\section{\bf 1. Introduction.}
Finding the entanglement entropy associated with a spatial surface
presents an interesting computational challenge, {\it e.g.}\
[\putref{LNST,CaandH,Solodukhin2}], that has attracted some recent
interest, {\it e.g.}\ [\putref{HandW}], not least because of the introduction of
holographic notions, [\putref{RyandT,Solodukhin,MyandS}]. Most attention is
given to even space--time dimensions, $d$, but interest attaches also to
odd $d$, [\putref{MyandS,MyandS2,HandW}], and this is the case I wish to
address in this paper for the very special situation of spherical
surfaces and conformally invariant scalar fields.

In a previous work, [\putref{Dowhyp}], I have analysed the universal
logarithmic term that occurs in the expansion of the entanglement entropy
associated with even spheres in the special case when the conically
deformed Euclidean space--time is the cyclically factored sphere,
S$^d/\mbox{\open\char90}_q$. Then the spatial surface is a $(d-2)$--sphere (with
vanishing extrinsic curvature). Conformal transformations can be used to
move between different manifolds, in particular between spheres, planes
and cylinders.

A universal term is one that is regularisation independent. In even
dimensions, for a cut--off regularisation, this takes the form of a
logarithm, while, for dimensional and $\zeta$--function\ regularisation, it shows up as
a pole. The `problem' is that for odd dimensions there is nothing
analogous (for conformal fields), indeed, in the last two regularisations
there are no divergences at all and, connectedly, the integrated
conformal anomaly vanishes.

In [\putref{RyandT}] (see equn.(7.11)), it is suggested that, for odd
dimensions, the constant term in the entropy, {\it i.e. } the one independent of
any introduced cutoff, is such a universal term. Myers and Sinha,
[\putref{MyandS2}], have taken up this suggestion in their attempt to
produce a $c$--theorem in higher dimensions and have commented on its
validity. If there is no divergence, then the entire expression should,
presumably, be considered universal.

If boundaries or singularities exist then the relevant $C_{d/2}$
coefficient (determining the conformal anomaly) is non--zero and a
`proper' universal term might occur. This would not be so for the
(periodic) factored sphere, S$^d/\mbox{\open\char90}_q$ (or `periodic lune'), which was
the case discussed exclusively in [\putref{Dowhyp}] and will occupy me
mostly here too.

In the present paper I wish to look at the entire expression for the
entropy. I use $\zeta$--function\ regularisation and there is, in general, a pole term,
related to the integrated conformal anomaly, $C_{d/2}$, and a finite
remainder given by a functional determinant. I anticipate that the
results have interest beyond their entropic bearing.

As before, I will obtain the periodic results by combining those for
DN--lunes, which are lunes with Dirichlet (D) {\it or} Neumann (N)
conditions on the boundary. Adding these gives the periodic lune,
[\putref{ChandD,Down}]. Geometrically, the DN--lune is obtained by
factoring the sphere by the dihedral group, $[q]$, of order $2q$.
\footnotecheck\defaultoption[]\@footnote{ This is generated by reflections in two hyperplanes in the
embedding space.} When $q=1$, the resulting fundamental domain is the
DN--hemisphere.

In [\putref{Dowhyp}], I showed that the conformal anomaly on the even
dimensional periodic lune, S$^d/\mbox{\open\char90}_q$, {\it i.e. } the conformal heat--kernel
coefficient, $C_{d/2}(q)$, had an extremum at the ordinary sphere, $q=1$.
I will show that the same applies to the effective action (or functional
determinant) for odd dimensions.

\section{\bf2. The lune}

The $d$--lune can be defined inductively by giving its metric in the
nested form
  $$
  ds^2_{d-lune}=d\theta_d^2+\sin^2\theta_d\,\,ds^2_{(d-1)-lune}\,,\quad
  0\le \theta_i\le\pi\,,
  \eql{lune}
  $$
which is iterated down to the $1$--lune of metric $d\theta_1^2$ with
$0\le\theta_1\le\beta$. The angle $\theta_1$ is referred to as the polar angle
and conventionally written $\phi$. The angle of the lune is $\beta$.

The boundary of the lune comprises two pieces corresponding to $\phi=0$
and $\phi=\beta$. The metric (\puteqn{lune}) shows immediately that these are
unit $(d-1)$--hemispheres because {\it their} polar angle, $\theta_2$, runs
only from $0$ to $\pi$. Conditions, typically Dirichlet and Neumann, can
be applied at the boundary. The boundary parts intersect, with a constant
dihedral angle of $\beta$, in a $(d-2)$--sphere, of unit radius, which
constitutes a set of points fixed under O(2) rotations parametrised by
$\phi$.

It can be seen that the 2--lune submanifold, with coordinates $\theta_1$ and
$\phi$, has a wedge singularity at its north and south poles of. These
poles are at $\theta_2=0$ and $\theta_2=\pi$ and are the 0--hemispheres of a
0--sphere. In the $d$--lune, the submanifolds, $\theta_2=0$ and $\theta_2=\pi$,
are the $(d-2)$--hemispheres of the $(d-2)$--sphere of fixed points just
mentioned.

