# About Notations in Multiway Array Processing

Jérémy E. Cohen

###### Abstract

This paper gives an overview of notations used in multiway array processing.
We redefine the vectorization and matricization operators to comply with some
properties of the Kronecker product. The tensor product and Kronecker product
are also represented with two different symbols, and it is shown how these
notations lead to clearer expressions for multiway array operations. Finally,
the paper recalls the useful yet widely unknown properties of the array normal
law with suggested notations.

## Introduction

Since tensors have become a popular topic in data science and signal
processing, a large number of papers have been published to describe at best
their different properties [1, 2, 3, 4, 5, 6, 7]. Yet a consensus on notations
has not been found. Some authors refer to the Tucker and Kruskal operators [8,
2] while others never use these notations but make use of the n-way product
[3, 9]. There is at least three different unfolding methods in the literature
[1, 3, 10]. The Kronecker product alone has two notations in the community
[11, 10, 1].

The tensor product itself is almost never used, even though it is the very
foundation of tensor algebra. Instead authors sometimes refer to the outer
product $\circ$, which may be seen as a tensor product of vectors in the
canonical basis of each vector space. But the tensor product allows for wider
generalization since it applies to tensors of any order and without referring
to any basis. Some authors suggest the use of two different symbols for the
Kronecker product $\mathop{\boxtimes}$ and the general tensor product
$\mathop{\otimes}$ [11]. But the same symbol has been used historically and
through the literature [12].

The Kronecker product is the expression of the tensor product for matrices
when a basis has been given. Yet using the same symbol leads to confusion when
manipulating arrays and multilinear operators at the same time. Moreover,
manipulating the arrays with matricizations and vectorizations exactly means
that the difference between a general tensor product and a basis-dependent
Kronecker product is of crucial importance. This leads also to redefining some
well known operations, namely the matricization and the vectorization.

The main goal of this paper is therefore to set notations for array and
manipulations on arrays that are consistent with one another, and with
notations used in quantum physics and algebraic geometry where tensors have
been used for decades [13]. We recommend that the vectorization and
matricization operators are computed so that the tensor products and Kronecker
products do not have to be swapped versions of each others, which is
compatible with definitions from algebraic geometry [10]. In other words, the
suggested notations will lead to

$\text{vec}\left(\boldsymbol{a}\mathop{\otimes}{\boldsymbol{b}}\right)=\boldsymbol{a}\mathop{\boxtimes}\boldsymbol{b},$

where $\mathop{\otimes}$ is the outer product, and $\mathop{\boxtimes}$ is the
Kronecker product. This differs from the usual equality
$\text{vec}\left(\boldsymbol{a}\mathop{\otimes}{\boldsymbol{b}}\right)=\boldsymbol{b}\mathop{\boxtimes}\boldsymbol{a}$,
which leads to unnecessarily complicated equations.

The first section introduces some notations for all operators used extensively
in multiway array processing. Then some useful properties of tensors are given
in Section II using the suggested notations. Finally Section III exposes the
array normal law as defined by Hoff [9]. It is shown that the array normal law
can be a handy tool for multiway array analysis.

##  I A Few Definitions

###  I-A About tensor products

First let us define the tensor product as given in [14].

###### Definition 1

Let $\mathcal{E}$, $\mathcal{F}$ be two vector spaces on a field
$\mathcal{K}$. There is a vector space
$\mathcal{E}\mathop{\otimes}\mathcal{F}$, called the tensor product space, and
a bilinear mapping

$\mathop{\otimes}~{}:~{}\mathcal{E}\times\mathcal{F}\rightarrow\mathcal{E}\mathop{\otimes}\mathcal{F}$

so that for every vector space $\mathcal{G}$ and for all bilinear mapping $g$
from $\mathcal{E}\times\mathcal{F}$ to $\mathcal{G}$, there is one and only
one linear mapping $h$ from $\mathcal{E}\mathop{\otimes}\mathcal{F}$ to
$\mathcal{G}$ defined by $g(x,y)=h(x\mathop{\otimes}y)$. This extends to
multilinear mappings.

In other words, the tensor product of vector spaces
$\left(\mathcal{E}_{i}\right)_{i\leq N}$ builds one (linear) vector space
$\bigotimes_{i}\mathcal{E}_{i}$.

