THE ANALYTIC TORSION OF THE FINITE METRIC CONE OVER A COMPACT MANIFOLD

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1 THE ANALYTIC TORSION OF THE FINITE METRIC CONE OVER A COMPACT MANIFOLD L. HARTMANN AND M. SPREAFICO Abstract. We give explicit formulas for the L analytic torsion of the finite metric cone over an oriented compact connected Riemannian manifold. We provide an interpretation of the different factors appearing in these formulas. We prove that the analytic torsion of the cone is the finite part of the limit obtained collapsing one of the boundaries, of the ratio of the analytic torsion of the frustum to a regularising factor. We show that the regularising factor comes from the set of the non square integrable eigenfunctions of the Laplace Beltrami operator on the cone.. Introduction and statement of the main result Let M, g be a compact connected oriented Riemannian manifold without boundary of dimension n with metric g. Let denotes the Hodge-Laplace operator on M relative to the metric g. Then, has a non negative discrete spectrum Sp and the zeta function of is well defined by ζs, q = λ s, λ Sp + q for Res > n, and by analytic continuation elsewhere, and is regular at s = 0. The analytic torsion T M, g of the pair M, g is defined by log T M, g = n q qζ 0, q. q= If the manifold M has a boundary M, the Laplace operator is assumed to be defined by suitable boundary conditions BC [8, Section 3]. In such a case, it is convenient to split the logarithm of the analytic torsion into two parts, the first being a global term and the second a local one, defined on a neighborhood of the boundary [4, Section 3]. The first term coincides with the Reidemeister torsion [4] of either M of the pair M, M, with the Ray and Singer homology basis [8], by the Cheeger Müller theorem [4] [5], so. the second term splits as log T abs M, g = log τm, g + log T bound,abs M, log T rel M, g = log τm, M, g + log T bound,rel M, log T bound,bc M = 4 χ M log + A BM,BC M, Date: May, Mathematics Subject Classification: 58J5.

2 L. HARTMANN AND M. SPREAFICO where χ is the Euler characteristic, and the last term is called the anomaly boundary term, and was described in the more general case in [] and []. It is clear that what is necessary in order to define the analytic torsion T M, g is that the spectrum of the Hodge-Laplace operator satisfies some assumptions that guarantee the possibility of defining the zeta function and of proving its regularity at s = 0. It is also clear that this follows by some spectral properties of the Hodge-Laplace operator. There are several approaches to describe these properties. We will follow the one of Spreafico, introduced in [] and []: so we require that Sp + is a graded regular sequence of spectral type of non positive order, as defined in [, Definitions.,.6]. The fact that this is true for the Hodge-Laplace operator on a compact manifold is well know. The given definition of the analytic torsion extends easily considering forms with values in some vector bundle V ρ associated to some orthogonal representation ρ of the fundamental group of W [8, Section ]. Under our approach what is necessary is that the spectrum of the resulting operator is a regular sequence of spectral type of non positive order, and again this is well known. Since the results of this paper are independent of these extensions, we will consider the simpler case of the Hodge-Laplace operator itself. An other possible generalization of the given definition of analytic torsion, and this is the case that we will consider here, is when the underlying space is no longer a compact manifold, but some type of open manifold, or manifold with singularities. In this paper we consider the case of the cone over a manifold CW, as defined below. Definition.. Let W, g be an oriented compact connected Riemannian manifold of dimension m without boundary with metric g. Let 0 l l be real numbers. Consider the space C [l,l ]W = [l, l ] W with the metric defined for x > 0 when l = 0 dx dx + x g. We call C [l>0,l ]W the finite metric frustum over W ; we call C 0,l] W the finite metric cone over W, and we denote it by C l W. It is clear that in order to obtain a suitable extension of the definition of the analytic torsion to the cone, we need first a suitable definition of the Hodge-Laplace operator. Spectral analysis on cones was developed by Cheeger [5] [6], that in particular showed that the formal Hodge-Laplace operator has a self adjoint extension on the space of square integrable forms. A complete set of solutions of the eigenvalues equation for can be described in terms of a complete discrete resolution of the Hodge-Lapalce operator on the section of the cone, see Lemma 3. [5, Section 3]. Considering square integrable forms and applying the boundary conditions, we obtain an explicit description of the spectrum of in terms of the spectrum of the Hodge-Laplace operator on the section, see Lemmas 3.. Remark.. It is important to observe here that beside the usual either absolute or relative boundary conditions, due to the presence of a non empty boundary, when the section W has even dimension m = p, further boundary conditions, called ideal boundary conditions, where introduced by Cheeger [5], as we recall here. Assume that there exists a decomposition H p W = V a V r, where V a and V r are maximal self annihilating subspaces for the cup product paring. Then, a p dimensional form belongs to the domain of the Laplace operator if its components in V a and V r satisfy Neumann and Dirichlet conditions respectively at x = 0 see [5] pg. 580 for details. Observe that our determination of the spectrum of the cone is obtained assuming this decomposition and these conditions. Moreover, where ever not explicitly stated, the above decomposition and the

3 ANALYTIC TORSION OF CONES 3 ideal conditions will be assumed. We conclude this remark recalling that ideal boundary conditions are necessary to guarantee Poincaré duality on the cone. We can then prove that Sp + is a regular sequence of spectral type and therefore the analytic torsion of the cone is well defined. We are in the position of extending the above definition of the analytic torsion to this setting, and we call the resulting object the L analytic torsion of the cone, T abs,ideal C l W we use the same notation and we restrict to absolute BC, since the relative torsion follows by Poincaré duality [0, Section 4]. Of course it is not obvious at all if the invariant obtained in this way as some geometric or topological meaning. Since the spaces of the L harmonic forms on W as proved to be isomorphic to the intersection cohomology of W, the natural candidate for the L analytic torsion is the intersection torsion. It was proved in [] that indeed L analytic torsion and intersection torsion of the cone over an odd dimensional manifold coincide. The case of an even dimensional section is not clear yet, due to some difficulty in producing a natural definition for the intersection torsion in this case. There is work in progress in this direction. Here we tackle the analytic side of the problem. The main purpose of this work is to compute T abs,ideal C l W, or more precisely to give formulas for it in terms of other either geometric or spectral invariants, and in particular invariants of the section W, g. A key point here is the following. The description of the spectrum of the Hodge- Laplace operator on the cone in terms of the spectrum of the Hodge-Laplace operator on the section makes possible to express or decompose all the spectral functions in particular the zeta function and the logarithmic Gamma function, see Subsection 5 on the cone in terms of the spectral functions on the sections, and to apply the technique introduced in [] in order to tackle the derivative at zero of a class of double series. This is the main technical point, and in fact this is the reason that permits to obtain the final results. We did follow the same approach first in [9], where we gave the formula for the torsion of the cone over a sphere, and then in [0], where we considered as section any compact connected manifold of odd dimension. These results are superseded by the formulas obtained in the present work, where the section can have any dimension. We stress the fact that, as observed in [0], the calculation are exactly the same and there is no more difficulty to deal with the even dimensional section case that with the odd dimensional one. Moreover, while the formulas in the odd dimensional section case have a clear geometric interpretation in terms of intersection torsion, due to [], the even dimensional section case is still quite obscure. However, due to the recent interest in the subject see [6], we decided to present the formulas for the general case and the details of the calculation. Indeed, the presentation followed in this paper adds some insights in the possible interpretation of the result, as we explain at the end of this section, but first we present our first main result in the following theorem. Theorem.. Let W m, g be an oriented compact connected Riemannian manifold of dimension m with metric g. The L analytic torsion T abs,ideal C l W of the cone over W with absolute boundary and ideal conditions is as follows, where r q = rkh q W, and m cex,q,n, α q and are defined in Lemma 3., p : log T abs C l W p = p q p qr q log l + log T W, g p q r q logp q + A BM,abs C l W,