All this is for arbitrary angle, $\beta$. If $\beta=\pi/q$, $q\in\mbox{\open\char90}$, the
lune can be identified with the fundamental domain of the dihedral group
action on the sphere. As mentioned previously, restricting to just the
rotational subgroup doubles the fundamental domain to a periodic lune of
angle $2\pi/q$, the edges now being identified. This is equivalent to
adding the sets of $N$ and $D$ modes, [\putref{Down}] and produces a
conical singularity of deficit $2\pi-2\beta$. It corresponds to the
introduction of the uniformising angle, $\phi'=2\pi\phi/2\beta$, which runs
from $0$ to $2\pi$ as $\phi$ runs from $0$ to $2\beta$. The deformation of
the $d$--sphere, (\puteqn{lune}), can be traced to that of the 1--lune, or
$\phi'$--circle, which has unit radius but circumference, $2\beta$.

It is sometimes convenient, for visualisation purposes, to think of the
$d$--sphere as embedded in $\mbox{\open\char82}^{d+1}$ with the lune pictured as the
curved, outer surface of a `hyper--wedge'. The edges of the lune are
great unit $(d-1)$--hemispheres, half the intersections of the sphere
with two $d$--flats (hyperplanes) mutually intersecting in a
$(d-1)$--flat, the `axis' of the `rotation' that takes one $d$--flat into
the other with rotation angle, $\beta$. This axis intersects the sphere in
two `poles', isometric in total to a $(d-2)$--sphere and a singular
region. It is a fixed point set for those elements of O$(d+1)$ that
maintain the axis of rotation, {\it i.e. }\ an O$(2)$ subgroup.

If $\beta=\pi/q$, $q\in\mbox{\open\char90}$, the lune, and its $q$ reflections in the
hyperplanes, tessellate the $d$--sphere. Combining a lune with its
neighbouring reflection gives the fundamental domain for the rotational
(cyclic) subgroup, $\mbox{\open\char90}_q$, of the dihedral group.

To make the geometry a little clearer, I look at the $d=3$ case and write
out the Cartesian embedding of the unit 3--lune, ${\cal L}^3_\beta$, in my
notation,
  $$\eqalign{
  x_1&=\cos\theta_3\cr
   x_2&=\sin\theta_3\cos\theta_2\cr
    x_3&=\sin\theta_3\sin\theta_2\cos\phi\cr
    x_4&=\sin\theta_3\sin\theta_2\sin\phi\cr
  }
  $$
with $0\le\phi\le\beta$. Translations in $\phi$ correspond to rotations in
the $x_3$--$x_4$ plane with the fixed point $x_3=x_4=0$ giving the set of
fixed points on ${\cal L}^3_\beta$ as the unit circle, $x_1^2+x_2^2=1$ for all
openings, $\beta$.

The example of $\beta=\pi$ is easiest to picture. The boundary pieces of
${\cal L}^3_\pi$, at $\phi=0$ and $\phi=\pi$, form, respectively, the
$x_3\ge0$ and $x_3\le0$ hemispheres of the 2-sphere, $x_4=0$. The
boundaries of these pieces coincide with the fixed point unit circle
$x_1^2+x_2^2=1$ and join, with a dihedral angle of $\pi$, to form an
equator of the 2--sphere with poles, $x_3=\pm1$.

\section{\bf 3. Conformal Transformations}

The conformal transformations relevant here are those considered in
[\putref{ApandD2}], {\it i.e. } those between the original sphere, S$^d$, the
Euclidean plane, $\mbox{\open\char82}^d$, and the cylinder, $\mbox{\open\char82}\times$ S$^{d-1}$. The
latter would give the entropy on a $(d-1)$--sphere where the separating
surface is the equatorial $(d-2)$--sphere. This has vanishing extrinsic
curvature, whereas the second mapping yields the entropy on a flat
$(d-1)$ space with surface an ordinary $(d-2)$--sphere which does have
extrinsic curvature `generated' by the conformal transformation.
Actually, this geometrical circumstance was behind the calculations of
[\putref{Dowker2,Dowker1}] and [\putref{Solodukhin}] and I am, in reality,
going over part of this ground again.

In the particular example of the 3--lune, consider the section $x_4=0$ to
be a space section. Project it from the S$^3$ pole $x_3=1$ onto the
equatorial plane, $x_3=0$. The fixed point circle projects to a unit
circle, centre the origin, the inside of which is the projection of the
hemisphere with negative $x_3$ and the outside is the projection of that
with positive $x_3$.

The log coefficient is conformally invariant, and so it is immaterial
where it is evaluated. However the entropy, being determined by the
effective action, will generally change (in a well defined fashion).