This definition is not basis dependent, which means that the tensor product
applies not solely to arrays, and that real-valued tensors are items from a
tensor space $\bigotimes_{i\leq N}\mathds{R}^{n_{i}}$. When a basis is given
for the tensor space, this represents a tensor by an array of coordinates. Two
well known basis-dependent representation for the tensor product are the outer
product and the Kronecker product [3, 12]. First let us define the Kronecker
product of two arrays :

###### Definition 2

The Kronecker product of two arrays $\boldsymbol{A}\in\mathds{R}^{p_{1}\times
q_{1}}$ and $\boldsymbol{B}\in\mathds{R}^{p_{2}\times q_{2}}$ is denoted by
$\boldsymbol{A}\mathop{\boxtimes}\boldsymbol{B}\in\mathds{R}^{p_{1}p_{2}\times
q_{1}q_{2}}$ and is defined by:

$\boldsymbol{A}\mathop{\boxtimes}\boldsymbol{B}:=\left[\begin{array}[]{c|c|c|c}a_{11}\boldsymbol{B}&a_{12}\boldsymbol{B}&\dots&a_{1q_{1}}\boldsymbol{B}\\\
\hline\cr a_{21}\boldsymbol{B}&&&\\\ \hline\cr\vdots&&&\\\ \hline\cr
a_{p_{1}1}\boldsymbol{B}&&&a_{p_{1}q_{1}}\boldsymbol{B}\end{array}\right].$

We also give the definition of the outer product of two vectors in a given
basis.

###### Definition 3

The outer product of two vectors $\boldsymbol{a}\in\mathds{R}^{p_{1}}$ and
$\boldsymbol{b}\in\mathds{R}^{p_{2}}$ is denoted by
$\boldsymbol{a}\mathop{\otimes}\boldsymbol{b}\in\mathds{R}^{p_{1}\times
p_{2}}$ and is defined by :

$\left(\boldsymbol{a}\mathop{\otimes}\boldsymbol{b}\right)_{ij}=a_{i}b_{j}$

These definitions can be extended in a trivial manner to more than two arrays.

Ambiguity happens when the used tensor product maps to the matrix algebra, but
no basis is given. We then need to use the symbol $\mathop{\otimes}$ for this
tensor product. Moreover, the Kronecker product and the tensor product in
matrix algebra coincide when a basis is provided. It is shown in the rest of
this paper that this notation eases array manipulation significantly.

On the other hand, the outer product does not need a third symbol, since there
is no ambiguity with the tensor product. Whenever dealing with arrays, the
tensor product of vectors is to be read as an outer product, but in any other
case the outer product makes no sense. To comply with usual notations in
multilinear algebra, the symbol $\circ$ both proposed and later abandoned by
Comon [15, 16] should be dropped since it refers also to composition. Also,
the usual notation for the tensor product is $\mathop{\otimes}$, except in
part of the multiway array processing community.

Multiway array processing is based on applying multilinear operators, which
are themselves higher order tensors (see appendix A), on data arrays to mine
these data. A well-known example is the multilinear operator obtained by High
Order Singular Value Decomposition (HOSVD) [3] yielding compression for the
data array. However, when fixing a basis there is not a single definition of a
product of two tensors. In particular, if $\boldsymbol{U}_{i}$ are matrices in
$\mathds{R}^{q_{i}\times n_{i}}$, the operator $\bigotimes_{i\leq
N}\boldsymbol{U}_{i}$ acting on tensors in $\bigotimes_{i\leq
N}\mathds{R}^{n_{i}}$ cannot be expressed in a given basis with outer products
and Kronecker products [17]. Still for computing this product of tensors, some
authors defined the $k$-way product [2, 3] :

$\left(\boldsymbol{\mathcal{Y}}\bullet_{k}\boldsymbol{U}_{k}\right)_{i_{1},\dots,i_{N}}:=\sum_{j=1}^{n_{k}}{Y_{i_{1},\dots,i_{k-1},j,i_{k+1},\dots,i_{N}}\boldsymbol{U}_{i_{n},j}}$

The bullet symbol $\bullet$ should be preferred to the multiplication symbol
$\times$, since the latter may already have other meanings. Moreover, the
$k$-way product is the contraction of a matrix along its second mode with a
tensor along its $k$-th mode.

Now operators $\bigotimes_{i\leq N}\boldsymbol{U}_{i}$ are already extensively
used in quantum mechanics among others, and their action on a tensor is
denoted with an implicit multilinear product:

$\left(\bigotimes_{i=1}^{N}\boldsymbol{U}_{i}\right)\boldsymbol{\mathcal{Y}}:=\boldsymbol{\mathcal{Y}}\bullet_{1}\boldsymbol{U}_{1}\dots\bullet_{N}\boldsymbol{U}_{N}$

which is to be read as $N$ contractions for the second mode of each matrix in
the tensor product with the $i$-th mode of the tensor. This also provides a
computation mean for the application of an operator on multiway arrays. One
can simply sequentially unfold the tensor in each mode, multiply the resulting
matrix with the matrix operator $\boldsymbol{U}_{i}$ expressed in the right
basis, and refold the tensor. For matrices, the 2-mode product is indeed well
known :

