Character Integral Representation of Zeta function in AdS$_{d+1}$: I. Derivation of the general formula

The zeta function of an arbitrary field in $(d+1)$-dimensional anti-de Sitter (AdS) spacetime is expressed as an integral transform of the corresponding $so(2,d)$ representation character, thereby extending the results of arXiv:1603.05387 for AdS$_4$ and AdS$_5$ to arbitrary dimensions. The integration in the variables associated with the $so(d)$ part of the character can be recast into a more explicit form using derivatives. The explicit derivative expressions are presented for AdS$_{d+1}$ with $d=2,3,4,5,6$.


Introduction
The one-loop free energy is one of the simplest physical quantities capturing non-trivial quantum effects. It is divergent due to the contribution from the modes having infinite energy but can be neatly regularized by making use of the spectral zeta function, namely in the scheme of the zeta function regularization. The zeta function for any field (massive or massless, and of arbitrary spin) in four-dimensional Anti-de-Sitter (AdS) spacetime has been first calculated by Camporesi and Higuchi in [1] and generalized to higher dimensions by the same authors in [2,3].
Zeta functions in AdS are useful in the test of certain AdS/CFT dualities: the one-loop free energy or vacuum energy in AdS spacetime corresponds to the non-planar contribution of the CFT free energy on the boundary. Since the typical AdS theories under consideration contain infinitely many fields, computing their one-loop free energy is a non-trivial task. 1 When the AdS theory is a higher-spin gravity, the one-loop free energy is calculable in an analytic manner, even though the field content still contains infinitely many elements. An interesting observation made in [9,10] is that the summation of the zeta functions over the field content is convergent while that of the regularized one-loop free energy is divergent. This is interesting as it signifies that the zeta function regularization renders finite both the high energy divergence and the spectrum sum divergence.
The viability of the one-loop free energy computation in higher-spin gravities heavily relies on the simple structure of the spectrum: e.g. in the case of the non-minimal type-A theory, first constructed in four dimensions [11,12] and later extended to arbitrary dimensions in [13], the spectrum consists of massless fields of all integer spins. The summation over a field content becomes quite cumbersome if the content itself does not have a simple expression. This kind of difficulty was encountered in the computation of the one-loop free energy of the AdS fields dual to the operators tri-and quadri-linear in free conformal scalar fields [14]. In the absence of the single-trace condition -that is, the cyclic projection on the operators -the field content could be expressed in a few lines, which could be used to calculate the one-loop free energy although it required quite burdensome works. What is worse is that the field content with cyclic projection does not have any manageable expression, hence it seems impossible to proceed in this way.
A key observation to bypass this problem is that the field content of an AdS theory dual to a free CFT can be derived group theoretically. One of the most efficient and general methods for such a derivation is the use of Lie algebra character. In fact, the spectrum of the four-dimensional type-A higher-spin gravity was obtained using the so(2, 3) character -namely, the Flato-Fronsdal theorem [15] (later generalized in arbitrary dimensions in [16,17]). Since both the zeta function and the character are determined uniquely by the labels of so(2, d) representations, we can devise a linear map which send the character of an so (2, d) representation to the zeta function of the AdS field carrying the same representation. If the linear map itself does not depend on the labels of the representation, we can use it for 1 If instead one considers the change in free energy by taking an alternative boundary condition for one of the AdS fields, this technical difficulty does not arise. This quantity is matched to the double-trace deformation of the corresponding CFT [4][5][6][7][8].
