THE ANALYTIC TORSION OF THE CONE OVER AN ODD DIMENSIONAL MANIFOLD

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1 THE ANALYTIC TORSION OF THE CONE OVER AN ODD DIMENSIONAL MANIFOLD L. HARTMANN AND M. SPREAFICO Abstract. We study the analytic torsion of the cone over an orientable odd dimensional compact connected Riemannian manifold W. We prove that the logarithm of the analytic torsion of the cone decomposes as the sum of the logarithm of the root of the analytic torsion of the boundary of the cone, plus a topological term, plus a further term that is a rational linear combination of local Riemannian invariants of the boundary. We show that this last term coincides with the anomaly boundary term appearing in the Cheeger Müller theorem [4] [9] for a manifold with boundary, according to Brüning and Ma []. We also prove Poincaré duality for the analytic torsion of a cone.. Introduction and statement of the results Analytic torsion was originally introduced by Ray and Singer in [], as an analytic counter part of the Reidemeister torsion of Reidemeister, Franz and de Rham. The first important result in this context, nowadays known as the Cheeger-Müller Theorem, was achieved by W. Müller [9] and J. Cheeger [4], who proved that for a compact connected Riemannian manifold without boundary, the analytic torsion and the Reidemeister torsion coincide, as conectured by Ray and Singer in []. The next natural question along this line of investigation was to answer the same problem for manifolds with boundary. It was soon realized that the answer to such a question was an highly non trivial one. W. Lück proved in [6] that in the case of a product metric near the boundary the boundary term is topological, and depends only upon the Euler characteristic of the boundary. The answer to the general case required 0 more years of work, and is contained in a recent paper of Brüning and Ma [] see also [6]. The new contribution of the boundary, beside the topological one given by Lück, is called anomaly boundary term, has a quite complicate expression, but only depends on some local quantities constructed from the metric tensor near the boundary see Section.3 for details. The next natural step is to study the analytic torsion for spaces with singularities, and the simplest singular space is the cone over a manifold, CW. Cones and spaces with conical singularities have been deeply investigated by J. Cheeger in a series of works [4] [5] see also [0]. Due to this investigation, all information on L -forms, Hodge theory, and Laplace operator on forms on CW are available. Further information on the class of regular singular operators, that contains the Laplace operator on CW, are given in works of Brüning and Seely see in particular [3]. As a result it is not difficult to obtain a complete description of the eigenvalues of the Laplace operator on CW in terms of the eigenvalues of the Laplace operator on W. With all these tools available, namely on one side the formula for the boundary term, and on the other some representation of the eigenvalues of the Laplace operator on the cone, it is natural to tackle the problem of investigating the analytic torsion of CW. A possible extension of the Cheeger Müller theorem could follow, or 000 Mathematics Subect Classification: 58J5.

2 L. HARTMANN AND M. SPREAFICO not. Indeed, in case of conical singularity such an extension would require intersection R torsion more than classical R torsion see [7]. However, if the section is a rational homology manifold, then the two torsions coincide see [4], end of Section, and the classical Cheeger Müller theorem is expected to extend. If CW is the chain complex associated to some cell decomposition of W, then the algebraic mapping cone ConeCW gives the chain complex for a cell decomposition of CW. It is then easy to see that the R torsion of CW only depends on the choice of a base for the zero dimensional homology. Even if Poincaré duality does not hold, it does hold between top and bottom dimension, and therefore we can fix the base for the zero homology using the Riemannian structure and harmonic forms see [] Section 3, see also [3]. The result for the R torsion is τcw = VolCW. On the other side, one wants the analytic torsion. The analytic tools necessary to deal with the zeta functions appearing in the definition of the analytic torsion, constructed with the eigenvalues of the Laplace operator on CW, are available by works of M. Spreafico [7] [8] [30]. In these works, the zeta function associated to a general class of double sequences is investigated. In particular, a decomposition result is presented and formulas for the zeta invariants of a decomposable sequence are given see Section.4. This technique applies to the case of the zeta functions appearing in the definition of the analytic torsion on CW. This approach was used in [] to study the cone over an odd low dimensional sphere, and is applied here to the general case, proving a conecture stated in [] see Corollary.. This method was originally used to deal with the zeta determinant of the Laplace operator on a cone in [6] see also [3], and consequently in [9] [3] [3] and [] to study the zeta determinants of the Laplacian on forms and analytic torsion type invariants. In particular, in [3] a general formula for the analytic torsion of a cone is given. The formula is obtained using a method introduced by one of the authors of this paper in some older works [6] [8] and some results of M. Lesch [4] [5], and it is not particularly illuminating as it is stated, since essentially it is ust an application of the formulas given in those works. In the abstract of [3], it is stated that the result is obtained by generalizing some computational methods of M. Spreafico, however such generalization is already contained in [8] and [9], and an even further generalization is contained in the preprint [3], of which the author of [3] seems not to be aware. We are now ready to state the main results of this paper we referee to the on line version of this work [] for further developments and results, for we fix some notation. Let W, g be an orientable compact connected Riemannian manifold of finite dimension m without boundary and with Riemannian structure g. We denote by C l W the cone over W with the Riemannian structure dx dx + x g, on CW {pt}, where pt denotes the tip of the cone and 0 < x l see Section 3. for details. The formal Laplace operator on forms on CW {pt} has a suitable L -self adoint extension abs/rel on C l W with absolute or relative boundary conditions on the boundary C l W see Section 3.3 for details, with pure discrete spectrum Sp abs/rel. This permits to define the associated zeta function ζs, abs/rel = λ Sp + abs/rel λ s, for Res > m+. This zeta function has a meromorphic analytic continuation to the whole complex s-plane with at most isolated poles see Section 4 for details. It is then possible to define the analytic

3 ANALYTIC TORSION OF CONES 3 torsion of the cone the trivial representation of the fundamental group is assumed log T abs/rel C l W = m+ q qζ 0, q abs/rel. In this setting, we have the following results analogous results with relative boundary conditions also follow by Poincaré duality on the cone, proved in Theorem 4. below. Theorem.. The analytic torsion on the cone C l W on an orientable compact connected Riemannian manifold W, g of odd dimension p is log T abs C l W = p q+ rkh q W ; Q log p q l + log T W, l g + S C l W, where the singular term S C l W only depends on the boundary of the cone: S C l W = p p Res 0 Φ k+,q s k q h Res ζ s, s=0 k h q p + k, =0 k=0 h=0 s=+ where the functions Φ k+,q s are some universal functions, explicitly known by some recursive relations, and is the Laplace operator on forms on the section of the cone. It is important to observe that the singular term S C l W is a universal linear combination of local Riemannian invariants of the boundary, for the residues of the zeta function of the section are such linear combination see Section 7 for details. Theorem.. With the notation of Theorem., the singular term of the analytic torsion of the cone C l W coincides with the anomaly boundary term of Brüning and Ma, namely S C l W = A BM,abs C l W. See Section.3 for the definition of A BM,abs C l W. If W is an odd sphere, we have: Corollary.. The natural extension of Cheeger Müller theorem for manifold with boundary is valid for the cone over an odd dimensional sphere, namely log T abs C l S p = log τc l S p + A BM,abs C l S p. In particular, if a denotes the radius of the sphere, then A BM,abs C l Sa p p! = 4 p p! p k=0 p k!k + k =0 k + k! +!! ak+. The result in the corollary should be understood as a particular case of the still unproved general result that the analytic torsion and the intersection R-torsion of a cone coincide up to the boundary term, for the intersection R-torsion is the classical R-torsion for the cone over a sphere. We point out that we also have a purely combinatoric proof of the result stated in Corollary., independent from Theorem.. To contain space, we omit the proof here; it will appear somewhere else see also []. We conclude with a remark on the even dimensional case, namely when the dimension of the section W is even. It is clear enough that all the arguments used in the odd dimensional case go through also in the even dimensional case, and that the anomaly boundary term is the one of Brüning and Ma. So we obtain formulas for the analytic torsion as in the theorems above.