4 4 L. HARTMANN AND M. SPREAFICO log T abs,ideal C l W p p = q p q + r q + p 4 r p log l p q r q logp q + p q!! + p q n= m cex,q,n log + αq αq + χw log + A BM,abs C l W. Next, we give an interpretation of the formulas given in Theorem.. This is done in two steps. In the first, described in whole details in Section, we show that all the sums appearing in the formula of the torsion in the odd case, and some of these sums in the even case, coincide with the determinant of the change of basis between the basis of the harmonic forms on the cone and the basis of the harmonic forms on the sections, see Theorem. of Section. This is a classical geometric contribution in the Reidemeister torsion of the cone see [], and therefore this interpretation clarifies completely the appearance of these sums. In particular, this describes completely the L - analytic torsion of a cone over an odd dimensional manifold, and also suggests that the Euler part of the boundary contribution is anomalous in the analytic torsion see Remark. at the end of Section. In the other case, i.e. for a cone over an even dimensional manifold, other sums appear in the formula for the analytic torsion. An interpretation of these further sums, collected into the two terms B and B in Theorem., is the second step of our interpretation of the formulas for the analytic torsion, and is obtained in the final section of the paper. The interpretation is suggested after proving that the formulas for the torsion can be obtained as some limit case of the formula for the torsion of the frustum. We pospone the discussion of these results to the last section. We conclude this introductory section with a few remarks on the anomaly term appearing in the even case. We observe that the anomaly term can be also written in terms of residues of some spectral functions, see Subsection 8.., or Theorem. of [9]. Formulas for particular sections, as spheres of discs, are given in [9] and [8] [7]. Remark.. By duality: p q n= m cex,q,n log + αq = p q αq n= m cex,q,n log + α q. Remark.3. Using Proposition.9 of [] the first term in the last line of the second formula in the theorem reads p q n= m cex,q,n log + αq = p q ζ αq q0, α q ζ q0, α q, = p det ζ q q W,g + α q α q log, det ζ q W,g + α q + α q

5 ANALYTIC TORSION OF CONES 5 where W,g is the Hodge-Laplace operator on the section of the cone W, g, and ζ q s, x = m cex,q,n + x s. n=. A geometric interpretation of the analytic torsion The spaces of harmonic forms on the frustum with absolute an mixed BC were computed in []. The space of harmonic forms on the cone with absolute and relative BC were compute []. In particular, in the even case m = p the result changes if we assume the Cheeger ideal BC, as described in the introduction. A simple calculation, proceeding as in the proof of Lemma 4. of [], gives the following result. Lemma.. We have the following isomorphisms of vector spaces induced by the extension of the inclusion of the forms: { H q abs C H q W, 0 q p, lw = dim W = p, {0}, p q p, { H q abs C H q W, 0 q p, lw = dim W = p, {0}, p + q p +, H q W if 0 q p, H q abs,ideal C lw = V a if q = p, dim W = p, 0 if p + q p +, H q abs C [l,l ]W = H q W. Proceeding as in Section 3.5 of [], we have a the following commutative diagram of isomorphisms of vector spaces we give here the diagram for the cone, the one for the frustum is in [], for 0 q [ ] m, H q abs C lw C l W H m q+ C l W A m q+ C l W,rel H m q+ ĈlW, Cl P W H q C l W γq W rel q x m q dx γq A m q W P q γq H q W H m q W H m q Ŵ H q W where A is the de Rham map on the cone, P Poincarè duality, the Hodge isomorphism, and the factor γ q is the ratio between the L norm of the constant extension of a form ω on the cone and the L norm of ω the double dot indicates the costant extension: l γ q = ω C l C l W ω = x m q dx = lm q+ W 0 m q +. The same analysis on the frustum gives see [, Section 3.5] for all definitions and details Γ q = ω F C [l,l ]W ω W = Note that for all q < [ m l x m q dx = l ], { m+ q lim Γ q = γ q. l 0 +,l =l l m+ q l m+ q if m + q 0, ln l l, if m + q = 0.

6 6 L. HARTMANN AND M. SPREAFICO Recall that for a real vector space V we denote Λ dim V V by detv, and we call this line the determinant line of V det0 := R. If v = {v,..., v dim V } is a basis for V, we use the notation detv for v v dim V. For a finite dimensional graded vector space V = V q q Z = q Z V q, set m Det m n V := detv q q, q=n where V p = V and V p+ = V. This one dimensional vector space is called the determinant line of V. With this notation, if α is a graded basis for the harmonic forms on W, we have that the ratio between the determinant of the homology graded basis induced by an harmonic basis of the frustum and of its section, and of the cone and of this section are, respectively: Det α m F Detα = Det [ m ] 0 α Cl Det [ m ] 0 α = Det p 0 α Cl Det p 0 α = γ [ m ] q rq Γq, absolute BC p rp 4 p q rq γq, absolute BC p q rq γq, m = p, absolute, ideal BC. For completeness we gave here also the result without ideal BC here, however in the following the formulas are all given assuming ideal BC. A simple calculation completes the proof of the following corollary of Theorem. a similar formula holds in the case m = 0. Theorem.. The analytic torsion of the cone reads: log T abs,ideal C l W m = m ] Det[ 0 α log T W, g + log Det [ m ] 0 α + B m C l W + B m C l W, + 4 χw log + A BS,abs C l W where B m / C lw are the following anomaly terms vanishing when m = p is odd: p C l W = q r q logp q!! + B p p q n= m cex,q,n log + αq αq, B p C l W = χw log. 4 The analytic torsion of the frustum reads: Det α log T abs C [l,l ]W = log T W, g + log Detα + χw log + A BS,abs C [l,l ]W. Remark.. As announced in the introduction, the formula in Theorem. completely describe the analytic torsion of a cone over an odd dimensional manifold in terms of topological and geometric quantities. In particular, the splitting between global and boundary terms is evident compare with the formula in equation. for a regular manifold, and the discussion in Section 6. In the even case, the same analysis suggests the description of the torsion given in the formula, with two new