$\boldsymbol{M}\bullet_{1}\boldsymbol{U}_{1}\bullet_{2}\boldsymbol{U}_{2}=\boldsymbol{U}_{1}\boldsymbol{M}\boldsymbol{U}_{2}^{T}$

One last operation useful for multiway array processing is the (basis-
dependent) Khatri-Rao product :

$\boldsymbol{A}\odot\boldsymbol{B}=\left[\boldsymbol{A}_{:1}\mathop{\boxtimes}\boldsymbol{B}_{:1},\dots,\boldsymbol{A}_{:p}\mathop{\boxtimes}\boldsymbol{B}_{:p}\right]$

###  I-B About manipulation of arrays

A major claim of this paper is to modify the usual definition of the
vectorization operator for arrays [18]. For now, the usual definition states
that an array should be vectorized choosing elements by the reverse
lexicographic order of indices, i.e. columns after columns in the first mode,
sliding along the second mode, then the third mode and so on (see figure 1a).

Yet every tensor can be expressed as a sum of $R$ tensor products of vectors
by the following model, called Canonical Polyadic (CP) decomposition or
PARAFAC [19, 20] :

$\boldsymbol{\mathcal{Y}}=\sum\limits_{r=1}^{R}{\bigotimes_{i=1}^{N}\boldsymbol{a}_{r}^{(i)}}$

and using the columnwise vectorization is not compatible with the way the
Kronecker product and the CP decomposition are defined. To see this, take a
matrix $\boldsymbol{M}$ defined by an outer product
$\boldsymbol{M}=\boldsymbol{a}\mathop{\otimes}\boldsymbol{b}$. When
vectorizing $\boldsymbol{M}$ columnwise, it is well known that vectors
$\boldsymbol{a}$ and $\boldsymbol{b}$ are swapped to express $\boldsymbol{M}$
as a Kronecker product, since in its definition the Kronecker product makes
the second index vary first :
$\text{vec}(\boldsymbol{M})=\boldsymbol{b}\mathop{\boxtimes}\boldsymbol{a}$.
The point here is that once the Kronecker product has been defined, it fixes
the direction for the vectorization of any array in order to get a direct
property like
$\text{vec}\left(\bigotimes_{i}\boldsymbol{a}_{i}\right)=\bigboxtimes_{i}\boldsymbol{a}_{i}$.

Thus we give the following definition of the vectorization operator of an
array, along with some basic properties in Table I:

###### Definition 4

The vectorization of an array $\boldsymbol{\mathcal{Y}}$, denoted
$\text{vec}(\boldsymbol{\mathcal{Y}})$, is the isomorphism defined as the
following function of its CP decomposition:

$\displaystyle\mathds{R}^{n_{1}\times\dots\times n_{N}}$
$\displaystyle\rightarrow\mathds{R}^{\prod\limits_{i=1}^{N}{n_{i}}\times 1}$
$\displaystyle\text{vec}:\quad\sum\limits_{r=1}^{R}{\bigotimes_{i=1}^{N}\boldsymbol{a}_{r}^{(i)}}$
$\displaystyle\mapsto\sum\limits_{r=1}^{R}{\bigboxtimes_{i=1}^{N}\boldsymbol{a}_{r}^{(i)}}$

This definition means the elements in the array are taken in the lexicographic
order of their index, i.e. along the last mode, then along the mode $N-1$ and
so on (see figure 1b).

In the same spirit, we define the matricization (or the unfolding) of an array
similarly to what is called flattening in algebraic geometry [10]. Some
properties are given in table I.

###### Definition 5

The matricization of an array $\boldsymbol{\mathcal{Y}}$ along the $i$-th
mode, denoted $\left[\boldsymbol{\mathcal{Y}}\right]_{(i)}$ is the isomorphism
defined as the following function of its CP decomposition :

$\displaystyle\mathds{R}^{n_{1}\times\dots\times n_{N}}$
$\displaystyle\rightarrow\mathds{R}^{n_{i}\times\prod\limits_{j\neq
i}^{N}{n_{j}}}$
$\displaystyle\left[.\right]_{(i)}:\quad\sum\limits_{r=1}^{R}{\bigotimes_{j=1}^{N}\boldsymbol{a}_{r}^{(j)}}$
$\displaystyle\mapsto\sum\limits_{r=1}^{R}{\boldsymbol{a}_{r}^{(i)}\mathop{\otimes}\bigboxtimes_{j\neq
i}^{N}\boldsymbol{a}_{r}^{(j)}}$