the character of a reducible representation without decomposing it into irreducible pieces. Such a map was explicitly constructed for bosonic fields in AdS 4 and any fields in AdS 5 and was named as "character integral representation of zeta function (CIRZ)" in [14]. The CIRZ turned out to be very efficient in evaluating the one-loop free energies. Notably, the computation of the non-minimal type-A higher-spin gravity becomes almost trivial. Moreover, the CIRZ allowed to tackle the one-loop free energy computation of the stringy AdS theory dual to free matrix model CFTs in the N → ∞ limit [14,[18][19][20]. The CIRZ also proved useful in other vector model dualities: the "colored" higher-spin gravity (where the four-dimensional CIRZ was generalized to fermionic fields) [21] and the type-J higherspin theories (whose conjectured dual are free vector model based on a massless spin-j field) [22]. Some key elements of the CIRZ were also used in [23].
In this paper, we aim to derive the CIRZ in dimensions different from AdS 4 and AdS 5 . More precisely, we seek for the formula extending the CIRZ in AdS d+1 with arbitrary integer d ≥ 2. Although the dimensional dependence in higher-spin gravity is rather minimal, most of the results in the literature [9,10,[24][25][26][27] concern only specific dimensions for technical reasons, except in [23] where the results of the type-A higher-spin gravity are extended to arbitrary (non-integer) dimensions. From the viewpoint of physical applications, one might not need to care about higher dimensions yet, but it is at the same time tempting to obtain results with parametric dependence on d. As usual, generalities may provide new and valuable lessons on what is considered to be well-understood. It is actually the case here for the CIRZ: in the course of its derivation for general dimensions, we find many new insights on the zeta function, so(2, d) character and the relations between them.
The general CIRZ formula we obtain is an integral transformation of so(2, d) character and has a quite simple structure: it is in a sense even simpler than the original CIRZ expressions obtained in AdS 4 and AdS 5 in [14]. In order to deliver a flavor of our results, let us write down the expressions of the general CIRZ derived in Section 3. Firstly, for odd AdS dimensions d + 1 = 2r + 1 (or equivalently, even boundary dimensions d = 2r), we obtain dα n 2π i α n and α k := (α 0 , . . . , α k−1 , α k+1 , . . . , α r ) .
Secondly, for even AdS dimensions d+1 = 2r+2 (or equivalently, odd boundary dimensions d = 2r + 1), we obtain where ǫ = +1/ − 1 for bosonic/fermionic spectrum. The subscript 1 in the zeta function means that it is the primary contribution to the actual zeta function. The remaining contributions will be introduced later in this paper but they are irrelevant in all the cases that we are interested in. Both of the formulae (1.1) and (1.3) involve complex contour integrals in α n with n = 0, 1, . . . , r. The contour of each α n is a circle enclosing the origin counter-clockwise with |α 0 | < · · · < |α r |. We would like to stress that the expressions (1.1) and (1.3) allow for a large room for various complex integral tricks: in the companion paper [28], we will compute the zeta function of partially-massless higher-spin gravities in arbitrary dimensions using complex integrals. The contour integrals in α i appearing in the above formulae (1.1) and (1.3) may be evaluated by the residue theorem, and hence reduce to a linear combination of derivatives in α i . The expression in terms of the α i contour integral is compact and useful in many applications but it might be not explicit enough in other cases. For instance, if one wants to implement the formula in a computer program, the other expression in terms of the α i derivatives would be more convenient. We derive the latter expression by adapting the standard tools used to derive the dimension formula from the Weyl character formula. Applying the expressions to d = 2, 3, 4, 5, 6, we provide the explicit form of the CIRZ in the dimensions which are the most relevant for physical applications.
The organization of the paper is as follows. In Section 2, we start by reviewing the one-loop free energy and zeta function in AdS d+1 , then rewrite the latter as an integral of the dimension formula of an so(d + 2) representation. In Section 3, we derive the CIRZ formula in arbitrary dimensions, i.e. we show that the zeta function of any field in AdS d+1 can be written as an integral transform of its so(2, d) character. In Section 4, we spell out an alternative form of the CIRZ where the previously mentioned contour integrals are replaced by a linear combination of derivatives of the character. Section 5 contains a brief summary and concluding remarks of the paper. Finally, some definitions and technical details are presented in Appendix A and B.