7 ANALYTIC TORSION OF CONES 7 anomaly terms, called B and B. If now we read the formula for the torsion in the even case looking for the global and the boundary contributions, we find out part of the boundary contribution can be interpret as anomalous. This is justified as follow. Recall that the boundary term splits into two parts: one is the classical part, see the work of Lück [3] and depends on the Euler characteristic of the boundary, the other is the anomaly boundary term, computed by Brüning and Ma [, ]. Then, we realise that collecting together all the terms where the Euler characteristic appears, the classical boundary term in the cone coincides with the one of the frustum, even if the boundary is different. More precisely, it seems that the analytic torsion of the cone does not see that the boundary at x = 0 collapse to a point. Following this interpretation, we splitter the term into two parts: the first is χw log, 4 and is classical contribution of the boundary, the second one is B W, and is understood as an anomaly boundary term. 3. Spectral properties of the Hodge-Laplace operator Consider the metric in Definition. either on the cone C l W or on the frustum C [l>0,l ]W. Let ω Ω q C l W ω Ω q C [l>0,l ]W, set ωx, y = f xω y + f xdx ω y, with smooth functions f and f, and ω j ΩW. Then we denote operators acting on the section by a tilde, 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. Lemma 3. Cheeger [5]. Let {ϕ q har, ϕq cex,n, ϕ q ex,n} be an orthonormal basis of Ω q W consisting of harmonic, coexact and exact eigenforms of q. Let λ q,n denotes the eigenvalue of ϕ cex,n q and m cex,q,n its multiplicity. Let J ν be the Bessel function of index ν. Define α q = + q m, = λ q,n + α q.

8 8 L. HARTMANN AND M. SPREAFICO Then, assuming that is not an integer, the solutions of the equation u = λ u, with λ 0, are 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 the index is an integer the solutions must be modified including some logarithmic term see for example [3] for a set of linear independent solutions of the Bessel equation. Following [5] and [3], the formal Hodge-Laplace operator in equation 3. defines a concrete self adjoint operator with domain in the space of the square integrable forms on the cone see [9] for details. This operator that we denote by the same symbol has a pure point spectrum as described in the following lemmas, whose proof follows applying BC and square integrability to the solutions described in Lemma 3.. Lemma 3.. The positive part of the spectrum of the Hodge-Laplace operator on C l W, with absolute boundary conditions on C l W is as follows, where 0 q m+. If m = dim W = p, p : Sp + q abs = { m cex,q,n : ĵµ q,n,α q,k/l } {m cex,q,n : ĵµq,n,αq,k/l } n,k= {m cex,q,n : jµq,n,k/l } {m cex,q,n : jµq,n,k/l } n,k= n,k= {m har,q : ĵ αq,αq,k /l} {m har,q : ĵ αq,αq,k /l}. k= If m = dim W = p, p : Sp + {m q p,p+ cex,q,n : ĵµq,n,αq,k/l } abs = n,k= {m cex,q,n : jµq,n,k/l } {m har,q : ĵ αq,αq,k /l} { Sp + p abs = m cex,p,n : ĵµ p,n,α p,k/l } {m cex,p,n : jµp,n,k/l } { } m har,p : j /l k= n,k= n,k= k= k= n,k= {m cex,q,n : ĵµq,n,αq,k/l } {m cex,q,n : jµq,n,k/l } n,k= {m har,q : ĵ αq,αq,k /l}, n,k= {m cex,p,n : ĵµp,n,αp,k/l } {m cex,p,n : jµp,n,k/l } { m har,p : j /l } k= n,k= n,k= k= n,k= {m har,p : ĵ αp,αp,k /l} k=,

9 Sp + p+ abs = {m cex,p+,n : ĵ µp+,n,αp+,k/l } ANALYTIC TORSION OF CONES 9 n,k= {m cex,p,n : ĵµp,n,αp,k/l } n,k= {m cex,p,n : jµp,n,k/l } {m cex,p,n : jµp,n,k/l } n,k= n,k= { {m har,p+ : ĵ αp+,αp+,k /l} k= m har,p : j /l } { } k= m har,p : j /l, k= where the j µ,k are the positive zeros of the Bessel function J µ x, the ĵ µ,c,k are the positive zeros of the function Ĵµ,cx = cj µ x + xj µx, c R, α q and are defined in Lemma 3.. Lemma 3.3. The positive part of the spectrum of the Hodge-Laplace operator on C [l,l ]W, l > 0, with relative boundary conditions on C [l,l ]W and absolute boundary conditions on C [l,l ]W is 0 q m + : { Sp + q rel,abs = m cex,q,n : ˆf } { µ q,n,α q,kl, l m cex,q,n : ˆf } µ q,n,α n,k= q,kl, l n,k= { m cex,q,n : ˆf } { µ q,n, α q,kl, l m cex,q,n : ˆf µ q,n, α q,kl, l { m har,q : ˆf α q,α q,k l, l } k= where the ˆf µ,c,k a, b are the zeros of the function n,k= { m har,q : ˆf α q,α q,k l, l ˆF µ,c x; l, l = J µ l xcy µ l x + l xy µl x Y µ l xcj µ l x + l xj µl x, with real c 0, and α q and as defined in Lemma 3.. Remark 3.. Application of the BC in Lemma 3. would give the zeros of the function J µ l xcj µ l x + l xj µl x J µ l xcj µ l x + l xj µl x = sinπµ ˆF µ,c x; l, l, however, for the following analysis, it is much more convenient to work with the function Y instead that with the function J. Lemma 3.4. The positive part of the spectrum of the Hodge-Laplace operator on C [l,l ]W, l > 0, with absolute boundary conditions is 0 q m + : } { } Sp + q abs {m = cex,q,n : ˆv µ q,n,α q,k m cex,q,n : ˆv µ q,n,α q,k n,k= n,k= { } { } m cex,q,n : vµ q,n,k m cex,q,n : vµ q,n,k n,k= n,k= { } { } m har,q : ˆv α q,α q,k m har,q : ˆv α q,α q,k, where the v µ,k are the zeros of the function and the ˆv µ,c,k are the zeros of the function k= Υ µ x = J µ l xy µ l x Y µ l xj µ l x, ˆΥ µ,c x = cj µ l x+l xj µl xcy µ l x+l xy µl x cy µ l x+l xy µl xcj µ l x+l xj µl x, with real c 0, and α q and as defined in Lemma 3.. k= } k=, } n,k=

10 0 L. HARTMANN AND M. SPREAFICO We define the torsion zeta function by and the analytic torsion is: 4. Simplifying the torsion zeta function t M ns = n q qζs, q, q= log T M n, g = t M n 0. In this section, we consider the torsion zeta function for the cone with absolute BC and for the frustum with mixed and absolute BC respectively. We proceed to some simplifications of it. We present the proof in the case of the cone, the proof for the frustum is analogous. Lemma 4.. The torsion zeta function of the cone is: with t m Cone s = tm 0 s + t m s + t m s + t m 3 s, [ m t m 0 s = ls ] q Z q s Ẑq,+s + m Z q s Ẑq, s, t p p ls s = Z p s Ẑp,0s, t p s = 0 t m s = ls [ m ] q+ m har,q z q, s + m z q, s, t p 3 s = p+ m har,p l s 4 z p,+s + z p, s, t p 3 s = 0, where Ẑq,0 denotes Ẑq,± with α q = 0, and Z q s = n,k= m cex,q,n j s,k, Ẑ q,± s = n,k= m cex,q,n ĵ s,±α q,k, z q,±s = Proof. Rearranging the sums and isolating the case q = p, α p = m = p, t p Cone s = ls p q if m = p, p t p Cone s =ls q p+ ls + 4 n,k= n,k= m cex,q,n j s µ q,n,k ĵ s,α q,k + ls p q+ m cex,q,n j s µ q,n,k ĵ s,α q,k + ls m har,p k= j s + j s,k,k. p,q p k= j s ±α q,k. when m = p, we obtain: if q+ k= k= m har,q ĵ s α q,α q,k, m har,q ĵ s α q,α q,k