(1)i(2)j(3)ka) columnwise vectorization(3)i(2)j(1)kb) suggested vectorization
Figure 1: Two possibilities for vectorizing an array. If $Y_{ijk}$ is an
element in array $\boldsymbol{\mathcal{Y}}$ of size $I\times J\times K$, then
in the vectorized form $\boldsymbol{y}$ it is located at : a)
$y_{(k-1)IJ+(j-1)I+i}$ b) $y_{(i-1)JK+(j-1)K+k}$
$\boldsymbol{M}_{1}$$\boldsymbol{M}_{K}$…$\begin{aligned}
\boldsymbol{Y}_{(1)}&=&\left[\color[rgb]{0.625,0.11328125,0.20703125}\definecolor[named]{pgfstrokecolor}{rgb}{0.625,0.11328125,0.20703125}\boldsymbol{M}_{1}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}|\dots|\color[rgb]{0.625,0.11328125,0.20703125}\definecolor[named]{pgfstrokecolor}{rgb}{0.625,0.11328125,0.20703125}\boldsymbol{M}_{K}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\right]\\\
\boldsymbol{Y}_{(2)}&=&\left[\color[rgb]{0.819,0.756,0.378}\definecolor[named]{pgfstrokecolor}{rgb}{0.819,0.756,0.378}\boldsymbol{N}_{1}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}|\dots|\color[rgb]{0.819,0.756,0.378}\definecolor[named]{pgfstrokecolor}{rgb}{0.819,0.756,0.378}\boldsymbol{N}_{I}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\right]\\\
\boldsymbol{Y}_{(3)}&=&[\color[rgb]{0.819,0.756,0.378}\definecolor[named]{pgfstrokecolor}{rgb}{0.819,0.756,0.378}\boldsymbol{N}_{1}^{T}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}|\dots|\color[rgb]{0.819,0.756,0.378}\definecolor[named]{pgfstrokecolor}{rgb}{0.819,0.756,0.378}\boldsymbol{N}_{I}^{T}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}]\end{aligned}$where
$\boldsymbol{M}_{i}\in\mathds{R}^{I\times K}$ and
$\boldsymbol{N}_{i}\in\mathds{R}^{J\times
K}$$\boldsymbol{N}_{1}$$\boldsymbol{N}_{I}$⋮ Figure 2: Unfoldings of a three-
way array $\boldsymbol{\mathcal{Y}}$ of size $I\times J\times K$

This definition can be exploited to obtain the unfoldings of a three way array
$\boldsymbol{\mathcal{Y}}$ as shown in figure 2. For example, a compact MATLAB
R2014b code for computing the three unfoldings is:

$\displaystyle\boldsymbol{Y}_{(1)}$ $\displaystyle=$
$\displaystyle\text{reshape}\left(\text{permute}\left(\boldsymbol{\mathcal{Y}},[1,3,2]\right),n_{1},n_{2}n_{3}\right)$
$\displaystyle\boldsymbol{Y}_{(2)}$ $\displaystyle=$
$\displaystyle\text{reshape}\left(\text{permute}\left(\boldsymbol{\mathcal{Y}},[2,3,1]\right),n_{2},n_{1}n_{3}\right)$
$\displaystyle\boldsymbol{Y}_{(3)}$ $\displaystyle=$
$\displaystyle\text{reshape}\left(\text{permute}\left(\boldsymbol{\mathcal{Y}},[3,2,1]\right),n_{3},n_{1}n_{2}\right)$

##  II Some Useful Properties

When manipulating multiway arrays, there are multiple properties that may
simplify calculations by wide margins. Formula (16) is useful for obtaining a
compact formula for the Jacobian of a CP model or obtaining bounds on
estimation errors. Table I provides overall well known results [12], with our
suggested notations. Note that permutation of vectors or operators when
switching from tensor products to vectorized arrays do not appear anymore.

In what follows, some well-studied decompositions are revisited with the
notations we promote.

#### Exemple 1

The Tucker decomposition of a tensor $\boldsymbol{\mathcal{Y}}$

$Y_{i_{1}\dots i_{N}}=\sum\limits_{r_{1}\dots r_{N}}^{R_{1}\dots
R_{N}}{\left(\prod\limits_{k=1}^{N}{U_{i_{k}r_{k}}}\right)G_{r_{1}\dots
r_{N}}}$

can be expressed independently of any fixed basis as the following :

$\boldsymbol{\mathcal{Y}}=\left(\bigotimes_{i=1}^{N}\boldsymbol{U}_{i}\right)\boldsymbol{\mathcal{G}}$

where operators $\boldsymbol{U}_{i}$ may be supposed unitary for
identifiability of the model.