Zeta functions in AdS
In this section, we shall review the basics of the one-loop free energy and zeta function, and the integral expression of the zeta function in AdS d+1 obtained in [3]. After reexpressing the numerator of the integrand in terms of the dimension formula of an so(d+2) representation, we shall discuss how the first derivative of the zeta function can be expressed in a "spectral integral" form.

One-loop free energy and zeta function
The one-loop free energy of a quantum field is given by the logarithm of the one-loop path integral: where V is the space of the off-shell fields which are traceless and transverse, and the sign ǫ is +1 for a boson and −1 for a fermion. The operator + m 2 is what appears in the quadratic Lagrangian, so one can regard the field practically as a free one. When the field has a gauge symmetry, we have to subtract the corresponding ghost contribution. The one-loop free energy (2.1) can be related to the zeta function where the trace is convergent for a sufficiently large value of Re(z). Once we obtain ζ [m 2 ;V] (z), the log det formula can be related to the zeta function by analytically continuing the value of z to zero as where we have used the zeta function regularization and the last expression is the finite part of the free energy. The UV divergence of the free energy corresponds to ζ [m 2 ;V] (0). The zeta function can be also related to the integrated propagator in the coincidence point limit: As we have just seen, both the free energy Γ (1) [m 2 ;V] and the propagator G [m 2 ;V] can be obtained from the zeta function (2.2), hence this allows us to focus on the zeta function for a given field space V with the mass squared m 2 .
In AdS d+1 , free fields can be classified by the irreducible representations (irreps) they carry for the isometry algebra so(2, d). The massive and massless [29][30][31] irreps 2 are the lowest-weight modules labeled by [∆; Y], where ∆ is the lowest eigenvalue of the energy operator generating so(2), and Y := (s 1 , . . . , s r ) is the highest weight of the rotational symmetry so(d) classifying the traceless and transverse tensors and can be interpreted as the spin of the field. The zeta function for the module [∆; Y] has been calculated in [2,3] (see also e.g. [25,26] for a review) and its expression is The volume of the d-sphere and the (regularized) volume of the (d + 1)-dimensional AdS spacetime are given respectively by where R is the radius of the AdS d+1 . The function µ Y (u) appearing in the numerator of the integrand is given by where ℓ k = s k + d 2 − k, and the sign ǫ is positive for bosonic fields (s k ∈ N) and negative for fermionic ones (s k ∈ 1 2 N). The combination dim is related to the dimension formula (which is also referred to as Weyl dimension formula) of an so(d + 2) irrep: • For even d = 2r, the dimension of the so(d This can be expressed in terms of the dimension of the so(d) irrep Y as dim so(d+2) , (2.11) and can also be related to the dimension of the so Making use of the above information, the zeta function can be written as with the function ρ ǫ (u), . (2.15)

Spectral integral form of the zeta function
The zeta function ζ [m 2 ;V] (z), defined generally as (2.2), enjoys a simple identity, which is nothing but the spectral version of the Hurwitz zeta function identity, The identity (2.16) simply implies where F.p. refers to the finite part in the limit z → 0, i.e. the constant term in the Laurent expansion in z. These two formulae provide the derivatives with respect to m 2 of the UV divergent and finite part of the free energy. The second equation (2.20) can be viewed as the regularized version of the formal expression, Now considering the AdS background, the identity (2.16) becomes and we also have relations analogous to (2.20). These identities prove useful since it is easier to study the zeta function ζ [∆;Y] (z) near z = 1 than z = 0. In the following, we shall make use of the identity (2.22) to show how a "spectral integral" form of the ζ ′ [∆;Y] (0) can be obtained. In the context of AdS 2r+1 /CFT 2r correspondence, this spectral integral formula was used to show the direct relation between ζ ′ [∆;Y] (0) and conformal anomaly coefficients. It first appeared in the case of totally symmetric representations in [35] and subsequently mixed symmetry representations in AdS 7 [36], then generalized to arbitrary representations in [26]. Below, we provide a short derivation of the spectral integral formula in AdS d+1 for both even d = 2r and odd d = 2r + 1.