The standard anomaly equation for the conformal variation of the
effective action, $W$, is, in $d$--dimensions,
  $$
  \delta W[e^{-2\omega}\,g]=C_{d/2}[e^{-2\omega}\,g;\delta\omega]\,,
  \eql{canom}
  $$
where $C_{d/2}[g;f]$ is the local heat--kernel coefficient averaged
against a test function, $f$
  $$
  C_{d/2}[g;f]=\int_{\cal M} C_{d/2}(g;x)\,f(x)\,.
  $$

Equation (\puteqn{canom}) can be integrated if the local coefficient is
known. For arbitrary dimension we do not have this luxury and, in any
case, the evaluation is complicated. It has been done only for $d=2,3,4$
and $6$. However, life is made easy in odd dimensions, if no boundary is
present (as is the case for the {\it periodic} lune) for then the
right--hand side of (\puteqn{canom}) is sero. Therefore the effective
actions on conformally related spaces are the same, and likewise for the
entropies according to the standard recipe given in the next section.

\section{\bf 4. Entropy}

According to the general prescription of Callan and Wilczek,
[\putref{CaandW}], the entropy is given in terms of the effective action,
$W^{(B)}$, on the Euclidean manifold deformed by a conical singularity of
angle $2\pi/B$ as,
  $$
   S=-(B\,\partial_B+1)\,W^{(B)}\bigg|_{B=1}\,.
   \eql{entropy}
  $$

In the present case, the deformed manifold, ${\cal M}_q$, is the periodic
lune, S$^d/\mbox{\open\char90}_q$, so that $B=q$. The field theory under consideration is
the scalar field conformally coupled in $d$ dimensions and the effective
action for this has been calculated in [\putref{Dow3,Dowjmp}]. The
evaluation of the derivative in (\puteqn{entropy}) involves a continuation
in $q$ from the integers \footnotecheck\defaultoption[]\@footnote{ The theory for any real $q$ is easily
worked out using eigenfunctions.}. In our previous work this was
straightforward as the relevant quantities, the heat--kernel
coefficients, were polynomial in $q$. For the effective action, the
matter is a little less obvious.

In terms of the $\zeta$--function, $\zeta(s,q)$, on ${\cal M}_q$, the unrenormalised
effective action is, generally,
  $$
  W^{(q)}={1\over2}\lim_{s\to0}{\zeta(0,q)\over s}-
  {1\over2}\zeta'(0,q)+\zeta(0,q)\log\,L+X\,,
  \eql{effact}
  $$
where $X$ is a possible finite correction and $\zeta(0,q),=C_{d/2}(q)$, the
conformal anomaly. The third term, where $L$ is a scaling length, can be
considered a concomitant of the ultraviolet pole divergence, (in this
version of $\zeta$--function\ regularisation). Taken together they correspond to the
log term that arises in the cut--off method. Conventionally, the pole
divergence would be removed by renormalisation. The `area law' for the
entanglement entropy is not recovered nor the other divergences
encountered in a cut--off or a lattice approach. This is not a worry as
these are non--universal terms anyway.

Generally, when $\zeta(0)$ vanishes, its role is, in some ways, taken over
by the derivative, $ \zeta'(0)$, this then also being conformally
invariant. Hence, as a working hypothesis, when treating the periodic
lune, I set $X$ to zero (it is a conformal invariant) and take the
entropy,
  $$
   S={1\over2}\bigg(1+q{\partial\over\partial q}\bigg)\zeta'(0,q)\bigg|_{q=1}\,,
   \eql{entropy2}
  $$
to be universal, in accordance with [\putref{RyandT}] and [\putref{MyandS2}]
and now have to address the calculation of the derivative.


\section{\bf 5. The log/divergence term}

As mentioned, in the present instance where $d$ is odd, the conformal
anomaly on the periodic lune is zero,
  $$
  C_{d/2}(q)=C^N_{d/2}(q)+C^D_{d/2}(q)=0\,,
  $$
and there is no pole nor log term. Nevertheless, for completeness, I
examine the individual coefficients as they would be required for
analysis on a DN--lune singly.

The required $C_{d/2}(q)$ coefficients are given by the expressions in
[\putref{Dowhyp}], where the original references can be found. One has,
  $$
  C^N_{d/2}(q)= -C^D_{d/2}(q)={1\over 2q\, d!}
  \bigg(B^{(d)}_d(d/2-1\big|\,q,{\bf 1})+
  B^{(d)}_d(d/2\big|\,q,{\bf 1})\bigg)\,,
  \eql{cees3}
  $$
The symmetry of the generalised Bernoulli polynomials,
  $$
   B^{(d)}_n(d-1+q-x|\,q,{\bf 1})=(-1)^n\,B^{(d)}_n(x|\,q,{\bf 1})\,,
  $$
should be noted.