Thus finding the Tucker decomposition of a tensor means finding linear
operators $\boldsymbol{U}_{i}$ in $\mathds{R}^{n_{i}\times R_{i}}$ so that the
multilinear operator $\left(\bigotimes_{i=1}^{N}\boldsymbol{U}^{T}_{i}\right)$
projects the tensor on its true subspace $\bigotimes_{i}\mathds{R}^{R_{i}}$,
in the same spirit as Principal Component Analysis.

Also from Table I, unfoldings can be easily obtained:

$\displaystyle\boldsymbol{Y}_{(i)}$
$\displaystyle=\left(\boldsymbol{U}_{i}\otimes\bigotimes_{j\neq
i}^{N}\boldsymbol{U}_{j}\right)\boldsymbol{G}_{(i)}$
$\displaystyle=\boldsymbol{U}_{i}\boldsymbol{G}_{(i)}\left(\bigboxtimes_{j\neq
i}^{N}\boldsymbol{U}_{j}^{T}\right)$

Of course for actual computation of the action of
$\left(\bigotimes_{i=1}^{N}\boldsymbol{U}_{i}^{T}\right)$ on
$\boldsymbol{\mathcal{Y}}$, all the $i$-way
$\boldsymbol{\mathcal{Y}}\bullet_{i}\boldsymbol{U}_{i}^{T}$ products may first
be computed sequentially before matricizing the tensor.

#### Example 2

In the first section, the CP decomposition of a tensor was defined. With our
notations, this model can be easily expressed as follows:

$\boldsymbol{\mathcal{Y}}=\sum\limits_{r=1}^{R}{\bigotimes_{i=1}^{N}\boldsymbol{a}_{r}^{(i)}}=\left(\bigotimes_{i=1}^{N}\boldsymbol{A}_{i}\right)\boldsymbol{\mathcal{I}}$

where
$\boldsymbol{A}_{i}=\left[\boldsymbol{a}_{1}\dots\boldsymbol{a}_{R}\right]$
and $\boldsymbol{\mathcal{I}}$ is the diagonal tensor with entries equal to
$1$. The CP model can be described with unfoldings and in a vectorized form
using (15) from table I :

$\displaystyle\boldsymbol{Y}_{(i)}$
$\displaystyle=\boldsymbol{A}_{i}\boldsymbol{\mathcal{I}}_{(i)}\left(\bigboxtimes_{j\neq
i}^{N}\boldsymbol{A}_{j}\right)^{T}=\boldsymbol{A}_{i}\left(\bigodot_{j\neq
i}^{N}\boldsymbol{A}_{j}\right)^{T}$
$\displaystyle\text{vec}(\boldsymbol{\mathcal{Y}})$
$\displaystyle=\sum\limits_{r=1}^{R}{\bigboxtimes_{i=1}^{N}\boldsymbol{a}_{r}^{(i)}}=\left(\bigodot_{i=1}^{N}\boldsymbol{A}_{i}\right)\mathds{1}$

where $\mathds{1}$ is a vector of ones.