4 4 L. HARTMANN AND M. SPREAFICO However, in the even dimensional case some further term appears: this was described in some details for W = S in [3]. Since we do not have a clear understanding of this new term yet, we prefer to omit the non particularly illuminating formulas for the even dimensional case here.. Preliminaries and notation We introduce in this section some notation necessary in the following. compact connected oriented Riemannian manifold. As usual W, g is a.. Manifolds with boundary. If W has a boundary W, then there is a natural splitting near the boundary of ΛW as direct sum of vector bundles ΛT W N W, where N W is the dual to the normal bundle to the boundary. Locally, let x denote the outward pointing unit normal vector to the boundary, and dx the corresponding one form, then near the boundary we have the collar decomposition Coll W = ɛ, 0] W, and if y is a system of local coordinates on the boundary, then x, y is a local system of coordinates in Coll W. The metric tensor decomposes near the boundary in this local system as g = dx dx + g x, where g x is a family of metric structure on W such that g 0 = i g, where i : W W denotes the inclusion. The smooth forms on W near the boundary decompose as ω = ω tan + ω norm, where ω norm is the orthogonal proection on the subspace generated by dx, and ω tan is in C W Λ W. We write ω = ω + dx ω, where ω C W Λ W, and. ω = dx ω. Define absolute and relative boundary conditions by B abs ω = ω norm W = ω W = 0, B rel ω = ω tan W = ω W = 0. Note that, if ω Ω q W, then B abs ω = 0 if and only if B rel ω = 0, B rel ω = 0 implies B rel dω = 0, and B abs ω = 0 implies B abs d ω = 0. Let Bω = Bω Bd + d ω. Then the operator = d + d with boundary conditions Bω = 0 is self adoint, and if Bω = 0, then ω = 0 if and only if d + d ω = 0. Note that B correspond to { ωnorm. B abs ω = 0 if and only if W = 0, dω norm W = 0,.3 B rel ω = 0 if and only if { ωtan W = 0, d ω tan W = 0,.. The form valued zeta functions and the analytic torsion. The Laplace operator q with boundary conditions B abs/rel has a pure point spectrum Sp q abs/rel consisting of real non negative eigenvalues. The sequence Sp + q abs/rel is a totally regular sequence of spectral type accordingly to Section.4, and the forms valued zeta function is the associated zeta function, defined by ζs, q abs/rel = ζs, Sp + q abs/rel = λ Sp + q abs/rel when Res > m. The analytic torsion T abs/relw, g; ρ of W, g with respect to the representation ρ : π W Ok, R is defined by log T abs/rel W, g; ρ = m q qζ 0, q abs/rel. q= λ s,

5 ANALYTIC TORSION OF CONES 5 The following duality holds for analytic torsion [6].4 log T abs W, g; ρ = m+ log T rel W, g; ρ. We will omit the representation in the notation whenever we mean the trivial representation. Next, recall some classical results of Hodge theory in order to define closed, coclosed, exact and coexact zeta functions. We restrict ourselves to the case of a manifold without boundary see [] for the case of manifold with boundary. Setting H q W, E ρ = {ω Ω q W, E ρ ω = 0}, the space of the q-harmonic forms, we have the Hodge decomposition.5 Ω q W, E ρ = H q W, E ρ dω q W, E ρ d Ω q+ W, E ρ. This induces a decomposition of the eigenspace of a given eigenvalue λ 0 of q into the spaces of closed forms and coclosed forms: E q λ = E q λ,cl E q λ,ccl, where E q λ,cl = {ω Ωq W, E ρ ω = λω, dω = 0}, Define exact forms and coexact forms by E q λ,ex = {ω Ωq W, E ρ ω = λω, ω = dα}, E q λ,ccl = {ω Ωq W, E ρ ω = λω, d ω = 0}. E q λ,cex = {ω Ωq W, E ρ ω = λω, ω = d α}. Note that, if λ 0, then E q λ,cl = E q q λ,ex, and E λ,ccl = E q λ,cex, and we have an isometry.6 φ :E q λ,cl E q λ,cex, φ :ω d ω, λ whose inverse is λ d. Also, the restriction of the Hodge star defines an isometry : d Ω q+ W dω m q W, and that composed with the previous one gives the isometries:.7 λ d : E q λ,cl E m q+ λ,cex, λ d : E q λ,ccl E m q λ,ex. By the very definition, we have ζs, q = dim E q λ λ s = ζ cl s, q + ζ ccl s, q, λ Sp + q where ζ cl s, q = λ Sp + q dim E q λ,cl λ s, ζ ccl s, q = λ Sp + q dim E q λ,ccl λ s. Since, by.6, ζ cl s, q = ζ ccl s, q, we obtain from the above relations the following formulas for the torsion of a closed m dimensional manifold W : log T W, g; ρ = m q qζ 0, q = q= m q ζ cl0, q = m q ζ ccl0, q. q=

6 6 L. HARTMANN AND M. SPREAFICO In particular, using again duality, for an odd dimensional manifold W of dimension m = p,.8 p log T W, g; ρ = q ζ cl0, q + p ζ cl0, p q= p = q ζ ccl0, q + p ζ ccl0, p..3. The Cheeger Müller theorem for manifolds with boundary, and the anomaly boundary term of Brüning and Ma. In case of a smooth orientable compact connect Riemannian manifold W, g with boundary W, for any representation ρ of the fundamental group for simplicity assume rkρ =, the analytic torsion is given by the Reidemeister torsion plus some further contributions. It was shown by J. Cheeger in [4] that this further contribution only depends on the boundary, and W. Lück proved the following formula in the case of a product metric near the boundary, where χx denotes the Euler characteristic of X [6] log T abs W, g; ρ = log τw ; ρ + χ W log. 4 In the general case a further contribution appears, that measures how the metric is far from a product metric. A formula for this new anomaly boundary contribution is contained in some recent result of Brüning and Ma []. More precisely, in [] equation 0.6 is given a formula for the ratio of the analytic torsion of two metrics, g and g 0. Using their notation for Z/ graded algebras, we identify an antisymmetric endomorphism φ of finite dimensional vector space V over a field of characteristic zero with the element ˆφ = m,k= φv, v k ˆv ˆv k, of Λ V. For the elements φv, v k are the entries of the tensor representing φ in the base {v k }, and this is an antisymmetric matrix. Now assume that r is an antisymmetric endomorphism of Λ V. Then, R k = rv, v k is a tensor of two forms in Λ V. We extend the above construction identifying R with the element.9 ˆR = m rv, v k ˆv ˆv k,,k= of Λ V Λ V. This can be generalized to higher dimensions. In particular, all the construction can be done taking the dual V instead of V. Accordingly to [], we define the following forms.0 S = m i ω i ω 0 0k ê k k= i Ω = m i Ω,kl ê k ê l, ˆΘ = k,l= m k,l= Θ kl ê k ê l. Here, ω are the connection one forms, and Ω, = 0,, the curvature two forms associated to the metrics g 0 and g, respectively, while Θ is the curvature two form of the boundary with the metric induced by the inclusion, and {e k } m k=0 is an orthonormal base of T W with respect to the metric g. Then, set. B = B e ˆΘ u S 0 Γ uk k + S k du. k=