11 ANALYTIC TORSION OF CONES Using Hodge duality on coexact q-forms on the section, we obtain the following identities: m cex,q,n = m cex,m q,n, λ q,n = λ m q,n, α q = α m q, = µ m q,n, and hence the first term in the previous equations reads: l s m q n,k= [ m = ls ] m cex,q,n j s µ q,n,k ĵ s,α q,k m cex,q,n j s µ q,n,k ĵ s,α q,k + m j s µ q,n,k ĵ s, α q,k 4. q n,k= [ m ] + term with q =, where the last term appears only if [ ] m is an integer, namely if m = p is odd, and has the following form. Since when q = p, m = p, α p = 0, and j ν,0,k = λ p,n = j ν,k, then [ m ] p ls term with q = = m cex,p,n j s µ p,n,k ĵ s µ p,n,0,k. n,k= The sign in front to the second term in equation 4. is the key difference between the even and the odd case. Next consider the term involving the harmonics, i.e. the sums k p s = p q+ Consider the function Since k= m har,q ĵ s α q,α q,k, kp s = p,q p Ĵ αq,α q x = α q J αq x + lxj α q x, q+ zz µz = zz µ+ z + µz µ z, and zz µz = zz µ z µz µ z, where Z is either J + or J, it follows that Ẑ αq,α q z := α q Z αq z + zz α q z = zz αq+z = zz αq z, Ẑ αq,α q z := α q Z αq z + zz α q z = zz αq z = zz αq z. k= m har,q ĵ s α q,α q,k. This permit to simplify Ĵ α q,α q as follows. If α q is negative, then Ĵ αq,α q x = xj αq x, and this means that ĵ αq,α q,k = j αq,k, for these q. If is positive, then Ĵ αq,α q x = xj αq x, and this means that ĵ αq,α q,k = j αq,k, for these q. When α q = 0 and this happens only if m = p, q = p, we have Ĵ0,0x = xj ± x, and hence we can use either j,k or j,k. Next,

12 L. HARTMANN AND M. SPREAFICO if m = p, α q is negative for 0 q p, α p = 0, and α q is positive for p q p, whence k p s = p q+ k= m har,q j α s + m har, q,k p j s + k= α,k p q+ if m = p, α q is negative for 0 q p, and α q is positive for p q p, whence k p s = p q+ k= m har,q j α s + q,k p q=p q+ q=p k= m har,q j s, α q,k k= m har,q j s, α q,k and since by Poincarré duality m har,q = m har,m q, and α m q = α q, we have the thesis note that when m = p, α p =. For the frustum, we define for c > 0 the function F ν x; l, l = J ν l xy ν l x Y ν l xj ν l x, and let f ν,k l, l denote its zeros. Then, we have the following result. Lemma 4.. The torsion zeta function of the frustum with mixed BC is with t m Frustum,mix s = wm 0 s + w m s + w m s + w m 3 s, w m 0 s = [ m ] q ˆD q, s; l, l ˆD q,+ s; l, l + m ˆD q,+ s; l, l ˆD q, s; l, l, w p s = p ˆDp,0 s; l, l ˆD p,0 s; l, l w m s = [ m ] q+ d q s; l, l + m d q s; l, l,, w p s = 0, w p 3 s = p+ d ps; l, l, w p 3 s = 0, where ˆD p,0 is given either by ˆD p,+ or ˆD p,, since α p = 0, when m = p, and ˆD q,± s; l, l = n,k= m cex,q,n ˆf s,±α q,k l, l, d q s; l, l = m har,q Lemma 4.3. The torsion zeta function on the frustum with absolute BC reads t m Frustum,abs s = ym 0 s + y m s + y m s + y m 3 s, k= f s α q,k l, l.

13 ANALYTIC TORSION OF CONES 3 where: y m 0 s = [ m ] q E q s Êq,+s + m E q s Êq, s, y p s = p y m s = [ m ] E p s Êp,0s, y p s = 0, q+ m har,q e q, s + m e q, s, y p 3 s = p+ m har,p e p, s, y p 3 s = 0, where Êq,0 denotes Êq,± with α q = 0, and E q s = m cex,q,n υ s µ, Ê q,n,k q,± s = n,k= n,k= 5. Zeta determinants m cex,q,nˆυ s,±α q,k, e q, s = k= υ s α q,k. We recall in this section the main points of the technique that we will use to compute the derivative at s = 0 of the zeta functions appearing in the torsion zeta functions introduced in Section 4. This section is essentially contained in Section 4 of [9], to which we refer for details, and based on []. Given a sequence S = {a n } n= of spectral type, we define the zeta function by ζs, S = when Res > es, and by analytic continuation otherwise, and for all λ ρs = C S, we define the Gamma function by the canonical product, 5. Γ λ, S = + λ gs j λ j j= j e a j 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 5.. 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= the sequence u κ n S n = u has spectral sector Σ κ θ,c, and is a totally regular sequence n k= of spectral type of infinite order for each n; a s n,

14 4 L. HARTMANN AND M. SPREAFICO 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. Define the following functions, Λ θ,c = { z C argz c = θ }, oriented counter clockwise: 5.3 Φ σh s = 0 t s e πi Λθ,c λt λ φ σ h λdλdt. By Lemma 3.3 of [], for all n, we have the expansions: p log Γ λ, S n /u κ n a αj,0,n λ αj + a k,,n λ k log λ, 5.4 φ σh λ j=0 k=0 p b σh,α j,0 λ αj + b σh,k, λ k log λ, j=0 for large λ in D θ,c. We set see Lemma 3.5 of [] l A 0,0 s = a 0,0,n 5.5 A j, s = n= a j,,n n= k=0 bσh,0,0u σ h n h=0 l bσh,j,u σ h n h=0 u κs n, u κs n, 0 j p, where the notation means that only the terms such that ζs, U has a pole at s = σ h appear in the sum. Theorem 5.. Let S be spectrally decomposable over U as in Definition 5.. 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 + l Res Φ σh s Res ζs, U, κ s=0 s=σ h h=0 ζ 0, S = A 0,0 0 A 0,0 + γ κ + κ l h=0 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 5.. 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.

