilbert space L 2 pCq contains Σ µ EpS ev 0 q as a non dense subspace. Next result is known (see [7] Theorem 6.4) Theorem 1.1. The following facts hold (i) The spectrum of the scaling action of R˚on the orthogonal space to Σ µ EpS ev 0 q in L 2 pC µ q is contained in the set of the imaginary parts of the zeros of the Riemann zeta function ζpzq on the critical line ℜpzq " 1 2 . (ii) Let s ą 0 be a real number such that ζp 1 2`i sq " 0. Then any circle C whose length is an integral multiple of 2π s is a ζ-cycle, and the spectrum of the action of R˚on pΣ µ EpS ev 0 qq K contains s. Theorem 1.1 states that for a countable and dense set of values of L P R ą0 , the Hilbert spaces HpLq :" pΣ µ EpS ev 0 qq K are non-trivial and, more importantly, that as L varies in that set, the spectrum of the scaling action of R˚on the family of the HpLq's is the set Z of imaginary parts of critical zeros of the Riemann zeta function. In fact, in view of the proven stability of ζ-cycles under coverings, the same element of Z occurs infinitely many times in the family of the HpLq's. This stability under coverings displays the Scaling Site S " r0, 8q¸Nˆas the natural parameter space for the ζ-cycles. In this paper, we show (see Section 5) that after organizing the family HpLq as a sheaf over S and using sheaf cohomology, one obtains a spectral realization of critical zeros of the Riemann zeta function. The key operation in the construction of the relevant arithmetic sheaf is given by the action of the multiplicative monoid Nˆon the sheaf of smooth sections of the bundle L 2 determined by the family of Hilbert spaces L 2 pC µ q, µ " exp L, as L varies in p0, 8q. For each n P Nˆthere is a canonical covering map C µ n Ñ C µ , where the action of n corresponds to the operation of sum on the preimage of a point in C µ under the covering. This action turns the (sub)sheaf of smooth sections vanishing at L " 0 into a sheaf L 2 over S . The family of subspaces Σ µ EpS ev 0 q Ă L 2 pC µ q generates a closed subsheaf ΣE Ă L 2 and one then considers the cohomology of the related quotient sheaf L 2 {ΣE. In view of the property of R˚-equivariance under scaling, this construction determines a spectral realization of critical zeros of the Riemann zeta function, also taking care of eventual multiplicities. Our main result is the following Theorem 1.2. The cohomology H 0 pS , L 2 {ΣEq endowed with the induced canonical action of R˚is isomorphic to the spectral realization of critical zeros of the Riemann zeta function, given by the action of R˚, via multiplication with λ is , on the quotient of the Schwartz space SpRq by the closure of the ideal generated by multiples of ζ`1 2`i s˘.
This paper is organized as follows. Section 2 recalls the main role played by the (image of the) map E in the study of the spectral realization of the critical zeros of the Riemann zeta function. In Section 3 we show the identification of the Hochschild homology HH 0 of the noncommutative space QˆzA Q with the coinvariants for the action of Qˆon the Schwartz algebra, using the (so-called) "wrong way" functoriality map π ! associated to the projection π : A Q Ñ QˆzA Q . We also stress the relevant fact that the Fourier transform on adèles becomes canonical after passing to HH 0 of the adèle class space of the rationals. The key Proposition 3.3 describes the invariant part of such HH 0 as the space of even Schwartz functions on the real line and identifies the trace map with the map E. Section 4 takes care of the two vanishing conditions implemented in the definition of E and introduces the operator ∆ " Hp1`Hq (H being the generator of the scaling action of Ro n SpRq ev ) playing the role of the Laplacian and intimately related to the prolate operator. Finally, Section 5 is the main technical section of this paper since it contains the proof of Theorem 1.2.

The map E and the zeros of the zeta function
The adèle class space of the rationals QˆzA Q is the natural geometric framework to understand the Riemann-Weil explicit formulas for L-functions as a trace formula [3]. The essence of this result lies mainly in the delicate computation of the principal values involved in the distributions appearing in the geometric (right-hand) side of the semi-local trace formula of op.cit. (see Theorem 4 for the notations) (later recast in the softer context of [13]). There is a rather simple analogy related to the spectral (left-hand) side of the explicit formulas for a global field K (see [4] Section 2 for the notations) which may help one to realize how the sum over the zeros of the zeta function appears. Here this relation is simply explained. Given a complex valued polynomial P pxq P Crxs, one may identify the set of its zeros as the spectrum of the endomorphism T of multiplication by the variable x computed in the quotient algebra Crxs{pP pxqq. It is well known that the matrix of T , in the basis of powers of x, is the companion matrix of P pxq. Furthermore, the trace of its powers, readily computed from the diagonal terms of powers of the companion matrix in terms of the coefficients of P pxq, gives the Newton-Girard formulae. 1 If one transposes this result to the case of the Riemann zeta function ζpsq, one sees that the multiplication 1 This is an efficient way to find the power sum of roots of P pxq without actually finding the roots explicitly. Newton's identities supply the calculation via a recurrence relation with known coefficients.