7 ANALYTIC TORSION OF CONES 7 Taking g = g, and g 0 an opportune deformation of g, that is a product metric near the boundary, and a flat vector bundle F, the formula of [] reads log T absw, g ; ρ T abs W, g 0 ; ρ = B. Note that the right side of this equation is as expected a local quantity, and is well defined if there exists a regular collar neighborhood of the boundary. If this is the case, we define the Brüning and Ma anomaly boundary term with absolute BC by. A BM,abs W = B, and we have.3 log T abs W, g; ρ = log τw ; ρ + 4 χ W log + A BM,abs W..4. Zeta determinants. This section is essentially contained in Section 4 of [], to which we refer for details. Given a sequence S = {a n } n= of spectral type, we define the zeta function by ζs, S = a s n, when Res > es, and by analytic continuation otherwise, and for all λ ρs = C S, we define the Gamma function by the canonical product,.4 Γ λ, S = + λ gs λ = e a n. a n n= Given a double sequence S = {λ n,k } n,k= of non vanishing complex numbers with unique accumulation point at the infinity, finite exponent s 0 = es and genus p = gs, we use the notation S n S k to denote the simple sequence with fixed n k, we call the exponents of S n and S k the relative exponents of S, and we use the notation s 0 = es, s = es k, s = es n ; we define relative genus accordingly. Definition.. Let S = {λ n,k } n,k= be a double sequence with finite exponents s 0, s, s, genus p 0, p, p, and positive spectral sector Σ θ0,c 0. Let U = {u n } n= be a totally regular sequence of spectral type of infinite order with exponent r 0, genus q, domain D φ,d. We say that S is spectrally decomposable over U with power κ, length l and asymptotic domain D θ,c, with c = minc 0, d, c, θ = maxθ 0, φ, θ, if there exist positive real numbers κ, l integer, c, and θ, with 0 < θ < π, such that: { } λn,k n= W W has spectral sector Σ θ,c, and is a totally regular sequence the sequence u κ n S n = u κ n k= of spectral type of infinite order for each n; the logarithmic Γ-function associated to S n /u κ n has an asymptotic expansion for large n uniformly in λ for λ in D θ,c, of the following form.5 log Γ λ, u κ n S n = l h=0 φ σh λu σ h n + L l=0 P ρl λu ρ l n log u n + ou r0 n, where σ h and ρ l are real numbers with σ 0 < < σ l, ρ 0 < < ρ L, the P ρl λ are polynomials in λ satisfying the condition P ρl 0 = 0, l and L are the larger integers such that σ l r 0 and ρ L r 0.

8 8 L. HARTMANN AND M. SPREAFICO Define the following functions, Λ θ,c = { z C argz c = θ }, oriented counter clockwise:.6 Φ σh s = 0 t s e πi Λθ,c λt λ φ σ h λdλdt. By Lemma 3.3 of [30], for all n, we have the expansions: p log Γ λ, S n /u κ n a α,0,n λ α + a k,,n λ k log λ,.7 φ σh λ =0 k=0 p b σh,α,0 λ α + b σh,k, λ k log λ, =0 for large λ in D θ,c. We set see Lemma 3.5 of [30] l A 0,0 s = a 0,0,n.8 A, s = n= a,,n n= h=0 l h=0 b σh,0,0u σ h n b σh,,u σ h n k=0 u κs n, u κs n, 0 p. Theorem.. Let S be spectrally decomposable over U as in Definition.. Assume that the functions Φ σh s have at most simple poles for s = 0. Then, ζs, S is regular at s = 0, and ζ0, S = A 0, 0 + κ l h=0 ζ 0, S = A 0,0 0 A 0,0 + γ κ + κ l h=0 Res Φ σh s Res ζs, U, s=0 s=σ h l h=0 Res 0 Φ σh s Res ζs, U + s=0 s=σ h Res Φ σh s Res ζs, U s=0 s=σ h l Res h=0 s=0 Φ σh s Res 0 s=σ h ζs, U, where the notation means that only the terms such that ζs, U has a pole at s = σ h appear in the sum. Remark.. We call regular part of ζ0, S the first term appearing in the formula given in the theorem, and regular part of ζ 0, S the first two terms. The other terms we call singular part. Corollary.. Let S = {λ,n,k } n,k=, =,..., J, be a finite set of double sequences that satisfy all the requirements of Definition. of spectral decomposability over a common sequence U, with the same parameters κ, l, etc., except that the polynomials P,ρ λ appearing in condition do not vanish for λ = 0. Assume that some linear combination J = c P,ρ λ, with complex coefficients, of such polynomials does satisfy this condition, namely that J = c P,ρ λ = 0. Then, the linear combination of the zeta function J = c ζs, S is regular at s = 0 and satisfies the linear combination of the formulas given in Theorem.. We conclude recalling some formulas for the zeta determinants of some simple sequences. The results are known to specialists, and can be found in different places. We will use the formulation

9 ANALYTIC TORSION OF CONES 9 of [5]. For positive real l and q, define the non homogeneous quadratic Bessel zeta function by s ν,k zs, ν, q, l = l + q, k= for Res >. Then, zs, ν, q, l extends analytically to a meromorphic function in the complex plane with simple poles at s =,, 3,.... The point s = 0 is a regular point and.9 z0, ν, q, l = ν +, z 0, ν, q, l = log πl I νlq q ν. In particular, taking the limit for q 0, z 0, ν, 0, l = log πl ν+ ν Γν Geometric setting and Laplace operator 3.. The finite metric cone. Let W, g be an orientable compact connected Riemannian manifold of finite dimension m without boundary and with Riemannian structure g. Embedding W in the opportune Euclidean space R k, and R k in some hyperplane of R k+h, with opportune h, disconnected from the origin, a geometric realization of the cone CW is the given by the set of the finite length l line segments oining the origin to the embedded copy of W. Let x the euclidean geodesic distance from the origin. We equip CW {p} with the Riemannian structure dx dx + x g, and we denote by C 0,l] W the space 0, l] W with this metric. We denote by C l W the compact space C 0,l] W = C 0,l] W {p}, and we call it the completed finite metric cone over W. We call the subspace {l} W of C l W, the boundary of the cone, and we denote it by C l W. This is of course diffeomorphic to W, and isometric to W, l g. Following common notation, we will call W, g the section of the cone. Also following usual notation, a tilde will denotes operations on the section of course g = g, and not on the boundary. All the results of Section. are valid. In particular, given a local coordinate system y on W, then x, y is a local coordinate system on the cone. We now give the explicit form of, d and. See [5] and [0] Section 5 for details. If ω Ω q C 0,l] W, set ωx, y = f xω y + f xdx ω y, with smooth functions f and f, and ω ΩW. Then a straightforward calculation gives 3. ωx, y = x m q+ f x ω y + q x m q f xdx ω y, dωx, y = f x dω y + x f xdx ω y f xdx dω y, d ωx, y = x f x d ω y m q + x f x + x f x ω y x f xdx d ω y, ωx, y = xf x m qx x f x ω y + x f x ω y x f x dω y + dx x f x ω y + ω y xf x m q + x x f x +m q + x f x x 3 f x d ω y.