15 ANALYTIC TORSION OF CONES 5 Corollary 5.. Let S j = {λ j,n,k } n,k=, j =,..., J, be a finite set of double sequences that satisfy all the requirements of Definition 5. of spectral decomposability over a common sequence U, with the same parameters κ, l, etc., except that the polynomials P j,ρ λ appearing in condition do not vanish for λ = 0. Assume that some linear combination J j= c jp j,ρ λ, with complex coefficients, of such polynomials does satisfy this condition, namely that J j= c jp j,ρ λ = 0. Then, the linear combination of the zeta function J j= c jζs, S j is regular at s = 0 and satisfies the linear combination of the formulas given in Theorem 5.. 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 of [9]. For positive real l and q, define the non homogeneous quadratic Bessel zeta function by zs, ν, q, l = k= j ν,k l + q s, 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 5.6 z0, ν, q, l = ν +, z 0, ν, q, l = log πl I νlq q ν. In particular, taking the limit for q 0, 5.7 z 0, ν, 0, l = log πl ν+ ν Γν Decomposition of the torsion zeta function Inspection of the formulas in the lemmas of Section 4 shows that the torsion zeta function is the finite sum of some simple and some double series: t Cone,abs s = t 0 s + t s + t s + t 3 s, t Frustum,mixed s = w 0 s + w s + w s + w 3 s, t Frustum,abs s =y 0 s + y s + y s + y 3 s, where t 0, t, w 0, w, and y 0, y are double series, and the others simple series. Simple series can be treated by using the formulas appearing at the end of Section 5. In order to deal with the double series, applying the spectral decomposition Theorem 5., the derivative at zero of t 0, t, w 0, w, and y 0, y decomposes into two parts, called regular and singular contribution see Remark 5., and this gives the following decomposition of the analytic torsion in the following we assume l > 0, log T abs,ideal C l W = t 0,reg0 + t,reg0 + t 0,sing0 + t,sing0 + t 0 + t 30, log T mixed C [l,l ]W = w 0,reg0 + w,reg0 + w 0,sing0 + w,sing0 + w 0 + w 30, log T abs C [l,l ]W = y m 0,reg s + ym,reg s + ym 0,sing s + ym,sing s + ym s + y m 3 s.

16 6 L. HARTMANN AND M. SPREAFICO On the other side, for the frustum, that is a manifold, we also have the decomposition log T mixed C [l,l ]W = log τc [l,l ]W, {l } W + 4 χ C [l,l ]W log + A BM,mixed C [l,l ]W, log T abs C [l,l ]W = log τc [l,l ]W + 4 χ C [l,l ]W log + A BM,abs C [l,l ]W. We will prove in Section 9 that 6. log τc [l,l ]W, {l } W + χw log = w 0,reg0 + w,reg0 + w 0 + w 30, A BM,mixed C [l,l ]W = w 0,sing0 + w,sing0, log τc [l,l 6. ]W + χw log = y 0,reg0 + y,reg0 + y 0 + y 30, A BM,abs C [l,l ]W = y 0,sing0 + y,sing0. This suggests to introduce a similar decomposition for the cone log T abs,ideal C l W = log T global C l W + 4 χ C lw log + A BM,abs C l W, with log T global C l W χ C lw log = t 0,reg0 + t,reg0 + t 0 + t 30, A BM,abs C l W = t 0,sing0 + t,sing0. We will call the first term, namely global plus Euler, the regular part of the torsion, and the second term the anomaly boundary term. We will prove in Section 0 that the notation is justified by the fact that t 0,sing 0 + t,sing 0 coincides with the term that we would obtain if we would apply the formula of Brüning and Ma for the cone as if it were a regular manifold. 7. Calculations I: application of Definition 5. In both the cases of the cone and of the frustum, we prove that the double series are spectrally decomposable according to Definition 5. see detail in [0] on the same simple sequence, that we discuss now. The simple sequence is U q = {m q,n : } n=. This is a totally regular sequence of spectral type with infinite order, eu q = gu q = m = dim W, and the associated zeta functions is s ζs, U q = ζ cex, q + αq. The possible poles of ζs, U q are at s = m h, h = 0,, 4,..., and the residues are completely determined by the residues of the function ζ cex s, q [0, Lemma 5.]. We give in this section complete calculations for the cone and for the frustum with mixed BC, we omit the calculations for the frustum with absolute BC that are very similar to ones for the frustum with mixed BC.

17 ANALYTIC TORSION OF CONES The cone. We consider the double series Z q s = m cex,q,n j s µ, Ẑ q,n,k q,± s = n,k= n,k= m cex,q,n ĵ s,±α q,k, appearing in Lemma 4.. These are the zeta functions associated to the double sequences S q = {m cex,q,n : jµ } and q,n,k Ŝq,± = {m cex,q,n : ĵµ q,n,±α q,k }. It is easy to see that these sequences have power κ =. By Definition 5., the relevant spectral function associated to S q is the function compare with equation 5.: log Γ λ, S q = log + λ jµ q,n,k k= = log I µq,n λ + log λ log log Γ +, and the relevant spectral function associated to Ŝq,± is the function: log Γ λ, Ŝq,± = log + λ ĵµ q,n,±α q,k k= where for π < argz π = log Î,±α q λ + log λ log µq,n Γ + log Îν,cz = e π iν Ĵ ν,c iz. ± α q, First, we need uniform asymptotic expansions of the function log Γ for large n. Such expansions can be obtained from those of the Bessel function see [0, Section 5.] for details. For large n, uniformly in λ, compare with equation 5. log Γ λ, S q /µ q,n = log λ log Γ + λ 7. log λ + log + λ + log π + log λ 4 + m φ q,j λ j= µ j q,n + Oµ p q,n, where φ q,j λ is the coefficient of µ j q,n in the asymptotic expansion of for the definition of the function U and W see Lemma 7. below for large, i.e. 7. log + log k= + k= U k λµ k q,n U k λµ k q,n = j=, φ q,j λµ j q,n.

18 8 L. HARTMANN AND M. SPREAFICO and 7.3 log Γ λ, Ŝq,±/µ q,n = log λ log Γ + + log ± α q λ µq,n log λ + log + λ + log π 4 log λ + m j=0 ˆψ q,j,± λ µ j q,n + Oµ p q,n, where ˆψ q,j,± λ is the coefficient of µ j q,n in the asymptotic expansion of log + W ±αq,kλµ k q,n, for large, i.e. log + k= k= W ±αq,kλµ k q,n therefore the complete coefficient of µ j q,n in log Γ λ, Ŝq,±/µ q,n is = j= ˆψ q,j,± λµ j q,n, 7.4 ˆφq,j,± λ = ˆψ q,j,± λ + j+ ±α q j. j Second, we need asymptotic expansions of the function log Γ and of the functions φ q,j and ˆφ q,j,± for large λ. The expansion of log Γ can be obtained from those of the Bessel function [0, pg.s 64, 64] compare with equation log Γ λ, S q /µ q,n = log π + + log Γ + + log log + log λ + Oe µq,n λ, 7.6 log Γ λ, Ŝq,±/µ q,n = λ + + log π + log λ + log log log µq,n Γ ± α q + Oe µq,n λ. About the functions φ q,j λ and ˆφ q,j,± λ we have the following facts. Lemma 7.. For j = 0, φ q,0 λ = log π + log λ, 4 ˆφ q,0,± λ = log π log λ. 4 For j > 0, the functions φ q,j λ and ˆφ q,j,± λ are polynomial of order 3j in λ, whose parity only depends on the index j, and with minimal monomial of degree j, and satisfy the following