by P pxq is replaced here with the map while the role of T (the multiplication by the variable) is played by the scaling operator uB u . These statements may become more evident if one brings in the Fourier transform. Indeed, let f P SpRq ev be an even Schwartz function and let wpf qpuq " u 1{2 f puq be the unitary identification of f with a function in L 2 pR˚, d˚uq, where d˚u :" du{u denotes the Haar measure. Then, by composing w with the (multiplicative) Fourier transform F : The function ψpzq is holomorphic in the complex half-plane H " tz P C | ℑpzq ą 1 2 u since f puq " Opu´N q for u Ñ 8. Moreover, for n P N, one has ż In the region ℑpzq ą 1 2 one derives, by applying Fubini theorem, the following equality ż Thus, for all z P C with ℑpzq ą 1 2 one obtains zqψpzq.
If one assumes now that the Schwartz function f fulfills ş R f pxqdx " 0, then ψp i 2 q " 0. Both sides of (2.2) are holomorphic functions in H: for the integral on the left-hand side, this can be seen by using the estimate Epf qpuq " Opu 1{2 q that follows from the Poisson formula. This proves that (2.2) continues to hold also in the complex half-plane H. Thus one sees that the zeros of ζp 1 2´i zq in the strip |ℑpzq| ă 1 2 are the common zeros of all functions FpEpf qqpzq, One may eventually select the even Schwartz function f pxq " e´π x 2`2 πx 2´1˘t o produce a specific instance where the zeros of FpEpf qq are exactly the non-trivial zeros of ζp 1 2´i zq, since in this case ψpzq " 1 4 π´1 4`i z 2 p´1´2izqΓ`1 4´i z 2˘.

Geometric interpretation
In this section we continue the study of the map E with the goal to achieve a geometric understanding of it. This is obtained by bringing in the construction the adèle class space of the rationals, whose role is that to grant for the replacement, in (2.1), of the summation over the monoid Nˆwith the summation over the group Qˆ. Then, up to the factor u 1{2 , E is understood as the composite ι˚˝π ! , where the map ι : QˆzAQ{ẐˆÑ QˆzA Q {Ẑˆis the inclusion of idèle classes in adèle classes and π : A Q {ẐˆÑ QˆzA Q {Ẑˆis induced by the projection A Q Ñ QˆzA Q . We shall discuss the following diagram The conceptual understanding of the map π ! uses Hochschild homology of noncommutative algebras. We recall that the space of adèle classes i.e. the quotient QˆzA Q is encoded algebraically by the cross-product algebra The Schwartz space SpA Q q is acted upon by (automorphisms of) Qˆcorresponding to the scaling action of Qˆon rational adèles. An element of A is written symbolically as a finite sum ÿ apqqU pqq, apqq P SpA Q q.
From the inclusion of algebras SpA Q q Ă SpA Q q¸Qˆ" A one derives a corresponding morphism of Hochschild homologies π ! : HHpSpA Q qq ÝÑ HHpAq.
Here, we use the shorthand notation HHpAq :" HHpA, Aq for the Hochschild homology of an algebra A with coefficients in the bimodule A. In noncommutative geometry, the vector space of differential forms of degree k is replaced by the Hochschild homology HH k pAq. If the algebra A is commutative and for k " 0, HH 0 pAq " A, so that 0-forms are identified with functions. Indeed, the Hochschild boundary map is identically zero when the algebra A is commutative. This result does not hold when A " A, since A " SpA Q q¸Qˆis no longer commutative. It is therefore meaningful to bring in the following Proposition 3.1. The kernel of π ! : HH 0 pSpA Q qq Ñ HH 0 pAq is the C-linear span E of functions f´f q , with f P SpA Q q, q P Qˆ, and where we set f q pxq :" f pqxq.
Proof. For any f, g P SpA Q q and q P Qˆone has x :" f U pq´1q, y :" U pqqg.
One knows ( [14] Lemma 1) that any function f P SpRq is a product of two elements of SpRq. Moreover, an element of the Bruhat-Schwartz space SpA Q q is a finite linear combination of functions of the form e b f , with e 2 " e. Thus any f P SpA Q q can be written as a finite sum of products of two elements of SpA Q q, so that (3.1) entails f´f q P ker π ! . Conversely, let f P ker π ! . Then there exists a finite number of pairs P´ÿ apqqU pqq¯:" ap1q.
We shall prove that for any pair x, y P A one has P prx, ysq P E. Indeed, one has This projection belongs to E in view of the fact that This completes the proof.