10 0 L. HARTMANN AND M. SPREAFICO 3.. Riemannian tensors on the cone. We give here the explicit form of the main Riemannian quantities on the cone. Recall that a tilde denotes quantities relative to the section, that we have local coordinate x, y,..., y m on C l W, and that the metric is g = dx dx + x g. Let {b k } m k= be a local orthonormal base of T W, and {b k }m k= the associated dual base. Then, e 0 = x, e 0 = dx, e k = x b k, e k = xb k, k m. Direct calculations give Cartan structure constants c k0 = 0,, k m, c 0kl = c k0l = δ kl x, k, l m, c kl = x c kl,, k, l m, and the Christoffel symbols are Γ 0kl = 0, k, l m, Γ 0k = Γ k0 = δ k x,, k m, Γ kl = Γ x kl,, k, l m. The connection one form matrix relatively to the metric g has components 3.4 ω,00 = 0, ω,0 = ω,0 = x e = b, m, ω,k = m Γ hk e h = x h= m Γ hk e h = h= To compute the curvature we calculate m dω,0 = l b dy l = l= l=0 m Γ hk b h = ω k,, k m. h= m l b k dy k dy l, l,k= where b = m k= b kdy k, and, for, k m, dω,k = d ω k ; while m m m ω ω k0 = ω ω 0k = ω,0l ω,lk = ω,0l ω,lk = b l ω lk, m m ω ω k = ω,l ω,lk = ω,0 ω,0k + ω,l ω,lk = b b k + ω ω k, l=0 for, k m. The curvature two form has components Ω,00 = 0, Ω,0 = l,k= l= m m l b k dy k dy l b l ω lk, m, l= Ω,k = d ω k b b k + ω ω k = Ω k b b k,, k m. l= l= Next, considering the metric g 0 = dx dx + g, similar calculations gives: 3.5 ω 0,0 = 0, 0 m, ω 0,k = ω k,, k m. By equations 3.4 and 3.5, S = m 3.6 e k e k = l l 3.7 S 0 = 0. k= m b k b k = k= m b k e k, We also need the curvature two form Θ on the boundary C l W. A similar calculation gives Θ k = Ω k. Note in particular that it is easy to verify the equation.6 of []: ˆΘ = i Ω S. k=

11 ANALYTIC TORSION OF CONES For S = l m,k= while i Ω k = Ω k b b k, gives b b k ˆb ˆb k, ˆΘ = l i Ω,k = l m,k= m,k= Ωk b b k ˆb ˆb k. Ω k ˆb ˆb k, 3.3. The Laplace operator on the cone and its spectrum. We study the Laplace operator on forms on the space C l W. This is essentially based on [5] and [3]. Let denote by L the formal differential operator defined by equation 3.3 acting on smooth forms on C 0,l] W, ΓC 0,l] W, ΛT C 0,l] W. We define in Lemma 3. a self adont operator acting on L C l W, Λ q C l W, and such that ω = Lω, if ω dom. Then, in Lemma 3., we list all the solutions of the eigenvalues equation for L. Eventually, in Lemma 3.3, we give the spectrum of. Lemma 3.. The formal operator L in equation 3.3 with the absolute/relative boundary conditions given in equations./.3 on the boundary C l W defines a unique self adoint semi bounded operator on L C l W, Λ q T C l W, that we denote by the symbol abs / rel, respectively, with pure point spectrum. Proof. Let L q denote the minimal operator defined by the formal operator L q, with domain the q-forms with compact support in C 0,l] W, namely doml q = Γ 0 C 0,l] W, ΛT C 0,l] W. The boundary values problem at the boundary x = l, i.e. C l W, is trivial, and gives the self adoint extensions stated. The point x = 0 requires more work. First, note that L q reduces by unitary transformation to an operator of the type 3.8 D + Ax x, D = i d dx, where Ax is smooth family of symmetric second order elliptic operators [3] pg More precisely, there exists a unitary transformation ψ q between the relevant spaces with the suitable L structures, see [3] for details. Under the transformation ψ q, L q has the form in equation 3.8, with Ax the constant smooth family of symmetric second order elliptic operators in ΓW, Λ q T W Λ q T W : q + m Ax = A0 = q m q q d q d q + m + q m + q Next, by its definition, Ax satisfies all the requirements at pg. 373 of [3], with p = in particular this follows from the fact that Ax is defined by the Laplacian on forms on a compact space. We can apply the results of Brüning and Seeley [3], observing that in the present case we are in what they call constant coefficient case Section 3 of [3]. By Theorem 5. of [3], the operator L extends to a unique self adoint bounded operator q. Note that this extension is the Friedrich extension by Theorem 6. of [3]. Note also that bundary condition at x = 0 are necessary in general in the definition of the domain of q, see L c, pg. 40 of [3] for these conditions. Eventually, by Theorem 5. of [3], the square here p =, so m = of the resolvent of q is of trace class. This means that the resolvent is Hilbert Schmidt, and consequently the spectrum of q is pure point, by the spectral theorm for compact operators. Note that we do not need the cut off function γ appearing in Theorem 5. of [3], since here 0 < x l.

12 L. HARTMANN AND M. SPREAFICO Lemma 3. [5]. Let {ϕ q har, ϕq cex,n, ϕ q ex,n} be an orthonormal base of ΓW, Λ q T W consisting of harmonic, coexact and exact eigenforms of q on W. Let λ q,n denotes the eigenvalue of ϕ q cex,n and m cex,q,n its multiplicity so that m cex,q,n = dim E cex,n q = dim E q ccl,n. Let J ν be the Bessel function of index ν. Define α q = + q m and µ q,n = λ q,n + αq. Then, assuming that µ q,n is not an integer, all the solutions of the equation u = λ u, with λ 0, are convergent sums of forms of the following six types: ψ q ±,,n,λ =xαq J ±µq,n λxϕ q cex,n, ψ q q ±,,n,λ =xαq J ±µq,n λx dϕ cex,n + x x αq J ±µq,n λxdx ϕ cex,n q ψ q ±,3,n,λ =xαq + x x αq J ±µq,n λx dϕ q cex,n + x αq J ±µq,n λxdx d q dϕ cex,n ψ q ±,4,n,λ =xαq + q J ±µq,n λxdx dϕ cex,n ψ q ±,E,λ =xαq J ± αq λxϕ q har ψ q ±,O,λ = xx αq J ± αq λxdx ϕ q har. When µ q,n is an integer the solutions must be modified including some logarithmic term see for example [34] for a set of linear independent solutions of the Bessel equation. Proof. The proof is a direct verification of the assertion, using the definitions in equations 3., 3., and 3.3. First, by Hodge theorem, there exist an orthonormal base of Λ q T W as stated. Thus, we decompose any form ω in this base. Second, we compute ω, using this decomposition and the formula in equation 3.3. This gives some differential equations in the functions appearing as coefficients of the forms. All these differential equations reduce to equations of Bessel type. Third, we write all the solutions using Bessel functions. A complete proof for the case of the harmonic forms can be found in [0] Section 5. Note that the forms of types and 3 are coexact, those of types and 4 exacts. The operator d sends forms of types and 3 in forms of types and 4, while d sends forms of types and 4 in forms of types and 3, respectively. The Hodge operator sends forms of type in forms of type 4, in 3, and E in 0. Corollary 3.. The functions + in Lemma 3. are square integrable and satisfy the boundary conditions at x = 0 defining the domain of rel/abs. The functions either are not square integrable or do not satisfy these conditions. Remark 3.. All the solutions are either not square or their exterior derivative are not square integrable. Requiring the last condition in the definition of the domain of rel/abs, it follows that there are not boundary conditions at zero. This was observed by Cheeger for harmonic forms when the dimension is odd in [5] Section 3.