19 ANALYTIC TORSION OF CONES 9 formulas: φ q,k λ ˆφ q,k,+ λ ˆφ q,k, λ ˆφq,k, λ ˆφ q,k,+ λ λ=0 λ=0 = 0, = 0. Proof. The case j = 0 follows by inspection of the formulas in equations 7. and 7.3. Next, assume j > 0. Recall the definition of the functions U j λ and V j λ [7, 7.0 and Ex. 7.]. Let u 0 w =, u j+ w = w w u jw + 8 w 0 5y u j ydy, v 0 w =, v j+ w = u j+ w w w u j w w w u w, then U j λ = u j, V j λ = v j. λ λ By these formulas, it is clear that the functions U j and V j are polynomials in λ with the stated properties. Since, [0, pg. 639] W ±αq,jλ = V j λ ± α q U j λ, λ the same is true for W j λ. Whence the first part of the statement follows by the very definition of the functions φ q,j λ and ˆφ q,j,± λ, since in the functions φ q,k λ and ˆφ q,k,± λ only U k and V k appear, and in the φ q,k λ and ˆφ q,k,± λ only U k and V k appear. Next consider the situation in more details: by definition where log + k= log U k λµ k q,n + k= = j= φ q,j λµ j q,n, V k λ ± α q U k λ µ k q,n = λ ˆφq,j,± λ = ˆψ q,j,± λ + j+ ±α q j, j j= ˆψ q,j,± λµ j q,n. Since, again by the very definition, V j 0 = U j 0 for all j, by comparing the two formulas above, we realize that, if λ = 0, in each odd term of the expansion of φ q,k λ ˆφ q,k,+ λ ˆφ q,k, λ in, i.e. in the coefficient of each j=k, the two terms coming from ˆφ q,j,± λ ± α q U j λ + j+ ±α q j, λ j cancel each other, while the term coming from φ q,j λ U k λ, will cancel out by the two terms coming from ˆφ q,j,± λ V k λ. This prove the first formula in the last statement. The proof of the second one is similar.

20 0 L. HARTMANN AND M. SPREAFICO Lemma 7.. For j > 0, the following functions of Φ q,j and ˆΦ q,j,± are regular at s = 0: Res Φ q,k s ˆΦ q,k,+ s ˆΦ q,k, s = 0, s=0 ˆΦq,k, s ˆΦ q,k,+ s = 0. Res s=0 Proof. The argument is the same for the functions ˆΦ and the function Φ, so just consider the last ones. By Lemma 7. and since see [0] 0 t s e πi Λθ,c λt λ 3j φ q,j λ = c k, λ k k=j { dλdt = λ k 0, k = 0, Γs+k Γks, k 0, we have that and this means that 3j Γs + k Φ q,j s = c k Γks, k= 3h Res Φ q,h s = c k = φ q,h 0, s=0 Res Φ q,h s = s=0 k=j 6h k=j c k = φ q,h 0. Applying the definition in equation 5.5, we see that all relevant b σh,0,0/ and ˆb σh,0,0/ vanishes. For j = σ h = m h, with h = 0,, 4,..., h m; and therefore when j = 0, b 0,0,0/ and ˆb 0,0,0/ do not appears in the sums, while when j > 0 there are neither logarithmic terms nor constant terms in the expansion for large λ of the functions φ q,j λ and ˆφ q,j,± λ by the previous lemmas. Thus, 7.7 A 0,0,q s = Â 0,0,q,± s = n= n= m cex,q,n a 0,0,n,q µ s q,n, A 0,,q s = m cex,q,n â 0,0,n,q,± µ s q,n, Â 0,,q,± s = n= n= m cex,q,n a 0,,n,q µ s q,n, m cex,q,n â 0,,n,q,± µ s q,n. where a 0,0,n,q and a 0,,n,q are the constant and the logarithmic term in the expansion in equation 7.5, and â 0,0,n,q,± and â 0,,n,q,± are the constant and the logarithmic term in the expansion in equation 7.6.

21 ANALYTIC TORSION OF CONES 7.8 Third, concerning the singular term, the functions defined in equation 5.3, are Φ q,j s = ˆΦ q,j,± s = 0 0 t s e πi Λθ,c λt λ φ q,jλdλdt, t s e πi Λθ,c λt λ ˆφ q,j,± λdλdt, where the φ and the ˆφ are given in equations 7. and 7.4, respectively. 7.. The frustum with mixed BC. Consider the double series s ˆD q,± s; l, l = m cex,q,n ˆf µ l q,n,±α q,k, l, n,k= associated to the double sequence ˆΘ s q,± l, l = {m cex,q,n : ˆf µ l q,n,±α q,k, l }. It is easy to see that this sequence as power κ =. By Definition 5., the relevant spectral function associated to ˆΘ q,± l, l is the function compare with equation 5.: log λ; ˆΘ q,± l, l = log + k= λ ˆf,±α q,k l, l = log Ĝµ n,q,±α q λ; l, l + log µ l q,n π + log l µq,n + lµq,n l µq,n ± α q l l µq,n lµq,n, l µq,n where for π < argz π Ĝ µ,cz; l, l = ˆF µ,ciz; l, l. We need uniform asymptotic expansions of the function log Γ for large n and for large λ. Such expansions can be obtained from those of the Bessel function. A long calculation gives the following results. For large n, uniformly in λ, compare with equation 5. log λ; ˆΘ q,± l, l /µ q,n = l λ l λ log l + l λ l + l λ 4 log l λ m l λ ˆψ q,j,± λ; l, l µ l q,n + log l µq,n j= + lµq,n l µq,n ± α q µ j q,n l l µq,n lµq,n + Oµ m l µq,n q,n, where ˆψ q,j,± λ; l, l is the coefficient of µ j q,n in the asymptotic expansion for large, i.e. of log + Ψ q,k,± l, l µ k q,n, where k= Ψ q,k,± l, l =sgnl l k U k l λ + sgnl l k W ±sgnl l α q,kl λ sgnl l h sgnl l k h U h l λw±sgnl l α q,k hl λ. k + h=

22 L. HARTMANN AND M. SPREAFICO In other words, log + k= Ψ q,k,± l, l µ k q,n = j= ˆψ q,j,± λ; l, l µ j q,n, therefore the complete coefficient of µ j q,n in log Γ λ, ˆΘ q,± l, l /µ q,n is 7.9 ˆφq,j,± λ; l, l = ˆψ q,j,± λ; l, l + j+ sgnl l α q j. j For large λ, compare with equation log Γ λ, Ŝ,±α q,kl, l /µ q,n = l l λ log l µ l q,n + log + O l µq,n λ l + lµq,n ± α q l µq,n l l µq,n lµq,n l µq,n Next consider the functions ˆφ q,j,± λ; l, l. By the following lemma we see that all the results given in Lemmas 7. and 7. hold for these functions. Lemma 7.3. The following relation holds: ˆφ q,j,± λ; l, l = sgnl l j φ q,j l λ + sgnl l j ˆφq,j,±sgnl l l λ. Proof. By definition in equations 7. and 7.4 log + U k λµ k q,n = φ q,j λµ j q,n. and where while by equation 7.9 where with log + k= j= ˆφ q,j,± λ = ˆψ q,j,± λ ± j+ α q, j k= W ±αq,k λµ k q,n = j= ˆψ q,j,± λµ j q,n, ˆφ q,j,± λ; l, l = ˆψ q,j,± λ; l, l + j+ sgnl l α q j, j log + k= Ψ q,k,± l, l µ k q,n = j= ˆψ q,j,± λ; l, l µ j q,n, Ψ q,k,± l, l =sgnl l k U k l λ + sgnl l k W ±sgnl l α q,kl λ sgnl l h sgnl l k h U h l λw±sgnl l α q,k hl λ. k + h=