Proposition 3.1 shows that the image of π ! : HH 0 pSpA Q qq Ñ HH 0 pAq is the space of coinvariants for the action of Qˆon SpA Q q, i.e. the quotient of SpA Q q by the subspace E. An important point now to remember is that the Fourier transform becomes canonically defined on the above quotient. Indeed, the definition of the Fourier transform on adèles depends on the choice of a non-trivial character α on the additive, locally compact group A Q , which is trivial on the subgroup Q Ă A Q . It is defined as follows The space of characters of the compact group G " A Q {Q is one dimensional as a Qvector space, thus any non-trivial character α as above is of the form βpxq " αpqxq, so that Therefore, the difference F β´Fα vanishes on the quotient of SpA Q q by E and this latter space is preserved by F α since F α pf q q " F α pf q q´1 .
3.1. HH, Morita invariance and the trace map. Let us recall that given an algebra A, the trace map induces an isomorphism in degree zero Hochschild homology which extends to higher degrees. If A is a convolution algebra of theétale groupoid of an equivalence relation R with countable orbits on a space Y , and π : Y Ñ Y {R is the quotient map, the trace map takes the following form The trace induces a map on HH 0 of the function algebras, provided one takes care of the convergence issue when the size of equivalence classes is infinite. If the relation R is associated with the orbits of the free action of a discrete group Γ on a locally compact space Y , the convolution algebra is the cross product of the algebra of functions on Y by the discrete group Γ. In this case, theétale groupoid is Y¸Γ, where the source and range maps are given resp. by spy, gq " y and rpy, gq " gy. The elements of the convolution algebra are functions f py, gq on Y¸Γ. The diagonal terms in (3.2) correspond to the elements of Y¸Γ such that spy, gq " rpy, gq, meaning that g " 1 is the neutral element of Γ, since the action of Γ is assumed to be free. Then, the trace map is This sum is meaningful on the space of the proper orbits of Γ. For a lift ρpxq P Y , with πpρpxqq " x the trace reads as In the case of Y " A Q acted upon by Γ " Qˆ, the proper orbits are parameterized by the idèle classes and this space embeds in the adèle classes by means of the inclusion ι : QˆzAQ Ñ QˆzA Q .
We identify the idèle class group C Q " QˆzAQ withẐˆˆR˚, using the canonical exact sequence affected by the modulus There is a natural section ρ : C Q Ñ AQ of the quotient map, given by the canonical inclusionẐˆˆR˚Ă A f QˆR " A Q . Next, we focus on theẐˆ-invariant part of SpA Q q. Then, with the notations of Proposition 3.1 we have Lemma 3.2. The following facts hold Trpf qpuq " 2 ÿ nPNˆf pnuq @u P R˚.
Proof. (i) By definition, the elements of the Bruhat-Schwartz space SpA Q q are finite linear combinations of functions on A Q of the form (S Q 8 is a finite set of places) where SpQ p q denotes the space of locally constant functions with compact support. An element of SpQ p q which is Zp -invariant is a finite linear combination of characteristic functions p1 Zp q p n pxq :" 1 Zp pp n xq. Thus an element h P SpA Q qẐˆis a finite linear combination of functions of the form With q " ś p´n p one has with ℓpxq :" f pqxq, ℓ " 1Ẑ b g, ℓ´f P EẐˆand the replacement of g with its even part 1 2 pgpxq`gp´xqq does not change the class of f modulo EẐˆ.
By Proposition 3.1 the Hochschild class in HH 0 pAq off is zero, thus Trpf q " 0. It follows from (3.4) that Epf qpuq " 0 @u P R˚. Then (2.2) implies that the function ψpzq " ş R˚f puqu 1 2´i z d˚u is well defined in the half-plane ℑpzq ą 1 2 where it vanishes identically, thus f " 0. The converse of the statement is obvious.
The next statement complements Proposition 3.1, with a description of the range of π ! : HH 0 pSpA Q qẐˆq Ñ HH 0 pAqẐˆ; it also shows that the map Epf qpuq " u 1{2 ř 8 n"1 f pnuq coincides, up to the factor u 1{2 2 , with the trace map (3.5). We keep the notations of Lemma 3.2 isomorphism, this means that π !´H H 0 pSpA Q qẐˆq¯is determined by the images of the elements of the subalgebra 1Ẑ b SpRq ev Ă SpA Q qẐˆ. Furthermore, one has the identity Proof. The first statement follows from Lemma 3.2 (i) and (iii). The second statement from (ii) of the same lemma.

The Laplacian ∆ " Hp1`Hq
This section describes the spectral interpretation of the squares of non-trivial zeros of the Riemann zeta function in terms of a suitable Laplacian. It also shows the relation between this Laplacian and the prolate wave operator.