13 ANALYTIC TORSION OF CONES 3 Lemma 3.3. The positive part of the spectrum of the Laplace operator on forms on C l W, with absolute boundary conditions on C l W is: { Sp + q abs = m cex,q,n : ĵµ q,n,α q,k/l } {m cex,q,n : µq,n,k/l } {m har,q : ĵ αq,αq,k /l} With relative boundary conditions: { Sp + q rel = m cex,q,n : µ q,n,k/l } n,k= k= n,k= n,k= {m cex,q,n : ĵµq,n, αq,k/l } {m cex,q,n : ĵµq,n,αq,k/l } {m cex,q,n : µq,n,k/l } {m har,q : ĵ αq,αq,k /l} k=. n,k= n,k= {m cex,q,n : µq,n,k/l } n,k= {m cex,q,n : ĵµq,n, αq,k/l } n,k= { m har,q : αq,k/l } k= { m har,q : α q,k /l} k=, n,k= where the µ,k are the zeros of the Bessel function J µ x, the ĵ µ,c,k are the zeros of the function Ĵ µ,c x = cj µ x + xj µx, c R, α q and µ q,n are defined in Lemma 3.. Proof. By the Lemma 3., Lemma 3. and its corollary, we know that the + solutions of Lemma 3. determine a complete system of square integrable solutions of the eigenvalues equation q u = λu, with λ 0, satisfying the boundary condition at x = 0. Since q abs/rel has pure point spectrum, in order to obtain a discrete resolution more precisely the positive part of it of q abs/rel, we have to determine among these solutions those that belong to the domain of q abs/rel, namely those that satisfy the boundary condition at x = l. By direct application of the BC we obtain the result. For example, for forms of type 3, we obtain the system { x α q J µq,n λx x=l = 0, x x α q + x x αq J µq,n λx λx αq J µq,n λx x=l = 0, that using classical properties of Bessel functions and their derivative, gives λ = µq,n,k/l. Lemma 3.4 [5], [0]. With the notation of Lemma 3., and a ±,q,n = α q ± µ q,n, then all the solutions of the harmonic equation u = 0, are convergent sums of forms of the following four types: ψ q ±,,n =xa±,q,n ϕ q ccl,n, ψ q q =xa±,q,n ±,,n dϕ ccl,n + a ±,q,nx a±,q,n dx ϕ q ccl,n, ψ q ±,3,n =xa±,q,n+ q dϕ ccl,n + a,q,nx a±,q,n+ dx ϕ q ccl,n, ψ q ±,4,n =xa±,q,n+ dx dϕ q ccl,n.

14 4 L. HARTMANN AND M. SPREAFICO Lemma 3.5. Assume dim W = p is odd. Then { H q abs C H q W, 0 q p, lw = {0}, p q p. { H q rel C {0}, 0 q p, lw = { x α q dx ϕ q, ϕ q H q W }, p + q p. Proof. First, by Remark 3., we need only to consider the + solutions in Lemma 3.4. The proof then follows by argument similar to the one used in the proof of Lemma 3.3. Let see one case in details. Consider ψ q +,,n = xa+,q,n ϕ q ccl,n, where a +,q,n = α q + µ q,n. In order that ψ q +,,n satisfies the absolute boundary condition., we need that dψ q +,,n norm = a +,q,n l a+,q,n dx dϕ q ccl,n = 0 Cl W and this is true if and only if a +,q,n = 0. The condition a +,q,n = 0 is equivalent to the conditions λ q,n = 0, and α q = α q. Therefore, ϕ q ccl,n is harmonic, 0 q p, and ψq +,,n = ϕq ccl,n. 4. Torsion zeta function and Poincaré duality for a cone Using the description of the spectrum of the Laplace operator on forms q abs/rel given in Lemma 3.3, we define the zeta function on q-forms as in Section., by ζs, q abs/rel = λ Sp + q abs/rel for Res > m+. The explicit knowledge of the behaviour of the large eigenvalues allows to completely determine the analytic continuation of the zeta function, by using the tools os Section.4. In particular, it is possible to prove that there can be at most a simple pole at s = 0. We will not do this here but the interested reader can compare with [30], because for our purpose it is more convenient to investigate the analytic properties of other zeta functions, resulting by a suitable different decomposition of the analytic torsion, as described here below. For we define the torsion zeta function by t abs/rel s = m+ q qζs, q abs/rel. q= It is clear that the analytic torsion of C l W is in the following we will use the simplified notation T C l W for T C l W, g; ρ log T abs/rel C l W = t abs/rel 0. Our first result is a Poincaré duality compare with Proposition.4, [6] and the result of [7]. Theorem 4.. Poincaré duality for the analytic torsion of a cone. Let W, g be an orientable compact connected Riemannian manifold of dimension m, without boundary, then λ s, log T abs C l W = m log T rel C l W. Proof. By Hodge duality in equation.7, the Hodge operator sends forms of type,, 3, 4, E, and O into forms of type 4, 3,,, O, and E, respectively. Moreover, sends q-forms satisfying absolute boundary conditions, as in equation., into m+ q-forms satisfying relative boundary