$\displaystyle\text{vec}(\boldsymbol{a}\mathop{\otimes}\boldsymbol{b})=\boldsymbol{a}\mathop{\boxtimes}\boldsymbol{b}$
(1)
$\displaystyle\text{vec}\left(\boldsymbol{A}\boldsymbol{X}\boldsymbol{B}^{T}\right)=\left(\boldsymbol{A}\mathop{\boxtimes}\boldsymbol{B}\right)\text{vec}\left(\boldsymbol{X}\right)$
(2)
$\displaystyle\left(\boldsymbol{A}\mathop{\otimes}\boldsymbol{B}\right)\left(\boldsymbol{C}\mathop{\otimes}\boldsymbol{D}\right)=\left(\boldsymbol{A}\boldsymbol{C}\mathop{\otimes}\boldsymbol{B}\boldsymbol{D}\right),\text{
if compatible}$ (3)
$\displaystyle\left(\bigotimes_{i=1}^{N}\boldsymbol{U}_{i}\right)\left(\bigotimes_{i=1}^{N}\boldsymbol{a}^{(i)}\right)=\left(\bigotimes_{i=1}^{N}\boldsymbol{U}_{i}\boldsymbol{a}^{(i)}\right)$
(4)
$\displaystyle\left(\boldsymbol{A}\mathop{\boxtimes}\boldsymbol{B}\right)\left(\boldsymbol{C}\odot\boldsymbol{D}\right)=\left(\boldsymbol{A}\boldsymbol{C}\odot\boldsymbol{B}\boldsymbol{D}\right),\text{
if compatible}$ (5)
$\displaystyle\left(\boldsymbol{A}\mathop{\otimes}\boldsymbol{B}\right)^{T}=\boldsymbol{A}^{T}\mathop{\otimes}\boldsymbol{B}^{T}$
(6)
$\displaystyle\boldsymbol{A}\boldsymbol{X}+\boldsymbol{XB}=\boldsymbol{D}\Leftrightarrow\left(\boldsymbol{A}\mathop{\boxtimes}\boldsymbol{B}\right)\text{vec}\left(\boldsymbol{X}\right)=\text{vec}\left(\boldsymbol{D}\right)$
(7)
$\displaystyle\text{vec}\left(\bigotimes_{i=1}^{N}\boldsymbol{a}^{(i)}\right)=\bigboxtimes_{i=1}^{N}\boldsymbol{a}^{(i)}\text{
in the same order}$ (8)
$\displaystyle\boldsymbol{\mathcal{Y}}=\boldsymbol{\mathcal{X}}\bullet_{i}\boldsymbol{U}\Leftrightarrow\boldsymbol{Y}_{(i)}=\boldsymbol{U}\boldsymbol{X}_{(i)}$
(9)
$\displaystyle\text{vec}\left(\left(\bigotimes_{i=1}^{N}\boldsymbol{U}_{i}\right)\boldsymbol{\mathcal{Y}}\right)=\left(\bigboxtimes_{i=1}^{N}\boldsymbol{U}_{i}\right)\text{vec}\left(\boldsymbol{\mathcal{Y}}\right)$
(10)
$\displaystyle\left[\left(\bigotimes_{j=1}^{N}\boldsymbol{U}_{j}\right)\boldsymbol{\mathcal{Y}}\right]_{(i)}=\boldsymbol{U}_{i}\boldsymbol{Y}_{(i)}\left(\bigboxtimes_{j\neq
i}^{N}\boldsymbol{U}_{j}^{T}\right)$ (11)
$\displaystyle\left[\left(\bigotimes_{j=1}^{N}\boldsymbol{U}_{j}\right)\boldsymbol{\mathcal{Y}}\right]_{(i)}=\left(\boldsymbol{U}_{i}\mathop{\otimes}\left(\bigotimes_{j\neq
i}^{N}\boldsymbol{U}_{j}^{T}\right)\right)\boldsymbol{Y}_{(i)}$ (12)
$\displaystyle\boldsymbol{\mathcal{Y}}=\sum\limits_{r=1}^{R}{\bigotimes_{j=1}^{N}\boldsymbol{a}_{r}^{(j)}}\Leftrightarrow\boldsymbol{Y}_{(i)}=\sum\limits_{r=1}^{R}{\boldsymbol{a}^{(i)}_{r}\mathop{\otimes}\bigboxtimes_{j\neq
i}^{N}\boldsymbol{a}_{r}^{(i)}}$ (13)
$\displaystyle\sum\limits_{r=1}^{R}{\boldsymbol{a}_{r}\mathop{\boxtimes}\boldsymbol{b}_{r}}=\left(\boldsymbol{A}\odot\boldsymbol{B}\right)\mathds{1},\text{
where }\boldsymbol{A}=\left[\boldsymbol{a}_{1}\dots\boldsymbol{a}_{R}\right]$
(14)
$\displaystyle\boldsymbol{Y}_{(i)}=\sum\limits_{r=1}^{R}{\boldsymbol{a}^{(i)}_{r}\mathop{\otimes}\bigboxtimes_{j\neq
i}^{N}\boldsymbol{a}_{r}^{(j)}}=\boldsymbol{A}_{i}\left(\bigodot_{j\neq
i}^{N}\boldsymbol{A}_{j}^{T}\right)$ (15)
$\displaystyle\frac{\partial\bigotimes_{j=1}^{N}\boldsymbol{a}_{j}}{\partial\boldsymbol{a}_{i}}=\boldsymbol{a}_{1}\mathop{\otimes}\dots\mathop{\otimes}\boldsymbol{a}_{i-1}\mathop{\otimes}\boldsymbol{I}\mathop{\otimes}\dots\mathop{\otimes}\boldsymbol{a}_{N}$
(16)
$\displaystyle\left|\bigotimes_{i=1}^{N}\boldsymbol{U}_{i}\right|=\prod_{i=1}^{N}\left|\boldsymbol{U}_{i}\right|^{\prod_{j\neq
i}^{N}n_{j}}$ (17) TABLE I: Some useful properties

Applying a multilinear transformation to the tensor results in operating
$\bigotimes_{i=1}^{N}\boldsymbol{W}_{i}$ on $\boldsymbol{\mathcal{Y}}$, which
has the following CP model :

$\left(\bigotimes_{i=1}^{N}\boldsymbol{W}_{i}\right)\boldsymbol{\mathcal{Y}}=\sum\limits_{r=1}^{R}{\bigotimes_{i=1}^{N}\boldsymbol{W}_{i}\boldsymbol{a}_{r}^{(i)}}.$

This kind of preprocessing is very useful if the observation noise is
correlated, or in some applications e.g. flexible coupled tensor decomposition
[21] where the decomposition may be computed in a transformed domain for one
of the mode.