4.1. The vanishing conditions. One starts with the exact sequence By implementing in the above sequence the evaluation δ 0 pf q :" f p0q, one obtains the exact sequence The next lemma shows that both SpA Q q 0 and SpA Q q 1 have a description in terms of the ranges of two related differential operators. For simplicity of exposition, we restrict our discussion to theẐˆ-invariant parts of these function spaces. Proof. (i) follows since GL 1 pA Q q is abelian, thus H commutes with the action of GL 1 pA Q q. Similarly Hf`f " 0 implies that xf pxq is constant and hence f " 0 for f P SpRq. Thus Hp1`Hq : SpRq Ñ SpRq 0 is injective. Let now f P SpRq ev with f p0q " 0. Then the function gpxq :" f pxq{x, gp0q :" 0, is smooth, g P SpRq odd and there exists a unique h P SpRq ev such that B x h " g. One has Hh " f so that p´1´Hq p h " p f . Thus if p f p0q " 0 one has p hp0q " 0 and there exists k P SpRq ev with Hk " p h. Then´p1`Hq p k " h and Hp1`Hq p k "´f . This shows that Hp1`Hq : SpRq ev Ñ SpRq ev 0 is surjective and an isomorphism.

The
Laplacian ∆ " Hp1`Hq and its spectrum. This section is based on the following heuristic dictionary suggesting a parallel between some classical notions in Hodge theory on the left-hand side, and their counterparts in noncommutative geometry, for the adèle class space of the rationals. The notations are inclusive of those of Section 3 Algebra of functions Cross-product by QD ifferential forms Hochschild homology Star operator ‹ ιˆF Differential d Operator H δ :" ‹d‹ Operator 1`H Laplacian ∆ :" Hp1`Hq Next Proposition is a variant of the spectral realization in [8,9].
Proposition 4.2. The following facts hold (i) The trace map Tr commutes with ∆ " Hp1`Hq and the range of Tr˝∆ is contained in the strong Schwartz space S pR˚q :" X βPR µ β SpR˚q, with µ denoting the Modulus.
(ii) The spectrum of ∆ on the quotient of S pR˚q by the closure of the range of Tr˝∆ is the set (counted with possible multiplicities) "´z´1 Proof. (i) The trace map of (3.5) commutes with ∆. By Lemma 4.1 (iii) the range of ∆ is SpRq ev 0 thus the range of E˝pHp1`Hqq is contained in S pR˚q (see [9], Lemma 2.51).
(ii) By construction, S pR˚q is the intersection, indexed by compact intervals J Ă R, of the spaces X βPJ µ β SpR˚q. The Fourier transform f ΠpN q " f @f P SpIq .
This direct sum decomposition commutes with ∆ since both ΠpN q and the conjugate of ∆ by the Fourier transform F are given by multiplication operators. The conjugate of H by F is the multiplication by´z, so that the conjugate of ∆ is the multiplication by´zp1´zq. The spectrum of ∆ is the union of the spectra of the finite-dimensional operators ∆ N :" ΠpN q∆ " ∆ΠpN q. By [9], Corollary 4.118, and the proof of Theorem 4.116, the finite-dimensional range of ΠpN q is described by the evaluation of f P SpIq on the zeros ρ P ZpN q of the Riemann zeta function which are inside the contour γ N , i.e. by the map where C pnρq denotes the space of dimension n ρ of jets of order equal to the order n ρ of the zero ρ of the zeta function. Moreover, the action of ∆ N is given by the matrix associated with the multiplication of f P SpIq by´zp1´zq: this gives a triangular matrix whose diagonal is given by n ρ terms all equal to´ρp1´ρq. Thus the spectrum of ∆ on the quotient of S pR˚q by the closure of the range of Tr˝∆ is the set (counted with multiplicities) "´ρ´1 Proof. This follows from Proposition 4.2 and the fact that for ρ P Ć Remark 4.4. The main interest of the above reformulation of the spectral realization of [8,9] in terms of the Laplacian ∆ is that the latter is intimately related to the prolate wave operator W λ that is shown in [10] to be self-adjoint and have, for λ " ? 2 the same UV spectrum as the Riemann zeta function. The relation between ∆ and W λ is that the latter is a perturbation of ∆ by a multiple of the Harmonic oscillator.

Sheaves on the Scaling Site and H 0 pS , L 2 {ΣEq
Let µ P R ą1 and Σ µ be the linear map on functions g : R˚Ñ C of sufficiently rapid decay at 0 and 8 defined by We shall denote with S ev 0 the linear space of real-valued, even Schwartz functions f P SpRq fulfilling the two conditions f p0q " 0 " ş R f pxqdx. The map (5.2) E : S ev 0 Ñ R, pEf qpuq " u 1{2 ÿ ną0 f pnuq is proportional to a Riemann sum for the integral of f . The following lemma on scale invariant Riemann sums justifies the pointwise "well-behavior" of (5.2) (see [7] Lemma 6.1) Lemma 5.1. Let f be a complex-valued function of bounded variation on p0, 8q. Assume that f is of rapid decay for u Ñ 8, Opu 2 q when u Ñ 0, and that ş 8 0 f ptqdt " 0. Then the following properties hold (i) The function pEf qpuq in (5.2) is well-defined pointwise, is Opu 1{2 q when u Ñ 0, and of rapid decay for u Ñ 8. (ii) Let g " Epf q, then the series (5.1) is geometrically convergent, and defines a bounded and measurable function on R˚{µ Z .