15 ANALYTIC TORSION OF CONES 5 conditions, as in equation.3. Therefore, using the explicit description of the eigenvalues given in Lemma 3.3, it follows that Sp q abs = Sp m+ q rel. Using the formulas in equations 3., 3., and 3.3, and the eigenforms in Lemma 3., a straightforward calculation shows that the forms of type, 3, and E are coexact, and those of type, 4, and O are exact, and that the operator d sends forms of type, 3, and E in forms of type, 4, and O, respectively, with inverse d. Then, set { F q ccl,abs = m ccl,q,n : ĵµ q,n,α q,k/l } {m ccl,q,0 : ĵ αq,αq,k /l} { F q cl,abs = n,k= k= m cl,q,n : ĵµ q,n,α q,k/l } {m cl,q,0 : ĵ αq,αq,k /l} {m ccl,q,n : µq,n,k/l } n,k=, n,k= k= {m cl,q,n : µq,n,k/l } n,k=. F q ccl,abs is the set of the eigenvalues of the coclosed q-forms with absolute boundary conditions, is the set of the eigenvalues of the closed q-forms with absolute boundary conditions. and F q cl,abs Since obviously Sp q abs = F q ccl,abs F q q cl,abs, and F ccl,abs = F q+ cl,abs, we have that t abs s = m+ q qζs, q abs = m+ q qζs, m+ q rel = m t rel s + m+ m + m+ q ζs, q = m t rel s + m+ m + q ζs, F q+ q + ζs, F cl,abs = m t rel s. ccl,abs rel Since by definition log T abs W = t abs 0, the thesis follows. 5. The torsion zeta function of the cone over an odd dimensional manifold In this section we develop the main steps in order to obtain the proof of our theorems. This accounts essentially in the application of the tools described in Section.4 to some suitable sequences appearing in the definition of the torsion. So our first step is precisely to obtain this suitable description. This we do in this section. In the next two subsections, we will make the calculations necessary for the proof of our main theorems. We proceed assuming dimw = p odd, and assuming absolute boundary condition; for notational convenience, we will omit the abs subscript.

16 6 L. HARTMANN AND M. SPREAFICO Lemma 5.. Here ν,k = ĵ ν,0,k. ts = ls p q n,k= p ls + ls n,k= m cex,q,n s µ q,n,k ĵ s µ q,n,α q,k ĵ s µ q,n, α q,k m cex,p,n s p q rkh q C l W ; Q Proof. Using the eigenvalues in Lemma 3.3 l s ζs, q = m cex,q,n ĵ s µ + q,n,α q,k n,k= + n,k= n,k= k= µ p,n,k µ p,n,k s s α q,k s α q,k. m cex,q,n ĵ s µ q,n,α q,k + k= k= n,k= m cex,q,n s µ q,n,k m cex,q,n s µ + q,n,k m har,q,0 ĵ s α + q,α q,k m har,q,0 ĵ s α. q,α q,k Since for each fixed q, with 0 q p, q q m cex,q,n ĵ s µ + q,n,α q,k q+ q + n,k= + q+ q + + q q = q It follows that ts = ls k= n,k= p q n,k= m har,q,0 ĵ s α q,α q,k n,k= m cex,q,n s µ q,n,k + q+ q + m cex,q,n s µ q,n,k n,k= + q + q+ n,k= m cex,q,n ĵ s k= µ q,n,α q,k m cex,q,n ĵ s µ q,n,α q,k n,k= m har,q,0 ĵ s α q,α q,k + q+ m cex,q,n s µ q,n,k k= m cex,q,n s µ q,n,k ĵ s µ q,n,α q,k + ls p q+ m har,q,0 ĵ s α q,α q,k. k= m har,q,0 ĵ s α q,α q,k. Next, by Hodge duality on coexact q-forms on the section see equation.7 λ q,n = λ p q,n, and recalling the definition of the constants α q and µ q,n in Lemma 3., we have that α q = + q p + = q p + = α p q, and µ q,n = µ p q,n. Thus, fixing q with 0 q p, q = q n,k= n,k= m cex,q,n s µ q,n,k ĵ s µ q,n,α q,k + p q m cex,q,n s µ q,n,k ĵ s µ q,n, α q,k n,k= m cex,q,n s µ q,n,k ĵ s µ q,n,α q,k ĵ s µ q,n, α q,k,

17 ANALYTIC TORSION OF CONES 7 while when q = p, α q = 0. Therefore, ts = ls p q p ls + + ls n,k= n,k= p q+ m cex,q,n s µ q,n,k ĵ s µ q,n,α q,k ĵ s µ q,n, α q,k m cex,p,n s µ p,n,k µ p,n,k s k= m har,q,0 ĵ s α q,α q,k, where ν,k = ĵ ν,0,k are the zeros of J ν. Eventually, consider the last sum in the previous equation. We will use some classical properties of Bessel function, see for example [34]. Recall m = dim W = p, and therefore α q = q p + is an integer. Moreover, α q is negative for 0 q < p. Fixed such a q, we study the function Ĵ α q,α q z = α q J αq z + zj α q z. Since zj µz = zj µ+ z + µj µ z it follows that Ĵ α q,α q z = zj αq+z = zj αq z, and hence ĵ αq,α q,k = αq,k. Next, fix q with p < q p, such that α q is a positive integer. Then, since zj µz = zj µ z µj µ z, the function Ĵα q,α q z = α q J αq z+zj α q z coincides with zj αq z, and hence ĵ αq,α q,k = αq,k. Note that when q = p, α p = 0 and hence αp,α p,k = 0,k =,k. Summing up, p q+ k= m har,q,0 ĵ s α q,α q,k = p q+ and since by Hodge duality m q,0 = m p q,0, q+ p = k= q+ m har,q,0 p = m har,q,0 s α q,k + p k= p + k= q+ q=p m har,q,0 α s + p q,k k= k= m har,p,0 s k= s α q,k s α q,k. Since m har,q,0 = rkh q C l W ; Q, this completes the proof. m har,q,0 α s, q,k,k + p q k= m har,p,0 s,k m har,q,0 s α q,k It is convenient to introduce the following functions. We set Z q s = m cex,q,n s µ, Ż q,n,k q s = m cex,q,n µ q,n,k s, 5. Z q,± s = n,k= n,k= m cex,q,n ĵ s µ q,n,±α q,k, z qs = n,k= k= s α q,k s α q,k,

18 8 L. HARTMANN AND M. SPREAFICO for 0 q p, and 5. and 5.3 Then, ts = ls = ls t p s = Z p s Żp s, t q s = Z q s Z q,+ s Z q, s, 0 q p. p q Z q s Z q,+ s Z q, s + ls p q rkh q C l W ; Qz q s p ls p q t q s ls p q rkh q C l W ; Qz q s, log T C l W = t log l 0 = + p Z p s Żp s p q+ r q z q 0 + q t q 0 p p q+ r q z q0 + q t q0, where r q = rkh q C l W ; Q. In order to obtain the value of log T C l W we use Theorem. and its corollary applied to the functions z q s, Z q s, Ż q s, Z q,± s. More precisely, the functions z q were studied at the end of Section.4, and we will study the functions t q in Sections 5. and 5., and eventually we sum up on the forms degree q in Section The functions t q s, 0 q p. In this section we study the functions t q s. For we apply Theorem. to the double sequences S q = {m q,n : µ q,n,k } n= and S q,± = {m q,n : µ q,n±α q,k } n=, since we have that Z q s = ζs, S q, Z q,± s = ζs, S q,±, where q = 0,,..., p, α q = p q. Note that the sequence S q coincides with the sequence S p analysed in Section 5., with q = p. So we ust need to study the other two sequences. First, we verify Definition.. For we introduce the simple sequence U q = {m q,n : µ q,n } n=. Lemma 5.. For all 0 q p, the sequence U q is a totally regular sequence of spectral type with infinite order, eu q = gu q = p, and s ζs, U q = ζ cex, q + αq. The possible poles of ζs, U q are at s = p h, h = 0,, 4,..., and the residues are completely determined by the residues of the function ζ cex s, q, namely: p k k+ Res ζs, U q = s Res ζ cex s=k+ =0 s=k++, q αq. Proof. By definition U q = {m cex,q,n : µ q,n } n=, where by Lemmas 3. and 3.3 µ q,n = λ q,n + αq, and the λ q,n are the eigenvalues of the operator q on the compact manifold W. Counting such eigenvalues according to multiplicity of the associated coexact eigenform, since the dimension of