Even d
For even values of d, the zeta function has the form, is an even polynomial of order 2r, the integral (2.23) is the same as one half of the integral from u = −∞ to u = ∞ with the same integrand. The integral is convergent in the region z > r + 1 2 , so we can close the contour by adding the infinite upper half-circle, then shrink it down to enclose the branch cut singularity (i.e. the line defined by Arg(u − i|∆|) = π), This contour integral is convergent for any value of z, hence we can directly replace z by the value we want. If we put z = 0, the integrand becomes analytic and we get If we put z = 1, the integrand has a simple pole at u = +i |∆| and gives where the absolute value |∆| appears as a result of sgn(∆). The result is even in∆ like the original form (2.23) hence insensitive to its sign. In physical term, the sign of∆ determines whether the underlying field takes Dirichlet or Neumann boundary condition. Since the two boundary conditions should give different results, we need to modify the above definition of the zeta function. By noticing that the expression (2.28) is not analytic on the imaginary axis of∆, we can consider another expression where we analytically continue the value of∆ from positive Re(∆) to negative one. This simply amounts to replacing |∆| by∆ in (2.28). At the level of the contour integral representation (2.24), this "new" definition of the zeta function corresponds to the modification, where the contour encircles counter-clockwise the branch cut starting at i∆ rather than i |∆| . The zeta function (2.29) is what has been used in the literature.

Odd d
For odd values of d, the zeta function has the form, is now an odd function. Similarly to the even d case, we can rewrite the above expression as a contour integral by adding to the real line the infinite radius upper-half circle. The function h Y (u) is analytic again but tanh ǫ (π u) has infinitely many simple poles on the imaginary axis. We can separate those contributions as Here, we take the prescription that the zeta function is analytic in∆. Due to the presence of infinitely many simple poles of tanh ǫ (πu), it is not easy to simplify further the above expression as opposed to the even d case. However, if we take the difference between ζ [∆;Y] (z) and ζ [d−∆;Y] (z), we can cancel the cumbersome contribution and end up with Note that we do not have a perfect cancellation due to the prescription of analytic continuation in ∆. Now taking z = 0 and z = 1 limit, we find The first equation (2.33) means that the UV divergence does not depend on the sign of∆, or in physical terms, the choice of the boundary conditions. Applying the second equation (2.34) to the zeta function identity (2.22), we reach the result, to express the zeta function (4.10) in terms of the so(2, d) character so that the dependence on ∆ and Y in ζ [∆;Y] enters only through the corresponding character χ so(2,d) (∆,Y) (see e.g. [37][38][39][40] for more details on so(2, d) characters). It turns out that it is sufficient to consider the character solely over (possibly reducible) generalized Verma modules of the conformal algebra. These representations, when irreducible, correspond to massive fields in AdS for which the one-loop partition function takes the form (2.1). The formalism we develop extends trivially to massless fields [18,41], which are described by irreducible representation defined as quotients of generalized Verma modules.