For odd $d$ the right--hand side of (\puteqn{cees3}) simplifies to a
constant, independent of q and, therefore,
  $$
  C^N_{d/2}(q)= -C^D_{d/2}(q)={1\over 2\, d!}\,
  B^{(d)}_d(d/2-1)\,,
  \eql{cees4}
  $$
which is proved in the Appendix.


\begin{ignore}
 According to (\puteqn{entropy}), I need the $q$--derivative of
$C_{d/2}(q)$ and employ the result
  $$
    q\,\partial_q\,{1\over q}\,B^{(d)}_{d}(x|\,q,{\bf1})=
    -{1\over q}\,B^{(d+1)}_{d}(x+q|\,q,q,{\bf1})\,,
    \eql{diff}
  $$
to obtain
  $$
  q\partial_q\,C^N_{d/2}(q)\bigg|_{q=1}=q\partial_q\,C^D_{d/2}(q)\bigg|_{q=1}=0
  \eql{diff2}
  $$
using the fact that
  $$
  B^{(d+1)}_d(d/2)=-B^{(d+1)}_d(d/2+1)\,,\quad d\quad{\rm odd}
  \eql{prop}
  $$
\end{ignore}

Thus again, as in [\putref{Dowhyp}], the $q$--derivative of the conformal
anomaly is zero (trivially in this case) at $q=1$. Numerical values are

  $$\eqalign{
  C^{D,N}_{d/2}(q)&=\pm{1\over2d!}\,B^{(d)}_d(d/2-1)\cr
  &=\pm{1\over(d-1)!}\,B_1\,B^{(d-1)}_{d-1}(d/2-1)\cr
  &= \pm   [-{1\over48} ,  -{19\over1440},  {17\over11520},  {271\over120960},
   -{367 \over1935360},   -{3233\over7257600},\ldots]\,,
  }
  \eql{ceed2}
  $$
for $d=3,5,\ldots$.

This $q$--independence can be seen in a different way. For a
$d$--dimensional manifold, ${\cal M}$, the heat--kernel coefficients, $C_k$,
in general take contributions from all submanifolds, of codimension 0
down to $d$. It is easily checked that the integrand of the
half--integral coefficient, $C_{k/2}$, for {\it even} codimension, has
dimensions of an inverse {\it odd} power of length. Its construction must
then necessarily involve powers of the extrinsic curvatures but these all
vanish for the fundamental domains of the group action on the sphere,
which applies here. In particular, the codimension two contribution to
$C_{d/2}$ is zero and this is the part that involves the `dihedral'
angle, $\pi/q$ between the boundary parts of the manifold, ${\cal M}$.

For example, in three dimensions, $C_{3/2}$ then has just an area
contribution and a trihedral corner, or vertex, contribution,
[\putref{ApandD2}], but, in the case of a dihedral action, the vertex
degenerates into a dihedral one, and its contribution vanishes. The
explicit local form of the area part remaining is,
  $$
  C_{3/2}=\pm{1\over384\pi}\sum_i\int_{\partial{\cal M}_i}dA\,\widehat R\,,
  \eql{cee32}
  $$
where $\widehat R=2$ is the intrinsic curvature of the boundary part
$\partial{\cal M}_i$, a 2-hemisphere. These parts add to a full S$^2$'s worth of
boundary (for all $q$) and (\puteqn{cee32}) evaluates to $\pm1/48$, in
agreement with (\puteqn{ceed2}).

\section{\bf 6. The derivative term}

I now consider the universal term in the entropy (\puteqn{entropy2})
(actually the whole entropy). For the periodic lune, the $\zeta$--function\ is the sum
of the N and D $\zeta$--functions,
  $$
  \zeta(s,q)=\zeta_N(s,q)+\zeta_D(s,q)\,.
  $$

The dihedral $\zeta$--functions\ have been calculated in [\putref{Dow3}], and in
[\putref{Dowjmp}], in terms of the Barnes $\zeta$--function, $\zeta_d(s,a|\,{\bf d})$. In
particular the derivatives at zero are, \footnotecheck\defaultoption[]\@footnote{ The first two terms on
the right--hand side are suggested by the factorisation of the
eigenvalues with the final term indicating a multiplicative anomaly.}
  $$\eqalign{
  \zeta'_N(0,q)&=\zeta'_d(0,d/2|\,q,{\bf 1})+\zeta'_d(0,d/2-1|\,q,{\bf 1})+M(a_N,q)\cr
  \zeta'_D(0,q)&=\zeta'_d(0,d/2+q|\,q,{\bf 1})+\zeta'_d(0,d/2-1+q|\,q,{\bf
  1})+M(a_D,q)\,,\cr
  }
  \eql{zetas}
  $$
where $M(a,q)$ is given by,
  $$
  M(a,q)=-
  \sum_{r=1}^{(d-1)/2}{H^O_{r-1}\over r\,2^{2r}}\,N_{2r}(d,q)\,,
  \eql{manom}
  $$
with $H^O_r$ the odd harmonic number,
  $$
  H^O_r=\sum_{k=0}^r{1\over{2k+1}}\,.
  $$
The $N_l$ are the residues of the Barnes function,
  $$
  \zeta_d(s+l,a|\,{\bf d})\rightarrow\,{N_l\over s}+R_l\,,\quad {\rm
  as}\,\,s\rightarrow 0\,.
  \eql{poles}
  $$
They are given by generalised Bernoulli polynomials (see later) and
depend on $d$, the argument, $a$, and the parameters, ${\bf d}$, which
here reduce, in effect, to just the number, $q$. The dihedral Neumann and
Dirichlet arguments, $a_N$ and $a_D$, are given by $(d-1)/2$ and
$(d-1)/2+q$, respectively.