##  III Second Order Statistics for Tensors

When dealing with Gaussian non-i.i.d. multiway arrays, the most natural way of
describing the distribution would be to give a definition to array normal
laws. This has been done for matrices [22], and recently for arrays of any
order [9]. The definition from [9] is the following :

###### Definition 6

Let $\boldsymbol{\mathcal{X}}$ be a multivariate random variable in
$\mathds{R}^{n_{1}\times\dots\times n_{N}}$. We say that
$\boldsymbol{\mathcal{X}}$ follows an array normal law of mean
$\boldsymbol{\mathcal{M}}$ and with tensor covariance
$\boldsymbol{\Gamma}=\bigotimes_{i=1}^{N}\boldsymbol{\Sigma}_{i}$ if and only
if

$p\left(\boldsymbol{\mathcal{X}}|\boldsymbol{\mathcal{M}},\boldsymbol{\Gamma}\right)=\frac{\text{exp}\left(-\frac{\|\boldsymbol{\Gamma}^{-\frac{1}{2}}(\boldsymbol{\mathcal{X}}-\boldsymbol{\mathcal{M}})\|^{2}}{2}\right)}{(2\pi)^{\prod_{i}{\frac{n_{i}}{2}}}\left|\boldsymbol{\Gamma}\right|^{\frac{1}{2}}}$

where
$\boldsymbol{\Gamma}^{-\frac{1}{2}}=\bigotimes_{i=1}^{N}\boldsymbol{\Sigma}_{i}^{-\frac{1}{2}}$
and $\boldsymbol{\Sigma}_{i}$ are symmetric. It is noted as
$\boldsymbol{\mathcal{X}}\sim\mathcal{A}\mathcal{N}\left(\boldsymbol{\mathcal{M}},\boldsymbol{\Gamma}\right)$

Array normal distributions are useful when manipulating the data arrays before
decomposing them. They suppose the covariance is expressed in every mode, i.e.
the covariance of the vectorized tensor can be expressed as a Kronecker
product of $N$ symmetric matrices, i.e. the covariance is separable.

For example, say it is needed to preprocess some tensor
$\boldsymbol{\mathcal{Y}}$, noisy version of $\boldsymbol{\mathcal{X}}$
corrupted by i.i.d. Gaussian noise, with the multilinear operator
$\bigotimes_{i\leq N}\boldsymbol{U}_{i}$. Then the array
$\boldsymbol{\mathcal{T}}=\left(\bigotimes_{i\leq
N}\boldsymbol{U}_{i}\right)\boldsymbol{\mathcal{Y}}$ follows the array normal
law

$\boldsymbol{\mathcal{T}}\sim\mathcal{AN}\left(\left(\bigotimes_{i=1}^{N}\boldsymbol{U}_{i}\right)\boldsymbol{\mathcal{X}},\bigotimes_{i=1}^{N}\boldsymbol{U}_{i}\boldsymbol{U}_{i}^{T}\right).$

Now with notations from two way array processing [22], the matrix law of the
$i_{th}$ unfolding is given by :

$\boldsymbol{T}_{(i)}\sim\mathcal{MN}\left(\boldsymbol{U}_{i}\boldsymbol{X}_{(i)}\bigboxtimes_{j\neq
i}^{N}\boldsymbol{U}_{j}^{T},\boldsymbol{U}_{i}\boldsymbol{U}_{i}^{T},\bigboxtimes_{j\neq
i}^{N}\boldsymbol{U}_{j}\boldsymbol{U}_{j}^{T}\right)$

and the normal law of the vectorized tensor is :

$\text{vec}(\boldsymbol{\mathcal{T}})\sim\mathcal{N}\left(\left(\bigboxtimes_{i=1}^{N}\boldsymbol{U}_{i}\right)\text{vec}(\boldsymbol{\mathcal{X}}),\bigboxtimes_{i=1}^{N}\boldsymbol{U}_{i}\boldsymbol{U}_{i}^{T}\right)$

A direct consequence is that preprocessing one mode of a noisy tensor modifies
the covariance of the noise in this mode but this does not affect the other
modes. In a least square procedure for fitting the CP decomposition, this
results in modifying the maximum likelihood estimate for the modified mode
only.