General dimensions
The character of the so(2, d) generalized Verma module V(∆; Y) takes the form, where P d is defined as with r = [d/2] and α = (α 1 , . . . , α r ). We first note that the dimension of the so(d + 2) irrep is given by the corresponding character evaluated at α = 0: where α = (α 0 , . . . , α r ) and r = [d/2]. Then, the so(d + 2) character can be related to that of so(d) using the following identity (see Appendix A for additional details on the identity): where α k = (α 0 , . . . , α k−1 , α k+1 , . . . , α r ) and Y ± = (s 1 , . . . , s r−1 , (±) d+1 s r ) . Another key trick is based on the identity, holding for Re(∆) > 0 and Re(z) > 0. Note here that for the convergence of the β integral, we ought to use e −β |∆| in the β integral, but in such a case the zeta function will be insensitive to the sign of∆ and becomes incapable of distinguishing different boundary conditions. Hence, like the discussion below (2.28), we first derive the formula assuming Re(∆) > 0 then analytically continue∆ to the negative Re(∆) region. Combining all these elements, we get where we used the factor e −β(∆− d 2 ) to reconstruct the so(2, d) characters of the irreps [∆; Y ± ] according to (3.1). We can simplify the above formula using the identity of the so(d) characters, (3.7) for k = 1, . . . , r. We obtain where the functionν ǫ (u, β, α) is defined to bẽ . (3.9) Note here that each summand in the second line of (3.8) diverges in the limit α → 0.
Only the sum of the r + 1 terms is regular in the limit α → 0. For this reason, we cannot exchange the order of the summation and the evaluation α = 0.
To further simplify the formula, it is convenient to replace the evaluation of the character at α = 0 by the contour integrals, dim so(d+2) where C k are contours encircling the origin counter-clockwise such that the contour C k lies inside of C k+1 : e.g. the circular contours with |α k | < |α k+1 | (see Figure 3.2). Figure 1. Example of "ordered" contours with r = 2.
The advantage of the contour integral representation is that now we can perform the contour integration before the summation -but keeping the order of contours fixed. In this way we get where we have interchanged the order of u and β integrations and µ(α) denotes the measure for (r + 1)-variable complex integral, Finally, exchanging the order of the u and α integrations, we obtain where the function ν ǫ (z, β, α) is defined to be . (3.14) Evaluation of the above function requires separate considerations for even and odd d.

AdS 2r+1
In even d = 2r, the function ν ǫ (z, β, α) can be evaluated as (3.15) This integral was evaluated in the region Re(α) = Im(β) = 0, β > 0 and Re(z) > 0, then analytically continued to other region. Since an additional factor of 1/Γ(z) is generated, we can express the result as where the function f [∆;Y] (z, β) is defined as Note that the contour of α k does not contain ±i β. The function β 2(z−1) has a branch cut on the negative real axis of β with the phase factor e 4πzi . Using this information, we can rewrite (3.16) as where the contour integral is defined by the Hankel contour (see Figure 2).

Re(β)
Im(β) Therefore, the first derivative of the zeta function reduces to whereas ζ [∆;Y] (0) = 0. Notice that, because the integrand of the above integral is devoid of branch cut, the Hankel contour can be deformed to a closed contour encircling the origin of the complex plane. In the end, we arrived at the relation between the zeta function ζ [∆;Y] (z) and the character χ so(2,d) (∆;Y) (β; α) for an arbitrary lowest weight representation [∆; Y] . Since the relation (3.19) is a linear map independent of the representation, we can apply the same formula to a generic reducible representation of the so(2, d) algebra, e.g. the representation corresponding to the field content of any theory.

Primary contribution
Let us consider first the primary contribution to the zeta function. It is given through λ ǫ (0, β, α) in the form of an integral, where we have used the fact that 0 F 1 ( 1 2 , − x 2 4 ) = cos x. The above integral is not convergent but can be considered as a distribution (see e.g. Chapter 6 of [42]). By multiplying the integrand with e −σ u , we can evaluate the integral in the region Re(σ) > |Re(α)| + |Im(β)|.
By taking the limit σ → 0 + , we obtain a finite result (hence the original integral is Abel summable [42]), We can analytically continue the above to other region of α, β. Plugging this into (3.23), we obtain the primary contribution of the zeta function as where the function f 1,[∆;Y] (β) is given by (3.29) We finally reached a formula where the relation itself does not involve any dependency on ∆ and Y.