23 ANALYTIC TORSION OF CONES 3 Now, log + k= Ψ q,k,± l, l µ k q,n = log + log + k= + and hence the thesis follows by the very definition. sgnl l k U k l λµ k q,n k= sgnl l k W ±sgnl l α q,kl λµ k q,n This means that the b σh,j,k in equation 5.5 are all disappearing, and hence A 0,0,± s; l, l = A 0,,± s; l, l = n= n= m cex,q,n a 0,0,n,± l, l µ s q,n, m cex,q,n a 0,,n,± l, l µ s q,n, where a 0,0,n,± l, l and a 0,,n,± l, l are the constant and the logarithmic term in the expansion in equation 7.0. Eventually, concerning the singular term, the functions defined in equation 5.3 are 7. ˆΦq,j,± s; l, l = where the ˆφ are given in equation t s e πi Λθ,c λt λ ˆφ q,j,± λ; l, l dλdt,, 8. Calculation II: application of Theorem 5. We now apply Theorem 5. and its corollary to compute the derivative at s = 0 of the functions Z, D, and E appearing in Lemmas 4., 4., and 4.3. According to Remark 5. we split the result in the regular and singular parts, denoted by the obvious subscript. In order to improve readability we will use in this section the simplified notation A and B for the two terms A 0,0 and A 0, defined in equation 5.5. We observe that, in all cases, when computing the regular part, all the terms that are equal in B q 0 and ˆB q,± 0 and in A q 0 and Âq,±0 respectively, cancel in the final formula for t m 0,reg 0, and similarly for the others. We thus introduce some regularized terms, denoted by a slashed letter, where only the relevant parts appear. 8.. The cone. We split the calculation into three parts: the regular contribution, the singular contribution and the contribution of the harmonics. We give all details for the cone The contribution of the regular part. By Theorem 5., we have Z q,reg 0 = B q 0, Ẑ q,±,reg 0 = ˆB q,± 0, Z q,reg0 = A q 0 B q0, Ẑ q,±,reg0 = Âq,±0 ˆB q,±0,

24 4 L. HARTMANN AND M. SPREAFICO where, by equation 7.7, A q s = B q s = n= B qs = Â q,± s = ˆB q,± s = m cex,q,n log π + log log µq,n Γ µ s q,n, n= n= ˆB q,±s = n= m cex,q,n + µ s q,n, m cex,q,n + µ s q,n log, m cex,q,n log π + n= n= m cex,q,n µ s q,n, m cex,q,n µ s q,n log. This gives in the following formulas we are taking the finite part A q 0 = B q 0 = n= B q0 = Â q,± 0 = ˆB q,± 0 = log log µq,n Γ + log ± α q µ s q,n, m cex,q,n log π + log log µq,n Γ, n= n= ˆB q,±0 = n= m cex,q,n +, m cex,q,n + log, m cex,q,n log π + n= n= m cex,q,n, m cex,q,n log. Thus, considering first t m 0,reg, by Lemma 4., we have t m 0,reg log log µq,n Γ + log ± α q, [ m ] 0 = q Z q,reg 0 Ẑq,+,reg0 + m Z q,reg 0 Ẑq,,reg0 log l, + [ m ] q Z q,reg0 Ẑ q,+,reg0 + m Z q,reg0 Ẑ q,,reg0.

25 ANALYTIC TORSION OF CONES 5 and hence t m 0,reg [ m ] 0 = q B q 0 ˆB q,+ 0 + m B q 0 ˆB q, 0 log l [ m ] m q A q 0 + B q0 Âq,+0 ˆB q,+0 [ m ] q A q 0 + B q0 Âq, 0 ˆB q, 0. We observe that all the terms that are equal in B q 0 and ˆB q,± 0 and in A q 0 and Âq,±0 respectively, cancel in the final formula for t m 0,reg 0. This is because the final formulas are Z q,reg0 Ẑq,+,reg0 + Z q,reg0 Ẑq,,reg0 = Z q,reg0 Ẑq,+,reg0 Ẑq,,reg0, Z q,reg 0 Ẑq,+,reg0 + Z q,reg 0 Ẑq,,reg0 = Z q,reg 0 Ẑq,+,reg0 Ẑq,,reg0, if m = p is odd, and Z q,reg0 Ẑq,+,reg0 Z q,reg0 Ẑ q,,reg0 = Ẑ q,+,reg0 + Ẑ q,,reg0, Z q,reg 0 Ẑq,+,reg0 Z q,reg 0 Ẑq,,reg0 = Ẑq,+,reg0 + Ẑq,,reg0, if m = p is even. Therefore, we can rewrite t m 0,reg [ m ] 0 = q B/ q 0 ˆB/ q,+ 0 + m B/ q 0 ˆB/ q, 0 log l [ m ] m q A/ q 0 + B/ q 0 Â/ q,+ 0 ˆB/ q,+ 0 [ m ] q A/ q 0 + B/ q 0 Â/ q, 0 ˆB/ q, 0.

26 6 L. HARTMANN AND M. SPREAFICO where and A/ q 0 = 0, B/ q 0 = m cex,q,n = 4 4 ζ cex0, q + αq = 4 ζ0, q, n= B/ q 0 = m cex,q,n log, n= Â/ q,± 0 = m cex,q,n log ± α q, µ n= q,n ˆB/ q,± 0 = m cex,q,n = 4 4 ζ cex0, q + αq = 4 ζ cex0, q, ˆB/ q,± 0 = n= m cex,q,n log, n= m cex,q,n = ζ cex 0, q + αq, n= m cex,q,n log = ζ cex0, q + αq. n= Thus, 8. [ m ] t m 0,reg 0 = q ζ cex 0, q + m ζ cex 0, q log l + [ m ] + m q [ m ] n= m cex,q,n log + α q q n= + m cex,q,n log n= m cex,q,n log α q + m cex,q,n log. n= Next, consider t p,reg. By Lemma 4., we have t p,reg 0 = p B p 0 ˆB p,0 0 log l A p 0 + B p 0 Âp,00 ˆB p,00. As above, all the terms that are equal in B q 0 and ˆB q,0 0 and in A q 0 and Âq,00 respectively, cancel in the final formula for t p,reg 0. Therefore, we can rewrite t p,reg 0 = p B/ p 0 ˆB/ p,0 0 log l A/ p 0 + B/ p 0 Â/ p,0 0 ˆB/ p,0 0.