The two lines in (\puteqn{zetas}) give the derivatives of the $\zeta$--functions\ for
Neumann and Dirichlet boundary conditions on the edges of a lune of angle
$\pi/q$. As described earlier, adding them gives the $\zeta$--function\ for a
(periodic) lune, or cone, of angle $2\pi/q$ that is, for S$^d/\mbox{\open\char90}_q$
which is the object of most interest.

According to (\puteqn{entropy}), I need the derivative of (\puteqn{zetas}) with
respect to $q$. This is non--controversial for the $M$ terms as they are
rational in $q$. For the Barnes function, I will assume that the
derivatives with respect to $s$ and $q$ commute and that $q$ can be
continued to 1. This can be justified from the contour expressions for
the Barnes multiple functions (see the Appendix).

There now follows a technical discussion of the derivatives of the Barnes
function needed in (\puteqn{zetas}).

\section{\bf 7. Derivatives of the Barnes $\zeta$--function}

Starting from the formal definition \footnotecheck\defaultoption[]\@footnote{ The continuation in $q$ is
visually obvious.},
  $$
  \zeta_d(s,a|\,q,{\bf1})=\sum_{\bf n=0}^\infty{1\over (a+qn_1+\ldots n_d)^s}
  \eql{barnes}
  $$
simple manipulation leads to the derivative at $q=1$,
  $$
  \partial_q\,\zeta_d(s,a|\,q,{\bf1})\bigg|_{q=1}=-{s\over d}\,\zeta_d(s,a|\,{\bf
  1})+s{a\over d}\,\zeta_d(s+1,a|\,{\bf1})\,,
  \eql{qdif}
  $$
assuming that $a$ is independent of $q$. Relevant for this is the
important recursion,
  $$
  \zeta_d(s,a+q|\,q,{\bf1})=\zeta_d(s,a|\,q,{\bf1})+\zeta_{d-1}(s,a|\,{\bf1})\,,
  \eql{recurs}
  $$
which, looking at (\puteqn{zetas}), relates the N and D expressions. In
particular it means that the Dirichlet contribution to the
$q$--derivative equals the Neumann one.

The derivative with respect to $s$ and the limit $s\to0$ can now be taken
in (\puteqn{qdif}) with a view to substitution into (\puteqn{zetas}). Thus,
interchanging the implied limits,
  $$\eqalign{
   \partial_q\,\zeta'_d(0,a|\,q,{\bf1})\bigg|_{q=1}&=-{1\over
   d}\,\zeta_d(0,a|\,{\bf1})+{a\over d}\,R_1(d,a)\cr
   &={(-1)^{d-1}\over d\,d!}\,B_d^{(d)}(a)+{a\over d}\,R_1(d,a)\,.
   }
   \eql{qdif2}
  $$
$R_1$ is the remainder at the $s=1$ pole, (\puteqn{poles}), when all
parameters equal 1. Henceforth, I do not display dependence on these
parameters unless necessary.

Before putting things together, I give some details on the residue and
remainder. The latter is given by Barnes as
  $$
  R_r(d,a)=(-1)^r\bigg({1\over(r-1)!}\,\psi^{(r)}_d(a)-N_r(d,q)\,H_{r-1}\bigg)
  \eql{remain}
  $$
where $H_r$ is the usual harmonic number,
  $$
  H_r=\sum_{k=1}^r{1\over k}\,,\quad H_0=0\,,
  $$
and the $\psi$--functions are defined in terms of the multiple
$\Gamma$--function,
  $$
  \psi_d^{(p)}(a)={\partial^p\over\partial a^p}\,\log\Gamma_d(a)\,.
  \eql{psi}
  $$
Hence the residue $R_1(d,a)=-\psi_d^{(1)}(a)\equiv-\psi_d(a)$ and the
derivative (\puteqn{qdif2}) becomes
  $$\eqalign{
   \partial_q\,\zeta'_d(0,a|\,q,{\bf1})\bigg|_{q=1}=
   {(-1)^{d-1}\over d\,d!}\,B_d^{(d)}(a)-{a\over d}\,\psi_d(d,a)\,.
   }
   \eql{qdif5}
  $$
It is shown in the Appendix that this can be rewritten more compactly as
   $$\eqalign{
   \partial_q\,\zeta'_d(0,a|\,q,{\bf1})\bigg|_{q=1}=\psi_{d+1}(a+1)\,.
   }
   \eql{qdif7}
  $$
This result will be employed shortly.