19 ANALYTIC TORSION OF CONES 9 the eigenspace of λ q,n are finite, and λ q,n n m for large n. This gives order and genus. The last formula follows expanding the powers of the binomial in the definition of the zeta function. Next, for c C, define the functions Ĵ ν,c z = cj ν z + zj νz. Recalling the series definition of the Bessel function [9] 8.40, we obtain that near z = 0 Ĵ ν,c z = + c z ν ν ν Γν. This means that the function z ν Ĵ ν,c z is an even function of z. Let ĵ ν,c,k be the positive zeros of Ĵν,cz arranged in increasing order. By the Hadamard factorization theorem, we have the product expansion + z ν Ĵ ν,c z = z ν Ĵ ν,c z z, ĵ ν,c,k and therefore Ĵ ν,c z = + c z ν ν ν Γν Next, recalling that when π < argz < π k= k= z ĵ ν,c,k. J ν iz = e π iν I ν z, J νiz = e π iν e π i I νz, we obtain Ĵν,ciz = e π iν ci ν z + zi νz. Thus, we define for π < argz < π 5.4 Î ν,c z = e π iν Ĵ ν,c iz, and hence 5.5 Î ν,±αq z = ±α q I ν z + zi νz = ± α q ν z ν ν Γν + z. ĵν,±α q,k Recalling the definition in equation.4 we have proved the following fact. Lemma 5.3. The logarithmic Gamma functions associated to the sequences S q,±,n have the following representations, when λ D θ,c, with c = min µ q,0, µ q,0,±α q, log Γ λ, S q,±,n = log + λ ĵµ q,n,±α q,k k= k= = log ε q,n,±α q λ + µ q,n log λ µ q,n log log Γµ q,n + log ± α q. µ q,n Proposition 5.. The double sequences S q,± have relative exponents p, p,, relative genus p, p, 0, and are spectrally decomposable over U q with power κ =, length l = p and domain D θ,c. The coefficients σ h appearing in equation.5 are σ h = h, with h =,,..., l = p.

20 0 L. HARTMANN AND M. SPREAFICO Proof. The values of the exponents and genus follow by classical estimates of the zeros of the Bessel functions [34], and zeta function theory. In particular, to determinate s 0 = p, we use the Young inequality and the Plana theorem as in [4]. Note that α >, since s =. As observed, the existence of a complete asymptotic expansion of the Gamma function follows by Lemma 5.3. This implies that S q,±,n are sequences of spectral type. A direct inspection of the expansions shows that S q,±,n are totally regular sequences of infinite order. The existence of the uniform expansion follows using the uniform expansions for the Bessel functions and their derivative given for example in [] 7.8 and Ex. 7., and classical expansion of the Euler Gamma function [9] We refer to [] Section 5 or to [3] Section 4 for details. This proves that S q,± are spectrally decomposable over U q, with power κ =. The length l of the decomposition is precisely p. For eu q = p, and therefore the larger integer such that σ h = h p is p. Remark 5.. Only the term with σ h =, σ h = 3,..., σ h = p namely h =, 4,..., p, appear in the formula of Theorem., since the unique poles of ζs, U q are at s =, s = 3,... s = p. Since we aim to apply the version of Theorem. given in Corollary., for linear combination of two spectrally decomposable sequences, we inspect directly the uniform asymptotic expansion of S q S q, S q,+. This give the functions φ σh. Lemma 5.4. We have the the following asymptotic expansions for large n, uniform in λ, for λ D θ,c, logγ λ, S q,n /µ q,n log Γ λ, S q,+,n /µ q,n log Γ λ, S q,,n /µ q,n = log I µq,n µ q,n λ + log Î µq,n,α q µ q,n λ + log Î µq,n, αq µ q,n λ log µ q,n log p = log λ + = α q µ q,n φ,q λ µ + O q,n µ q,n p Proof. Using the representations given in Lemmas 5.9 and 5.3, we obtain log Γ λ, S q,n /µ q,n log Γ λ, S q,+,n /µ q,n log Γ λ, S q,,n /µ q,n = log I µq,n µ q,n λ + log Î µq,n,α q µ q,n λ + log Î µq,n, αq µ q,n λ log µ q,n log α q µ q,n Recall the uniform expansions for the Bessel functions given for example in [] 7.8 pg. 376, and Ex. 7., +z e ν log z I ν νz = eν + +z p U z + πν + z 4 ν + O ν p, where U 0 w =,. =. U w = w w d dw U w + 8 w 0 5t U tdt,

21 ANALYTIC TORSION OF CONES with w = +z, and where I ννz = + z 4 e ν πνz V 0 w =, +z e ν log z + +z + p = V z ν + O ν p, V w =U w w w U w w w d dw U w. Using these expansions, we obtain the following expansion for Îν,±α q νz, Î ν,±αq νz = ±α q I ν νz + νzi ννz = ν + z e ν +z e ν log 4 π z + +z + p = W ±αq,z ν + O ν p, where W ±αq,z = V z ± This gives, αq U +z z. Thus, log Îν,±α q νz =ν + z + ν log z ν log + + z + log ν + 4 log + z p log πν + log + W ±αq,z ν + O ν p. log Γ λ, S q,n /µ q,n log Γ λ, S q,+,n /µ q,n log Γ λ, S q,,n /µ q,n p U λ = log λ log + = µ + O q,n µ p q,n p W p +αq, λ W + log + + O αq, λ µ p + log + q,n = µ q,n Expanding the logarithm as log + where a 0 =, a = l and l = a k= a z = = = = = l z, k a k l k, we have that µ q,n log Γ λ, S q,n /µ q,n log Γ λ, S q,+,n /µ q,n log Γ λ, S q,,n /µ q,n p = log λ + l λ + l + λ + l λ = p + l λ + l + λ + l = α q λ + + O µ p. q,n µ q,n µ + O q,n µ p q,n,