27 ANALYTIC TORSION OF CONES 7 where and Â/ p,0 0 = 0, ˆB/ p,0 0 = m cex,p,n = 4 4 ζ cex0, p + αp = 4 ζ cex0, p, n= ˆB/ p,0 0 = m cex,p,n log µ p,n. n= Observing that α p = + p p + = 0, µ p,n = λ p,n, and hence ζ cex s, p + αp = ζ cex s, p = m cex,p,n λ s p,n, Thus, ˆB/ p,0 0 = n= m cex,p,n log λ p,n = 4 ζ cex0, p. n= 8. t p,reg 0 = p ζ cex0, p log l 4 p ζ cex0, p The contribution of the singular part. By Lemma 7., the functions Φ q,j and Φ q,j,± are regular at s = 0 when m is odd, and the functions Φ q,j and Φ q,j,± are regular at s = 0 when m is even. By the location of the poles of ζs, U given at the beginning of Section 7, it follows that the formula for the singular part in Theorem 5. reduces to the single sums recall κ = and the zeta of U is regular at s = 0 Z q,sing 0 = 0, Ẑ q,±,sing 0 = 0, Z q,sing0 = m Φ q,j 0 Res ζs, U, Ẑ q,±,sing0 = m s=j j= j= ˆΦ q,j,± 0 Res ζs, U. s=j where the Φ are defined in equation 7.8. Thus, considering first t m 0,sing, by Lemma 4., we have [ m ] t m 0,sing 0 = q Z q,sing0 Ẑ q,+,sing0 + m Z q,sing0 Ẑ q,,sing0, and hence t m 0,sing 0 = 4 [ m ] q m j= Φ q,j 0 ˆΦ q,j,+ 0 + m Φ q,j 0 ˆΦ q,j, 0 Res ζs, U, s=j Second, consider t p,sing 0 as given in Lemma 4.. We have t p,sing 0 = p Z p,sing0 Ẑ p,0,sing0, and hence t p,sing 0 = p 4 m j= Φ p,j 0 ˆΦ p,j,0 0 Res ζs, U. s=j

28 8 L. HARTMANN AND M. SPREAFICO The contribution of the harmonics. The contribution of the harmonics can be computed using the formula in equation 5.7 at the end of Section 5. By the definition in Lemma 4., since we have t m s = ls [ m ] z q,± s = zs, ±α q, 0,, q+ m har,q zs, α q, 0, + m zs, α q, 0, ; by equations 5.7 and 5.6, t m 8.3 [ m 0 = q+ m har,q z0, α q, 0, + m z0, α q, 0, log l ] + = + [ m ] q+ m har,q z 0, α q, 0, + m z 0, α q, 0, [ m ] q+ m har,q α q + [ m ] q+ m har,q On the other side, Whence + m α q + log l log αq Γ α q + + m log αq Γ α q +. π π t p p+ ls 3 s = 4 m har,p zs, α p, 0, + zs, α p, 0,. t p 3 0 = p+ m har,p z0, α p, 0, + z0, α p, 0, log l + p+ m har,p 4 z s, α p, 0, + z s, α p, 0, = p+ 4 m har,p α p + + α p + log l + p+ 4 m har,p = p 4 m har,p log l + p+ 4 m har,p log Since α p = + p p =, this gives log αp Γα p + + log αp Γ α p + π π α p sinπα p. 8.4 t p 3 0 = p 4 m har,p log l + p m har,p log. 8.. The frustum with mixed BC. We omit the details, since the calculations are similar to the one performed for the cone. As above, we split into three parts.

29 ANALYTIC TORSION OF CONES The contribution of the regular part. According to Theorem 5.. we have where, according to equation 7.0  ± s; l, l = n= ˆB ± s; l, l = 0, and hence  ± 0; l, l = ˆD q,±,reg0; l, l = Âq,±0; l, l ˆB q,±0; l, l, m q,cex log l l + log n= ˆB ± 0; l, l = 0, Now, note that and hence, by Lemma 4., 8.5 and 8.6 l l µq,n m q,cex log l µ l q,n + log l l µq,n  0; l, l Â+0; l, l = n= + lµq,n ± α q l µq,n + lµq,n ± α q l µq,n l l µq,n l l µq,n m q,cex log l l = ζ cex 0, q log l l, [ m ] w m 0,reg 0 = q ˆD q,,reg0; l, l ˆD q,+,reg0; l, l + m ˆD q,+,reg0; l, l ˆD q,,reg0; l, l [ m ] = q Âq, 0; l, l Âq,+0; l, l + m Âq,+0; l, l Âq, 0; l, l = log l [ m ] q ζ cex 0, l q + m+ ζ cex 0, q, w p,reg 0 = p ˆD p,0,reg 0; l, l ˆD p,0,reg0; l, l = p log l l ζ cex 0, p. lµq,n µ s l µq,n q,n, lµq,n, l µq,n 8... The contribution of the singular part. Since the functions Φ q,j and Φ q,j,± are regular at s = 0, and by the location of the poles of ζs, U given at the beginning of Section 7, it follows that the formula for the singular part in Theorem 5. reduces to the single sums recall κ = and the zeta of U is regular at s = 0 ˆD q,±,sing 0; l, l = 0, ˆD q,±,sing 0; l, l = m j= ˆΦ q,j,± 0; l, l Res ζs, U. s=j

30 30 L. HARTMANN AND M. SPREAFICO where the Φ are defined in equation 7.. Thus, considering first w m 0,sing, by Lemma 4., we have and hence 8.7 [ m ] w m 0,sing 0 = q ˆD q,,sing0; l, l ˆD q,+,sing0; l, l + m ˆD q,+,sing0; l, l ˆD q,,sing0; l, l, [ m ] w m 0,sing 0 = q 4 Second, for w p,sing 0, we have and hence m j= ˆΦ q,j, 0; l, l ˆΦ q,j,+ 0; l, l + m ˆΦ q,j,+ 0; l, l ˆΦ q,j, 0; l, l w p,sing 0 = p 8.8 w p,sing 0 = p 4 p j= Res ζs, U, s=j ˆD p,0,sing 0; l, l ˆD p,0,sing0; l, l, ˆΦp,j,0 0; l, l ˆΦ p,j,0 0; l, l Res ζs, U. s=j The contribution of the harmonics. In order to determinate the contribution of the harmonics, we use the technique described in Section of [] to deal with simple sequences of spectral type in fact this was already in []. Recall that, by Lemma 4., we need d q0; l, l, where by definition d q s; l, l = m har,q k= and the f s ν,k l, l are the positive zeros of the function f s α q,k l, l, F ν z; l, l = J ν l zy ν l z Y ν l zj ν l z. Recalling the series definition of the Bessel functions, we obtain that near z = 0, F ν z = l ν πz l ν. This means that the function F ν z is an even function of z. By the Hadamard factorization Theorem, we have the product expansion and therefore Defining, for π < argz < π, F ν z = πz F ν z = πz l ν l ν l ν l ν + k= k= z, f ν,k l, l z fν,k l,, l G ν z = F ν iz,