I emphasise that I use Barnes' definitions of the multiple functions.

\section{\bf 8. Derivative of the multiplicative anomaly term}

I now compute the derivative of the multiplicative anomaly, $M$, terms in
(\puteqn{zetas}). I do not need to do this for the periodic lune, as these
terms cancel on addition, as shown next. However I give the results,
again for completeness.

I need the residues, $N_l$, given in terms of generalised Bernoulli
polynomials as
  $$\eqalign{
  N_l(d,q)&={(-1)^{d-l}\over(l-1)!(d-l)!}{1\over q}\,B^{(d)}_{d-l}(a|\,q,{\bf1})\cr
    &={1\over(l-1)!(d-l)!}{1\over q}\,B^{(d)}_{d-l}(d-1+q-a|\,q,{\bf1})\,.
    }
  \eql{residue}
  $$

If $l=2r$, as in (\puteqn{manom}), and $d$ is odd, the minus sign in
(\puteqn{residue}) shows that the $M$ terms in (\puteqn{zetas}) are opposite
for N and D conditions and would cancel if added, as advertised and as
noted in [\putref{Dow3,Dowjmp,DandKi2}].

I show that the $q$--derivative of $M(a_N,q)$, (\puteqn{manom}),  vanishes
at $q=1$ which circumstance follows directly from the general result,
[\putref{Barnesa}],
   $$
    q\,\partial_q\,{1\over q}\,B^{(d)}_{d-l}(a|\,q,{\bf1})=
    -{1\over q}\,B^{(d+1)}_{d-l}(a+q|\,q,q,{\bf1})\,.
    \eql{diff}
  $$

For, setting $a=a_N=(d-1)/2$ and $q=1$, the right--hand side equals
  $$
  -B^{(d+1)}_{d-2r}((d+1)/2)
  $$
which vanishes from a well known property of the Bernoulli polynomials
since $d-2r$ is odd.

It is interesting, and potentially more significant, to note that the
multiplicative anomaly, $M(a_N,q)$, is, in fact, independent of $q$, if
$d$ is odd, just like the conformal anomaly. For $d=3,5,\ldots$, one has,
  $$
  M(a_N,q)=-{1\over8},\,\,{5\over288},\,\,-{2303\over691200},
  \,\,{142601\over203212800},\,\ldots\,.
  \eql{manom2}
  $$

\section{\bf 9. The q--derivative on shell}

Formally differentiating the full $\zeta$--functions, (\puteqn{zetas}), with respect to
$q$ at 1, using (\puteqn{qdif7}) and the result of the previous section, I
find for the total Neumann $\zeta$--function, (the Dirichlet value is the same) that,
  $$\eqalign{
  \partial_q\zeta'_N(0,q)\bigg|_{q=1}=\psi_{d+1}(d/2)+\psi_{d+1}(d/2+1)\,.
  }
  \eql{qdif3}
  $$
The expression,
  $$\eqalign{
  \psi_d(a)={(-1)^{d-1}\over(d-1)!}B_{d-1}^{(d)}(a)\,\psi(a)-{1\over (d-1)!}
  \sum_{k=1}^{d-1}{(-1)^k\over k}\,B_{d-k-1}^{(d-k)}(d-a)\,B_{k}^{(k)}(a)\,,
  }
  \eql{psid2}
  $$
for the multiple $\psi$--function in terms of the standard digamma
function, $\psi(a)$, is derived in [\putref{dowgjms}], where other
references can be found.

Then, taking into account the antisymmetry for odd $d$,
  $$
  B^{(d+1)}_{d}(d/2)+B^{(d+1)}_{d}(d/2+1)=0\,,
  \eql{zero}
  $$
and the standard formula for the digamma function,
  $$
 \psi(1/2-n)= \psi(1/2+n)=-\gamma-2\log2+2H^O_{n-1}\,,\quad n\in\mbox{\open\char90}\,,
  $$
the transcendentals $\gamma$ and $\log2$ cancel leaving the purely algebraic
expression,
    $$\eqalign{
  =-{2\over d}&B^{(d+1)}_d(d/2+1)-\cr
  &(-1)^d\sum_{k=1}^{d}(-1)^k{B_{k-1}^{(k)}(d/2)
  \,B_{d+1-k}^{(d+1-k)}(d/2+1)\over d+1-k}
  -\cr&(-1)^d\sum_{k=1}^{d}(-1)^k
  {B_{k-1}^{(k)}(d/2+1)\,B_{d+1-k}^{(d+1-k)}(d/2)\over
  d+1-k}\,,\cr
  }
  \eql{qdif4}
  $$
which, unsurprisingly perhaps, evaluates to zero, by machine, dimension
by dimension.