We recall that a sheaf over the Scaling Site S " r0, 8q¸Nˆis a sheaf of sets on r0, 8q (endowed with the euclidean topology) which is equivariant for the action of the multiplicative monoid Nˆ [6]. Since we work in characteristic zero we select as structure sheaf of S the Nˆ-equivariant sheaf O whose sections on an open set U Ă r0, 8q define the space of smooth, complex-valued functions on U . The next proposition introduces two relevant sheaves of O-modules.
Proposition 5.2. Let L P p0, 8q, µ " exp L, and C µ " R˚{µ Z . The following facts hold (i) As L varies in p0, 8q, the pointwise multiplicative Fourier transform defines an isomorphism between the family of Hilbert spaces L 2 pC µ q and the restriction to p0, 8q of the trivial vector bundle L 2 " r0, 8qˆℓ 2 pZq. (ii) The sheaf L 2 on r0, 8q is defined by associating to an open subset U Ă r0, 8q the space F pU q " C 8 0 pU, L 2 q of smooth sections vanishing at L " 0 of the vector bundle L 2 . The action of Nˆon L 2 is given, for n P Nˆand for any pair of opens U and U 1 of r0, 8q, with nU Ă U 1 , by (5.5) F pU, nq : C 8 0 pU 1 , L 2 q Ñ C 8 0 pU, L 2 q, F pU, nqpξqpxq " σ n pξpnxqq.
Note that with µ " exp x one has ξpnxq P L 2 pC µ n q and σ n pξpnxqq P L 2 pC µ q. By construction one has: σ n σ m " σ nm , thus the above action of Nˆturns L 2 into a sheaf on S " r0, 8q¸Nˆ.
(iii) By Lemma 5.1 (i), Epf qpuq is pointwise well-defined, it is Opu 1{2 q for u Ñ 0, and of rapid decay for u Ñ 8. By (ii) of the same lemma one has It then follows from [7] (see p6.4q which is valid for z " 2πn L P R) that Since f P S ev 0 , with wpf qpuq :" u 1{2 f puq, the multiplicative Fourier transform Fpwpf qq " ψ, ψpzq :" ş R˚f puqu 1 2´i z d˚u is holomorphic in the complex half-plane defined by ℑpzq ą´5{2 [7]. Moreover, by construction S ev 0 is stable under the operation f Þ Ñ uB u f`1 2 f , hence wpS ev 0 q is stable under f Þ Ñ uB u f . This operation multiplies Fpwpf qqpzq " ψpzq by iz. This argument shows that for any integer m ą 0, z m ψpzq is bounded in a strip around the real axis and hence that the derivative ψ pkq psq is Op|s|´mq on R, for any k ě 0. By applying classical estimates due to Lindelof [11], (see [1] inequality (56)), the derivatives ζ pmq p 1 2`i zq are Op|z| α q for any α ą 1{4. Thus all derivatives B m L of the function (5.6), now re-written as hpL, nq :" L´1 2 ζ`1 2´2 πin L˘ψ p 2πn L q, are sequences of rapid decay as functions of n P Z. It follows that ΣEpf q is a smooth (global) section of the vector bundle L 2 over p0, 8q. Moreover, when n ‰ 0 the function hpL, nq tends to 0 when L Ñ 0 and the same holds for all derivatives B m L hpL, nq. In fact, for any m, k ě 0, one has ÿ n‰0 |B m L hpL, nq| 2 " OpL k q when L Ñ 0.
This result is a consequence of the rapid decay at 8 of the derivatives of the function ψ, and the above estimate of ζpzq and its derivatives. For n " 0 one has hpL, 0q " L´1 2 ζp 1 2 qψp0q. (iv) For any open subset U Ă r0, 8q the vector space C 8 0 pU, L 2 q admits a natural Frechet topology with generating seminorms of the form (K Ă U compact subset) One obtains a space of smooth sections C 8 0 pU, ΣEq Ă C 8 0 pU, L 2 q defined as sums of products ř h j ΣEpf j q, with f j P S ev 0 and h j P C 8 0 pU, L 2 q. The map σ n : L 2 pC µ n q Ñ L 2 pC µ q in (ii) is continuous, and from the equality σ n˝Σµ n " Σ µ it follows (here we use the notations as in the proof of (ii)) that the sections ξ P C 8 pU 1 , L 2 pC 1 qq which belong to C 8 pU 1 , ΣEpS ev 0 qq are mapped by F pU, nq inside C 8 pU, ΣEpS ev 0 qq. In this way one obtains a sheaf ΣE Ă L 2 of O-modules over S .