Secondary contribution
The second term z λ ′ ǫ (0, β, α) in (3.24) is linear in z, hence contributes to the derivative of zeta function as a residue: Remark that, since the second line of (3.31) is regular in β, only terms with negative powers in the Laurent series of λ ′ ǫ (0, β, α) can contribute to the residue in (3.30). To isolate these negative power terms, let us split tanh ǫ (πu) in the integrand of λ ǫ (z, β, α) given in (3.21) into two terms as tanh ǫ (πu) = 1 − 2 1 + ǫ e 2πu . (3.32) Focusing on the second term of (3.32), the u integral (3.21) gives a regular function of β, so all the negative power terms come from the first term "1". Now moving to the first term of (3.32), the corresponding integral can be evaluated for Re(z) > 0 as from which we obtain Here, O(1) means a regular function of β and it is related to the second term of (3.32). Again, both of the u integrals arising from the first and second terms in (3.32) are evaluated in the region where Re(α) = Im(β) = 0, then analytically continued to other values. In the end, we obtain a compact expression of the secondary contribution: (3.35) where the β contour now encloses anti-clockwise the branch cut from −i α k to i α k . Remark that if the character is even in β, then the whole integrand is even in β and the residue vanishes trivially. Therefore, in such a case, the secondary contribution of the zeta function vanishes. Furthermore, the secondary contribution is independent of ǫ, i.e. whether the spectrum is bosonic or fermionic.

Cross-check
In the above, we have derived the CIRZ in any dimensions. The derivation required many technical steps, hence it would be good if we can cross-check the formula in the end. In principle, if the zeta function of a field with an arbitrary mass and spin is reproduced from CIRZ by inserting the character of the corresponding field, it will be a sufficient test for our formula. In this section, we show how the spectral integral expressions for ζ ′ [∆,Y] (0) given in (2.28) and (2.35) for odd and even dimensional AdS can be reproduced starting from the CIRZ formula obtained in the previous section. This, besides being a check, displays interesting analogies between the spectral and Hurwitz zeta functions.

AdS 2r+1
Let us consider first the CIRZ formula (3.19) in odd dimensional AdS. By inserting the so(2, d) character, we obtain ,  By taking a derivative with respect to ∆, the formula becomes , (3.38) Using (3.2), (3.4) and (3.10), one can show that Therefore, the odd dimensional CIRZ formula (3.19) correctly reproduces the known result (2.27).

AdS 2r+2
In even dimensional AdS, we show how the difference of the zeta function can be obtained from the CIRZ (3.28). In fact, the derivation shows an interesting analogy with one wellknown property of Hurwitz zeta function: It can be derived from the integral representation, where the integral is along the Hankel contour. For Re(z) < 0 and a ∈ (0, 1), we can add to the contour an infinite clockwise circle. Then, the contour can be deformed to encircle the infinitely many simple poles arising from 1/ sinh β 2 : they are at β = 2π i n for all integer n = 0. Hence, we obtain from which we can easily derive (3.40). In particular, it allows to cancel the divergence of the zeta function at z = 1 and evaluate the finite part: Now let us move to the difference of the spectral zeta function. Since the difference of the character χ so(2,d) α) is even in β, the corresponding secondary contribution simply vanishes: Focussing on the primary contribution (3.28), we first find that the integrand function f 1,[∆;Y] reduces to by inserting the expression of χ so(2,d) (∆;Y) . By changing the order of the β and α integrals: we can focus first on the function ξ(z,∆) which plays an analogous role as the Hurwitz zeta function in the relation (3.40). In particular, ∂ ∂∆ ξ(z,∆, α) = −z ξ(z + 1,∆, α). We recast the integral (3.46) as the integral over the Hankel contour, (3.47) Like in the case of Hurwitz zeta function, we add to the contour an infinite clockwise circle and shrink it to enclose the infinite many simple poles arising, at this time, from sinh β±i α 2 : they are at β = (2π n ∓ α) i . By collecting the residues, we get which is divergent in the z → 1 limit. The divergence can be canceled by taking the difference,