An expression for the second $q$--derivative of the effective action is
derived in the Appendix. Its numerical evaluation shows an alternating
sign, being positive for $d=3$ (corresponding to a minimum for
$\zeta'(0,q)$ at $q=1$).

\section{\bf 10. The entropy}

According to this result and the definition, (\puteqn{entropy2}), the
entanglement entropy reduces to just half the quantity,
$\zeta'(0,q)\big|_{q=1}$, obtained by adding the ordinary ND--hemisphere
effective actions. These have been given above, (\puteqn{zetas}), and can be
formally expressed in terms of multiple $\Gamma$--functions,
    $$\eqalign{
   \zeta'_N(0,1)&=\log{\Gamma_d(d/2-1)\,\Gamma_d(d/2)\over\rho_d^2}+M(a_N,1)\cr
   \zeta'_D(0,1)&=\log{\Gamma_d(d/2+1)\,\Gamma_d(d/2)\over\rho_d^2}-M(a_N,1)\,,
   }
   \eql{zedash3}
  $$
where $\rho_d=\Gamma_{d+1}(1)$ is the multiple modular form,
[\putref{Barnesa}], and the $M$ term is given by (\puteqn{manom2}).

The full sphere effective action has a longish history, some of which is
detailed in [\putref{DandKi2}]. From the sum of the N and D expressions,
(\puteqn{zedash3}), (see [\putref{Dowjmp}] equns.(22),(23)), I find,
  $$
   \zeta'(0,1)=\log{\Gamma_d(d/2-1)\,\Gamma_d(d/2+1)\,\Gamma_d^2(d/2)\over\rho_d^4}\,.
   \eql{zedash5}
  $$

Explicit formulae can be obtained by expanding the Barnes $\zeta$--function\ in terms
of the Hurwitz $\zeta$--function\ with coefficients related to Stirling numbers,
[\putref{Barnesb}]. I just give some samples on the full sphere,
  $$\eqalign{
  \zeta'(0,1)&=-{3\over2}\,\zeta'_R(-2)-{1\over4}\log2\approx-0.127614\,,\quad d=3\cr
  &=-{5\over32}\,\zeta'_R(-4)-{1\over16}\,\zeta'_R(-2)+{1\over64}\log2\approx
  0.011486\,,\quad d=5\,.
  }
  \eql{zedash2}
  $$

Another form follows by rewriting (\puteqn{zedash5} as
  $$\eqalign{
   \zeta'(0,1)&=\log{\Gamma_{d+1}(d/2-1)\,\Gamma_{d+1}(d/2)\over
   \Gamma_{d+1}(d/2+2)\,\Gamma_{d+1}(d/2+1)}\cr
   &={2(-1)^{d+1}\over d!}\int_0^1 dz\,\,\pi z\,
  \tan\pi z\,\prod_{j=1}^{(d-1)/2}(z^2-(j-1/2)^2)
   }
   \eql{zedash4}
  $$
which is derived in [\putref{dowgjms}]. It corresponds to the $k=1$ case of
the GJMS operator, $P_{2k}$, which is just the usual conformal Laplacian.
The integrand can be expanded and contact made with (\puteqn{zedash2}) but
for numerical purposes (\puteqn{zedash4}) is adequate and I find for
$\zeta'(0,1)$,
  $$\eqalign{
  -0.001595\,,\quad &d=7\cr
  0.000262\,,\quad &d=9\cr
  -0.000047\,,\quad &d=11\,.
  }
  \eql{zedash6}
  $$
The values alternate in sign, like the conformal anomaly in even
dimensions.
\section{\bf 11. Conclusion}

The main technical result in the preceding is the vanishing of the
$q$--derivative of the effective action on the odd dimensional orbifold
S$^d/\mbox{\open\char90}_q$ at $q=1$, showing that the entanglement entropy associated
with a hyperspherical $(d-2)$--submanifold is essentially just the
effective action on the ordinary $d$--sphere, on certain assumptions
regarding universality.

I have not yet been able to prove the responsible Barnes--Bernoulli
identity, which, in terms of multiple functions reads,
  $$
  \psi_{d+1}(d/2)+\psi_{d+1}(d/2+1)=0
  $$
for all odd $d$.

The entropic significance of the numerical values on shell,
(\puteqn{zedash2}), (\puteqn{zedash6}), is unclear to me. Perhaps more
interesting would be results off criticality obtained, say, by adding a
mass term. Extension to other fields is also indicated and also to other
symmetric spaces.

\section{\bf Acknowledgments}

I thank Robert Myers for information and suggestions.