(v) Let ξ P H 0 pU, ΣEq. By hypothesis, ξ is in the closure of C 8 0 pU, ΣEpS ev 0 qq Ă C 8 0 pU, L 2 q for the Frechet topology. The Fourier components of ξ define continuous maps in the Frechet topology, thus it follows from (5.6) that the functions f n " Fpξqpnq are in the closure, for the Frechet topology on C 8 0 pU, Cq, of C 8 0 pU, Cqg n , where g n pLq :" ζ`1 2´2 πin L˘i s a multiplier of C 8 0 pU, Cq. This conclusion holds thanks to the moderate growth of the Riemann zeta function and its derivatives on the critical line. Conversely, let ξ P C 8 0 pU, L 2 q be such that each of its Fourier components Fpξqpnq belongs to the closure for the Frechet topology of C 8 0 pU, Cq, of C 8 0 pU, Cqg n . Let ρ P C 8 c pr0, 8q, r0, 1sq defined to be identically equal to 1 on r0, 1s and with support inside r0, 2s. The functions α k pxq :" ρppkxq´1q (k ą 1) fulfill the following three properties (1) α k pxq " 0, @x ă p2kq´1, α k pxq " 1, @x ą k´1.
For all m ą 0 there exists C m ă 8 such that |x 2m B m x α k pxq| ď C m k´1 @x P r0, 1s, k ą 1.
To justify (3), note that x 2 B x f ppkxq´1q "´k´1f 1 ppkxq´1q and that the derivatives of ρ are bounded. Thus one has |px 2 B x q m α k pxq| ď }ρ pmq } 8 k´m @x P r0, 8q, k ą 1 which implies (3) by induction on m. Thus, when k Ñ 8 one has α k ξ Ñ ξ in the Frechet topology of C 8 0 pU, L 2 q. This is clear if 0 R U since then, on any compact subset K Ă U , all α k are identically equal to 1 for k ą pmin Kq´1. Assume now that 0 P U and let K " r0, ǫs Ă U . With the notation of (5.7) let us show that p pn,mq K ppα k´1 qξq Ñ 0 when k Ñ 8. Since α k pxq " 1, @x ą k´1 one has, using the finiteness of p L´n}pB m L ξqpLq} L 2 kÑ8 Ñ 0.

Then one obtains
L´nB m L ppα k´1 qξqpLq " L´nppα k´1 qB m L ξqpLq`m Thus using (3) above and the finiteness of the norms p pn`2j,m´jq K pξq one derives: p pn,mq K ppα k´1 qξq Ñ 0 when k Ñ 8. It remains to show that α k ξ belongs to the submodule C 8 0 pU, ΣEq. It is enough to show that for K Ă p0, 8q a compact subset with min K ą 0, one can approximate ξ by elements of C 8 0 pU, ΣEq for the norm p p0,mq K . Let P N be the orthogonal projection in L 2 pC µ q on the finite-dimensional subspace determined by the vanishing of all Fourier components Fpξqpℓq for any ℓ, |ℓ| ą N . Given L P K and ǫ ą 0 there exists N pL, ǫq ă 8 such that (5.8) }p1´P N qB j L ξpLq} ă ǫ @j ď m, N ě N pL, ǫq. The smoothness of ξ implies that there exists an open neighborhood V pL, ǫq of L such that (5.8) holds in V pL, ǫq. The compactness of K then shows that there exists a finite N K such that It now suffices to show that one can approximate P N ξ, for the norm p p0,mq K , by elements of C 8 0 pU, ΣEq. To achieve this result, we let L 0 P K and δ j P C 8 c pR˚q, |j| ď N be such that ż R˚u 1{2 δ j puqd˚u " 0 @j, |j| ď N.
One construct δ j starting with a function h P C 8 c pR˚q such that Fphq´2 πj L0¯‰ 0 and acting on h by a differential polynomial whose effect is to multiply Fphq by a polynomial vanishing on all 2πj 1 L0 , j 1 ‰ j and at i{2. By hypothesis each Fourier component Fpξqpnq belongs to the closure in C 8 0 pU, Cq of the multiples of the function ζ`1 2´2 πin L˘. Thus, given ǫ ą 0 one has functions f n P C 8 0 pU, Cq, |n| ď N such that πin L˙f n pLq˙| ď ǫ @j ď m, |n| ď N.