Derivative expression of the CIRZ
In this section, we present a different expression of CIRZ in terms of derivatives in α i . As explained in Section 2, the combination dim µ Y (u) in the zeta function (2.5) is related to the Weyl dimension formula which can be obtained as a limit of the so(d + 2) characters. In Section 3, this limit was taken as a contour integral. The expression with contour integrals in α i variables is useful -see the companion paper [28] -but sometimes not explicit enough. For instance, if one wants to implement the CIRZ formula in a computer program, it will be more convenient to have an expression, where all the α i contour integrals are already evaluated using the residue theorem, involving α i derivatives of the so(d + 2) characters. In fact, for an expression in terms of α i derivatives, it is simpler to re-derive the CIRZ by taking the limit of so(d + 2) characters using a generalized L'Hôpital's rule. Below, we demonstrate how to obtain such an expression. The CIRZ for AdS 4 and AdS 5 originally presented in [14] are recovered as special cases.

General dimensions
In order to recover the Weyl character formula (3.3), we need to evaluate the so(d + 2) characters in the limit α → 0. It is actually subtle to perform the evaluation since the so(d + 2) characters take the form, and both the numerator and denominator vanish as α → 0 : The explicit expressions for the numerator N where the differential operator D Φ d+2 is given, in the notation ∂ i = ∂ α i , by Firstly, the denominator of (4.3) depends only on d : and can be explicitly evaluated as explained in Appendix B.1. The result reads (4.6) Secondly, the numerator of (4.3) can be recast, as explained in Appendix B.2, into where the differential operator D (n) is defined as Using (4.5) and (4.7), we can write the zeta function as where ρ ǫ (u) is given in (2.15). At this point, introducing the β-integral (3.5) will give rise to a factor e −β ∆ . By reconstructing the so(d + 2) character from e −β ∆ and N (d) Y and using the identity (A.8), we obtain .

Odd dimensional AdS
For d = 2r, the function ν ǫ (z, β, α) has been computed exactly as (3.15). The corresponding ϕ ǫ,n (β; z) is , (4.14) hence, the zeta function can be expressed as The β integral is convergent for Re(z) > d, but one can analytically continue z to other values. Since the only singularity of the integrand is at β = 0, we can deform the β integral to a complex integral with the contour encircling the origin counter-clockwise: In the end, the first derivative of the zeta function in AdS 2r+1 reads which contains the α i -derivatives instead of the contour integrals in (3.17).

Explicit expressions in low dimensions
In this section we spell out the explicit formulae for the zeta function in AdS 3 , AdS 5 and AdS 7 , and its primary and secondary contributions in AdS 4 and AdS 6 . In order to display the various formulae in a compact way, let us introduce .
(4.22) Then, the zeta function in AdS 2r+1 reads whose first derivative is given by On the other hand, the primary contribution to the zeta function in AdS 2r+2 reads Section 4, we derived a different CIRZ formula in terms of derivatives, generalizing the previous derivative expressions for AdS 4 and AdS 5 [14] to AdS 3 , AdS 6 and AdS 7 . This procedure also clearly generalizes to arbitrary dimensions. As outlined in the Introduction, the CIRZ is particularly useful to deal with theories with an infinite spectrum. When the spectrum can be captured by some CFT data, as is the case for partially massless higher-spin theories [43], one can compute the free energy of the theory without necessarily knowing the detailed decomposition of the spectrum. This will be done in the companion paper [28], where we will establish the matching of the one-loop corrections of partially massless higher-spin gravities with the 1/N corrections of the dual CFTs in any dimensions. The CIRZ method could also prove efficient in computing oneloop effects of a Kaluza-Klein tower. Indeed, the corresponding spectrum is obtained from the branching rule of the higher-dimensional field, and hence should be fully encompassed by the character of the latter.