\section{\bf Appendix}

In this appendix I give some relevant material concerning the Barnes
functions and start with the proof of the $q$--independence of the N and
D $C_{d/2}$ coefficients,(\puteqn{cees4}), which I will do algebraically. I
seek to show that the bracket in (\puteqn{cees3}) is linear in $q$ and write
down the standard polynomial,
  $$
  B^{(n)}_\nu(x|\,q,{\bf1})=\sum_{s=0}^\nu q^s\comb \nu s B_s
  B^{(n-1)}_{\nu-s}(x)\,,
  \eql{poly}
  $$
noting that the only term odd in $q$ comes from the $s=1$ term in the
sum. Next I apply the formula ([\putref{Norlund}] p.167),
  $$
   B^{(n)}_\nu(x+q|\,q,{\bf1})=B^{(n)}_\nu(x|\,-q,{\bf1})
   \eql{neg}
  $$
to get
  $$\eqalign{
  B^{(d)}_d(d/2-1|\,-q,{\bf1})&=B^{(d)}_d(d/2-1+q|\,q,{\bf1})\cr
  &=-\,B^{(d)}_d(d/2|\,q,{\bf1})
  }
  $$
after using symmetry for odd $d$. Hence the bracket in (\puteqn{cees3}) is
twice the odd part of $B^{(d)}_d(d/2|\,q,{\bf1})$ obtained from
(\puteqn{poly}), thus proving (\puteqn{cees4}).

Contour integrals provide an alternative starting point. Barnes
[\putref{Barnesa}] p.406 gives an expression for the derivative at zero,
\footnotecheck\defaultoption[]\@footnote{ $q'$ is introduced for notational flexibility.}
  $$
  \zeta'_d(0,a|q,q',{\bf 1})={-i\over2\pi}\int_C
  {e^{-az}\big(\log(-z)+\gamma\big)dz\over z(1-e^{-qz})(1-e^{-q'z})(1-e^{-z})^{d-2}}
  \eql{zint}
  $$
and so
  $$
  \partial_q\zeta'_d(0,a|q,q',{\bf 1})={i\over2\pi}\int_C
  {e^{-(a+q)z}\big(\log(-z)+\gamma\big)dz\over(1-e^{-qz})^2 (1-e^{-q'z})
  (1-e^{-z})^{d-2}}\,.
  \eql{dzint}
  $$

To compare, Barnes also gives the integral for the $\psi_d$--function,
obtained simply by differentiating (\puteqn{zint}) with respect to $a$,
  $$
  \psi_d(a|q,q',{\bf 1})={i\over2\pi}\int_C
  {e^{-az}(\log(-z)+\gamma)dz\over (1-e^{-qz})(1-e^{-q'z})(1-e^{-z})^{d-2}}
  \eql{psiint}
  $$
so we obtain very simply (set $q'=1$ in (\puteqn{dzint}) and $q'=q$ in
(\puteqn{psiint})),
   $$\eqalign{
  \partial_q\zeta'_d(0,a|q,{\bf 1})&=\,\psi_{d+1}(a+q|\,q,q,{\bf1})\cr
  }
  \eql{qdif6}
  $$
Evaluating at $q=1$ yields,
  $$\eqalign{
  \partial_q\zeta'(0,a|q,{\bf 1})\bigg|_{q=1}&=\psi_{d+1}(a+1)\cr
  &=-{a\over d}\,\psi_{d}(a)+{(-1)^{d-1}\over d\,d!}\,B^{(d)}_d(a)
  }
  \eql{qdif10}
  $$
after using Barnes' recursion,
   $$
  \psi_{d+1}(a+1)=-{a\over d}\,\psi_{d}(a)-{1\over d}\,\zeta_{d}(0,a)\,,
  \eql{psid}
  $$
and so I have regained the expression (\puteqn{qdif5}), derived earlier by a
different method. \footnotecheck\defaultoption[]\@footnote{ One could turn this around and use this
development to prove the recursion, (\puteqn{psid}).}

Higher derivatives can be deduced. For example
  $$
    \eqalign{
  \partial^2_q\zeta'_d(0,a|q,{\bf 1})&=2\psi^{(2)}_{d+2}(a+q|\,q,q,q,{\bf1})+
  \psi^{(2)}_{d+1}(a+2q|\,q,q,{\bf1})
  \cr
  }
  \eql{qdif8}
  $$
whence, setting $q=1$,
   $$
    \eqalign{
  \partial^2_q\zeta'_d(0,a|q,{\bf 1})\bigg|_{q=1}&=2\psi^{(2)}_{d+2}(a+1)+
  \psi^{(2)}_{d+1}(a+2)\cr
  &={\partial^2\over\partial a^2}\,\big(2\psi_{d+2}(a+1)+\psi_{d+1}(a+2)\big)
  \cr
  }
  \eql{qdif9}
  $$
expressed via the trigamma function. This process can be continued
easily.

Algebraically. the right hand side of (\puteqn{qdif3}) vanishes because of
the factorisation
  $$
  \psi_{d+1}(a)+\psi_{d+1}(a+1)=(a-d/2)\,
  \bigg(P_d(a)\,\psi(a-d)+{Q_d(a)\over a-d}\bigg)
  $$
for odd $d$ where $P_d(a)$ and $Q_d(a)$ are polynomials such that there
is no pole at $a=d$.

\vfil\break
 \vglue 20truept

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\bye