We now can find a small open neighborhood V of L 0 and functions φ j P C 8 pV q, |j| ď N such that This is possible because the determinant of the matrix M n,j pLq " Fpδ j q`2 πn L˘i s non-zero in a neighborhood of L 0 where M n,j pL 0 q is the identity matrix. The even functions d j puq on R, which agree with u´1 {2 δ j puq for u ą 0, are all in S ev 0 since ş R d j pxqdx " 2 ş R˚u 1{2 δ j puqd˚u " 0. One then has πn L˙ by (5.6), and by (5.9) one gets ÿ φ j pLqFpΣ µ pEpd j qqqpnq " ζˆ1 2´2 πin L˙f n pLq @L P V.
One finally covers K by finitely many such open sets V and use a partition of unity subordinated to this covering to obtain smooth functions ϕ ℓ P C 8 c p0, 8q, g ℓ P S ev 0 such that the Fourier component of index n, |n| ď N , of ř ϕ ℓ ΣEpg ℓ q is equal to ζ`1 2´2 πin L˘f n pLq on K. This shows that ξ belongs to the closure of We recall that the space of global sections H 0 pT , F q of a sheaf of sets F in a Grothendieck topos T is defined to be the set Hom T p1, F q, where 1 denotes the terminal object of T . For T " S and F a sheaf of sets on r0, 8q, 1 assigns to an open set U Ă r0, 8q the single element˚, on which Nˆacts as the identity. Thus, we understand an element of Hom S p1, F q as a global section ξ of F , where F is viewed as a sheaf on r0, 8q invariant under the action of Nˆ.
With the notations of Proposition 5.2 and for ξ P Hom S p1, L 2 q, we write p ξpL, nq :" Fpξqpnq for the (multiplicative) Fourier components of ξ. Then we have Proof. (i) Let ξ P Hom S p1, L 2 q: this is a global section ξ P C 8 0 pr0, 8q, L 2 q invariant under the action of Nˆ, i.e. such that σ n pξpnLqq " ξpLq for all pairs pL, nq. The Fourier components p ξpL, nq of any such section are smooth functions of L P r0, 8q vanishing at L " 0, for n ‰ 0, as well as all their derivatives. The equality σ n pξpLqq " ξpL{nq entails, for n ą 0, This shows that the p ξpL, nq are uniquely determined, for n ą 0 by the function p ξpL, 1q and, for n ă 0, by the function p ξpL,´1q. With gpLq " p ξpL, 0q one has: gpLq " n´1 2 gpL{nq for all n ą 0. This implies, since Q˚is dense in R˚and g is assumed to be smooth, that g is proportional to L´1 2 and hence identically 0, since it corresponds to a global section smooth at 0 P r0, 8q. This argument proves that γ is injective. Let us show that γ is also surjective. Given a pair of functions f˘P C 8 0 pr0, 8q, Cq we construct a global section ξ P H 0 pS , L 2 q such that γpξq " pf`, f´q. One defines ξpLq P L 2 pC µ q by by means of its Fourier components set to be p ξpL, 0q :" 0, and for n ‰ 0 by p ξpL, nq :" |n|´1 2 f signpnq pL{nq.
Since f˘pxq are of rapid decay for x Ñ 0, ř | p ξpL, nq| 2 ă 8, thus ξpLq P L 2 pC µ q. All derivatives of f˘pxq are also of rapid decay for x Ñ 0, thus all derivatives B k L pξpLqq belong to L 2 pC µ q and that the L 2 -norms }B k L pξpLqq} are of rapid decay for L Ñ 0. By construction σ n pξpLqq " ξpL{nq, which entails ξ P H 0 pS , L 2 q with γpξq " pf`, f´q.
(ii) Let ξ P H 0 pS , ΣEq. By Proposition 5.2 (v), the functions f˘" p ξpL,˘1q are in the closure, for the Frechet topology on C 8 0 pr0, 8q, Cq, of the ideal generated by the functions ζ`1 2¯2 πi L˘. Conversely, let ξ P H 0 pS , L 2 q and assume that γpξq is in the closed submodule generated by multiplication with ζ`1 2¯2 πi L˘. The Nˆ-invariance of ξ implies p ξpL, nq " |n|´1 2 p ξpL{|n|, signpnqq for n ‰ 0. Thus the Fourier components p ξpL, nq belong to the closure in C 8 0 pU, Cq of the multiples of the function ζ`1 2´2 πin L˘, then Proposition 5.2 (v) again implies ξ P H 0 pS , ΣEq. The action of R˚on the sheaf L 2 is given by the action ϑ on the Fourier components of its sections ξ. With µ " exp L, L P p0, 8q, n P N˚and λ P R˚, this is The following result explains in particular how the quotient sheaf L 2 {ΣE on S handles eventual multiplicities of critical zeros of the zeta function. Proof. We first show that the canonical map q : H 0 pS , L 2 q Ñ H 0 pS , L 2 {ΣEqq is surjective. Let ξ P H 0 pS , L 2 {ΣEqq: as a section of L 2 {ΣE on r0, 8q, there exists an open neighborhood V " r0, ǫq of 0 P r0, 8q and a section η P C 8 0 pV, L 2 q such that the class of η in C 8 0 pV, L 2 {ΣEq is the restriction of ξ to V . The Fourier components p ηpL, nq are meaningful for L P V . Since ξ is Nˆ-invariant, for any n P Nt he class of F pV {n, nqpηq, with F pV {n, nqpηqpLq :" σ n pηpnLqq (see (5.4)) is equal to the class of the restriction of η in C 8 0 pV {n, L 2 {ΣEq. We thus obtain ηpLq´F pV {n, nqpηq P C 8 0 pV {n, ΣEq Furthermore, the Fourier components of α " F pV {n, nqpηq are given by p αpL, kq " n 1 2 p ηpnL, nkq.