A Character identities
In this Appendix we shall derive the identity (3.4) used in the main text. The character formula for the so(d + 2) irrep labelled by the highest weight (s 0 , Y), with Y = (s 1 , . . . , s r ), reads with the denominator and the numerator Here, ℓ j := s j + d 2 − j and det[a ij ] denotes the determinant of a matrix whose matrix element is a ij . The indices i, j range from 0 to r. The determinants appearing in (A.3) may be expanded as an alternating sum of minors: .
(A.4) The above expression can be recast into recursive formulae in d as . (A.5) For d = 2r + 1, (A.5) is straightforward to prove. For d = 2r, the formula (A.4) can be first expanded as then the relative minus sign in the second line can be absorbed into the determinant by changing the last column from sin(α i ℓ r ) to sin(−α i ℓ r ).
Similarly, one can decompose the denominator of an so(d + 2) character as a multiple of the denominator of an so(d) character expressed in terms of α k : where the overall factor (−1) k comes from re-ordering part of the product containing the α k variables in D (d+2) (α). Note that (A.7) can be expressed as by using the definition of the function P d defined in (3.2). Combining (A.5) and (A.8), we can finally express the so(d + 2) characters in terms of so(d) characters as . (A.9)

B Generalized L'Hôpital's rule
In this Appendix, we will discuss some technical details of the generalized L'Hôpital's rule (4.3) (see e.g. [44] for a pedagogical introduction) which are crucial in obtaining the derivative expression of CIRZ in Section 4. The differential operator D Φ d+2 is defined as In the orthonormal basis, the set of positive roots of so(d + 2) reads where {e k } k=0, ..., r is a basis of unit orthonormal vector of R r+1 . As a consequence, when acting on an so(d + 2) character, we have ∂ θ = 1 2 (∂ α i ± ∂ α j ) for θ = e i ± e j , and ∂ θ = ∂ α k for θ = e k .

(B.3)
From now on, we will use the notation ∂ i = ∂ α i . Since the differential operator (B.1) acts on both the numerator and denominator, we can rescale this operator to eliminate the factor 1 2 from ∂ θ with θ = e i ± e j . The differential operator D Φ d+2 then takes the form (4.4).

B.1 Computing the denominator
According to (A.2), the denominator of the so(d + 2) character depends on the dimension d+ 2, but not the highest weight (s 0 , Y) of the representation. As a result, the denominator of the generalized L'Hôpital's rule is simply a d-dependent constant, which is denoted c d in (4.5). We will review a part of the derivation of the Weyl dimension formula, which can be found in e.g. [44] and enables us to obtain an explicit formula for c d . First, recall that the denominator of an so(d + 2) character can be expressed as the product, where ρ is the so(d + 2) Weyl vector: and , denotes the inner product on the root space. Since in this paper we have used e iθ , instead of e θ , for the variables of the so(d + 2) character, the constant c d is given by Since the Weyl vector of so(d + 2) in the orthonormal basis is we finally obtain c d as given in (4.6).

B.2 Simplifying the numerator
Using (A.5), we can express N (d+2) in terms of N (d) , then the numerator of the generalized L'Hôpital's rule becomes Here, we took advantage of the facts that and that for d = 2r the differential operator D Φ d+2 is invariant under the α k → −α k transformation for any k = 0, . . . , r. The differential operator D Φ d+2 can then be rewritten as: ∂ k [d = 2r + 1] , (B.14) where D Φ d |k is expressed in terms of the unit vectors {e 0 , . . . , e k−1 , e k+1 , . . . , e r } of R r . Expanding the last product, we can then isolate the derivative with respect to α k :  With this decomposition at hand, we can write where we have used the fact that the dependency on the angle α k is confined to the function σ d (α k u). On top of that, this enable us to extract the u dependent part according to ∂ k . Notice finally that the remaining summands involve all variables {α i } i=0,...,r except for α k . As a consequence, the terms∂ all produce the same contribution for different k = 0, . . . , r. We can therefore drop the subscript k in the above expressions. In the end, we obtain (4.7) with (4.8), where the factor u d−2r is from the additional ∂ k in (B.14) for d = 2r + 1.