22 L. HARTMANN AND M. SPREAFICO where we denote by l λ the term in the expansion relative to the sequence S thus the one containing the U z and by l ± λ the terms relative to S ± thus the ones containing the W ±αq,z. Setting 5.6 the result follows. φ q, λ = l λ + l + λ + l λ φ q, λ = l λ + l + λ + l α q λ +, Remark 5.. Note that there are no logarithmic terms log µ q,n in the asymptotic expansion of the difference of the logarithmic Gamma function given in Lemma 5.4. So we can apply Corollary.. Next, we give some results on the functions φ,q λ, and Φ,q s defined in equation.6. Lemma 5.5. For all and all 0 q p, the functions φ,q λ are odd polynomial in w = λ φ,q λ = k=0 a,q,k w k+, φ,q λ = k=0 a,q,k w k+ + α q. The coefficients a,q,k are completely determined by the coefficients of the expansion given in Lemma 5.4. Proof. This follows by direct inspection of the last equality in the statement of Lemma 5.4. Lemma 5.6. For all and all 0 q p, φ,q 0 = 0. Proof. The proof is by induction on. We will consider all the functions as functions of w = λ. We use the following hypothesis for the induction, for k : 5.7 φ k,q = 0, φ k,q = 0, 5.8 l k αk l+ q k = k, l k l+ k = 0, where the functions φ,q λ are defined in equation 5.6, and the function lλ in the course of the proof of Lemma 5.4. First, we verify the hypothesis for =. The formulas in equation 5.8 follow by the definition when k =. For those in equation 5.7, we have by definition when k = that φ,q λ = l λ + l + λ + l λ = U λ + V λ + V λ + α q α q U 0 λ and = +, λ λ 3 φ,q λ = l λ + l + λ + l λ + α q = U λ + V λ + U λ V λ = 3 λ + λ 3 λ 3 +, and hence formulas in 5.7 are also verified when k =. Second we prove that all formulas hold for k =. Recalling that U k = V k for all k, we have from the definition that l l+ = U α q U U α q U k= k U k l k l+ k α qu k l k + l+ k,

23 ANALYTIC TORSION OF CONES 3 and hence, using the hypothesis we obtain l l+ = α qu + U k α k q k= = α q k= + k= k α qu k l k α k q k k α q U k l k α qu + α q U 3 l + kα q U k l k + α q = α q α qu + k= α q U + α q U = α q, thus proving the first formula in 5.8 for k =. For the first formula in 5.7, φ,q = l + l + + l, and hence φ,q = k= k= k U k l k l + k l k k α q U k l k l+ k, and using the induction hypothesis, and the previous formula with k = ust proved, this means that φ,q = k= k U k αi k k k αi k αiu k = 0. k k=

24 4 L. HARTMANN AND M. SPREAFICO and the first formula in 5.7 with k = follows. hypothesis, we have For the second formula in 5.8, using the l l+ = U α q U U α q U k= k = α q U + + k= k + U k l k l+ k + k= k α q U k k= k l k α k q k U k α k+ q k + + α qu k l k+ α q U k l k + l+ k = α q U + α qu + α q l k + α q U k l k k= = α q U + α q U + l + ku k l k = α q U + α q + α q U Eventually, for the second formula in 5.7 = 0. k= φ,q = l + l + + l = α q = k= + k= k= k k α q U k k α q + U k l k l + k l k U k α k q k l k l+ k k= α q U k α k+ q = 0, Corollary 5.. For all and all 0 q p, 0 p, Res s=0 Φ +,q s = 0. Next, we determine the terms A 0,0 0 and A 0,0, defined in equation.8. Lemma 5.7. For all 0 q p, A 0,0,q s = A 0,0,q s A 0,0,q,+ s A 0,0,q, s = log α q µ q,n n= A 0,,q s = A 0,,q s A 0,,q,+ s A 0,,q, s = ζs, U q. m q,n µ s, q,n

25 ANALYTIC TORSION OF CONES 5 Proof. For S q equation.8 reads for S q,± : A 0,0,q s = A 0,,q s = A 0,0,q,± s = A 0,,q,± s = n= n= n= n= m cex,q,n m cex,q,n m cex,q,n m cex,q,n a 0,0,n,q a 0,,n,q a 0,0,n,q,± a 0,,n,q,± p b,0,0,q µ + µ s = q,n q,n, p b,0,,q µ + µ s = q,n q,n ; p b,0,0,q,± µ + µ s = q,n q,n, p b,0,,q,± µ + µ s We need the expansions for large λ of l λ, l ± λ, for =,,..., p, log Γ λ, S q,n/µ q,n and log Γ λ, S q,±,n /µ q,n. Using classical expansion for Bessel functions and their derivative see [3] or [0] for details, we obtain log Γ λ, S q,n /µ q,n = log π + µ q,n + log Γµ q,n + + = q,n log µ q,n µ q,n log µ q,n + q,n. log λ + Oe µq,n λ. For S q,±, by the same expansions in the definition of the function Î, equation 5.5, we obtain ze z Î ν,±αq z + b k z k + Oe z, π and hence log Γ λ,s q,±,n /µ q,n = µ q,n λ + log Γµ q,n + µ q,n This gives a 0,0,n,q = log π + µ q,n + a 0,,n,q = µ q,n +, a 0,0,n,q,± = log π + µ q,n a 0,,n,q,± = µ q,n, k= log π + log λ + log µ q,n log µ q,n µ q,n log ± α q + Oe µq,n λ. µ q,n log µ q,n µ q,n log log Γµ q,n +, log µ q,n log µq,n Γµ q,n + log ± α q, µ q,n

26 6 L. HARTMANN AND M. SPREAFICO while the b,0,0,q, b,0,0,q,± all vanish since the functions l λ, l ± λ do not have constant terms. Therefore, a 0,0,n,q a 0,0,n,q,+ a 0,0,n,q, = log a 0,,n,q a 0,,n,q,+ a 0,,n,q, =, α q µ q,n, and the thesis follows. Applying Theorem. and its corollary, we obtain the values of t q 0 and t q0. Proposition 5.. For 0 q p, t q 0 = t q,reg 0 + t q,sing 0, t q0 = t q,reg0 + t q,sing0, where t q,reg 0 = ζ0, U q = ζ cex 0, q + α q t q,reg0 = A q,0,0 0 A q,0,0, t q,sing0 = p Res 0 Φ +,q s Res ζs, U q = s=0 s=+ =0 Proof. By definition in equations 5. and 5.,, t q,sing 0 = 0, p =0 s Res 0 Φ +,q s Res ζ cex s=0 s=+, q + αq. t q 0 =Z q 0 Z q,+ 0 Z q, 0, t q0 =Z q0 Z q,+0 Z q, 0. where Z q s = ζs, S q, and Z q,± s = ζs, S q,±. By Proposition 5. and Lemma 5.4, we can apply Theorem. and its Corollary to the linear combination above of these double zeta functions. The regular part of Z q 0 Z q,+ 0 Z q, 0 is then given in Lemma 5.7, while the singular part vanishes, since, by Corollary 5., the residues of the functions Φ k,q s at s = 0 vanish. The regular part of Z q0 Z q,+0 Z q, 0 again follows by Lemma 5.7. For the singular part, since by Proposition 5., κ =, l = p, and σ h = h, with 0 h p, by Remark 5. we need only the odd values of h = +, 0 p, and this gives the formula stated for t p,sing The function t p s. In this section we study the function t p s. For we apply Theorem. to the double sequences S p = {m p,n : µ p,n,k } n= and Ṡp = {m p,n : µ p,n,k } n=, since Z p s = ζs, S p, Ż p s = ζs, Ṡp. Spectral decomposition is with respect to the simple sequence U p = {m p,n : µ p,n } n=. Since the method is essentially the same as in the previous subsection, we ust state the results here. Lemma 5.8. The sequence U p is a totally regular sequence of spectral type with infinite order, eu p = gu p = p, and ζs, U p = ζ cex s, p, with possible simple poles at s = p h, h = 0,, 4,....