Next step is to extend the functions ηpL,˘1q P C 8 0 pV, Cq to f˘P C 8 0 pr0, 8q, Cq fulfilling the following property. For any open set U Ă r0, 8q and a section β P C 8 0 pU, L 2 q, with the class of β in C 8 0 pU, L 2 {ΣEqq being the restriction of ξ to U , the functions p βpL,˘1q´f˘pLq belong to the closure in C 8 0 pU, Cq of the multiples of the function ζ`1 2¯2 πi L˘. To construct f˘one considers the sheaf G˘which is the quotient of the sheaf of C 8 0 pr0, 8q, Cq functions by the closure of the ideal subsheaf generated by the multiples of the function ζ`1 2¯2 πi L˘. Since the latter is a module over the sheaf of C 8 functions, it is a fine sheaf, thus a global section of G˘can be lifted to a function. By Proposition 5.2 (v), the Fourier components p ξ j pL,˘1q of local sections ξ j of L 2 representing ξ define a global section of G˘. The functions f˘are obtained by lifting these sections. By appealing to Lemma 5.3, we let φ P H 0 pS , L 2 q to be the unique global section such that γpφq " pf`, f´q. Then we show that qpφq " ξ. We have already proven that the restrictions to V " r0, ǫq are the same. Thus it is enough to show that given L 0 ą 0 and a lift ξ 0 P C 8 0 pU, L 2 q of ξ in a small open interval U containing L 0 , the difference δ " φ´ξ 0 is a section of ΣE. Again by Proposition 5.2 (v), it suffices to show that the Fourier components p δpL, nq are in the closure of the ideal generated by multiples of ζ`1 2´2 πin L˘. The Nˆ-invariance of ξ shows that F pU {n, nqpξ 0 q (see (5.5)) is a lift of ξ in U {n. Thus by the defining properties of the functions f˘one has { F pU {n, nqpξ 0 qp˘1q´f˘P C 8 pU, Cqζ˘, for ζ˘pLq " ζˆ1 2¯2 πi L˙.
With a similar argument and using the invariance of φ under the action of F pU {n, nq, one obtains that p δpnq is in the closure of the ideal generated by the multiples of ζ`1 2´2 πin L˘.
This sequence is equivariant for the action (5.11) of ϑ of R˚on the bundle L 2 . For h P L 1 pR˚, d˚uq one has (5.13) { pϑphqξqpL, nq " Fphqˆ2 πn L˙p ξpL, nq.
Φ is well defined since all derivatives of Φ˘pf qpLq tend to 0 when L Ñ 0 (any function f P SpRq is of rapid decay as well as all its derivatives). The exact sequence (5.12), together with Lemma 5.3, then gives an induced isomorphism γ : H 0 pS , L 2 {ΣEqq » pC 8 0 q 2 {`C 8 0 ζ`ˆC 8 0 ζ´˘.
In turn, the map Φ induces a morphism Φ : SpRq{pSpRqζq Ñ pC 8 0 q 2 {`C 8 0 ζ`ˆC 8 0 ζ´˘. By (5.13) this morphism is equivariant for the action of R˚. The map Φ is not an isomorphism since elements of its range have finite limits at 8. However it is injective and its range contains all elements of pC 8 0 q 2 which have compact support. Since ζ˘pLq " ζ`1 2¯2 πi L˘t ends to a finite non-zero limit when L Ñ 0,Φ is an isomorphism.
Remark 5.5. By a Theorem of Whitney (see [12], Corollary 1.7), the closure of the ideal of multiples of ζ`1 2`i s˘in SpRq is the subspace of those f P SpRq which vanish of the same order as ζ at every (critical) zero s P Z. Thus if any such zero is a multiple zero of order m ą 1, one finds that the action of R˚on the global sections of the quotient sheaf L 2 {ΣE admits a non-trivial Jordan decomposition of the form ϑpλqξ " λ is pξ`N pλqξq, with N pλq m " 0 and p1`N puqqp1`N pvqq " 1`N puvq for all u, v P R˚.