## Concluding remarks

This paper proposed a reworked set of notations for the multiway array
processing community. The notations commonly used are sometimes not compatible
with other communities, and the author believes the suggested notations
simplify calculations with arrays. The permutation induced by the column wise
vectorization operator is a source of error, as well as the confusion between
tensor products and the Kronecker product. Moreover, unfolding operators given
in the literature suggest a similar permutation, which is also unnecessary.
With proposed notations, there are no more permutations anywhere in the
computation of matricized and vectorized tensors. Finally, this framework
enables a simple definition of the multivariate normal distribution for arrays
of any order, which is useful for example in computing decompositions of noisy
preprocessed tensors.

Notations in this paper are of course one choice among others, and some
definitions are not the ones usually used. For digging further into tensors,
some nice surveys are available [1, 11, 23].

##  Appendix A Linear operators acting on tensors

We state and prove that linear operators acting on a tensor space of finite
dimension linear spaces is itself a tensor space. This justifies the notation
$\bigotimes_{i=1}^{N}\boldsymbol{U}_{i}$ suggested in this paper. A deeper
proof including infinite dimensions can be found in the excellent book by
Hackbusch [24].

Let $\mathcal{E}$, $\mathcal{E}^{\prime}$, $\mathcal{F}$ and
$\mathcal{F}^{\prime}$ be four finite vector spaces on a field $\mathcal{K}$,
of respective dimensions $n,n$ and $m,m$. Let $\otimes$ and $\otimes^{\prime}$
be two tensor products on $\mathcal{E}\times\mathcal{F}$ and
$\mathcal{E}^{\prime}\times\mathcal{F}^{\prime}$, and consider the following
mapping:

$\begin{array}[]{cccc}&\mathcal{L}(\mathcal{E},\mathcal{E}^{\prime})\times\mathcal{L}(\mathcal{F},\mathcal{F}^{\prime})&\rightarrow&\mathcal{L}(\mathcal{E}\otimes\mathcal{F},\mathcal{E}^{\prime}\otimes^{\prime}\mathcal{F}^{\prime})\\\
\otimes_{p}:&\quad(u,v)&\mapsto&u\otimes_{p}v:(x\otimes y)\mapsto
u(x)\otimes^{\prime}v(y)\end{array}$

where $\mathcal{L}(\mathcal{E},\mathcal{E}^{\prime})$ is the linear space of
linear operators mapping $\mathcal{E}$ to $\mathcal{E}^{\prime}$.

###### Theorem 1

$\mathcal{L}(\mathcal{E}\otimes\mathcal{F},\mathcal{E}^{\prime}\otimes^{\prime}\mathcal{F}^{\prime})$
associated with the bilinear mapping $\otimes_{p}$ is a tensor space, i.e.

$\mathcal{L}(\mathcal{E}\otimes\mathcal{F},\mathcal{E}^{\prime}\otimes^{\prime}\mathcal{F}^{\prime})=\mathcal{L}(\mathcal{E},\mathcal{E}^{\prime})\otimes_{p}\mathcal{L}(\mathcal{F},\mathcal{F}^{\prime}).$

To prove theorem 1, we only need to check that $\otimes_{p}$ maps one basis of
$\mathcal{L}(\mathcal{E},\mathcal{E}^{\prime})\times\mathcal{L}(\mathcal{F},\mathcal{F}^{\prime})$
to a free family of
$\mathcal{L}(\mathcal{E}\otimes\mathcal{F},\mathcal{E}^{\prime}\otimes^{\prime}\mathcal{F}^{\prime})$,
which yields injectivity. Then we will be able to conclude arguing that this
two spaces have the same dimension, so that by Green’s theorem, $\otimes_{p}$
is a bijective bilinear map from the carthesian product space to a linear
space, which is exactly what a tensor product is [14].

Let $\\{e_{i}\\}_{i}$, $\\{e^{\prime}_{i}\\}_{i}$, $\\{f_{j}\\}_{j}$ and
$\\{f^{\prime}_{j}\\}_{j}$ be some bases of
$\mathcal{E},\mathcal{E}^{\prime},\mathcal{F}$ and $\mathcal{F}^{\prime}$.
Define the following basis for
$\mathcal{L}(\mathcal{E},\mathcal{E}^{\prime})$:

$u_{i}(e_{j})=\delta_{ij}e^{\prime}_{i}$

where $\delta_{ij}$ is the kronecker symbol equal to 1 if and only if $i$
equals $j$. Define the basis $\\{v_{i}\\}$ similarly. Let us prove that
$\\{u_{i}\otimes_{p}v_{j}\\}_{ij}$ is a free family.

Given some $\\{\lambda_{ij}\\}_{ij}$, suppose for any $x\otimes y$ in
$\mathcal{E}\otimes\mathcal{F}$

$\sum\limits_{i,j}{\lambda_{ij}u_{i}(x)\otimes v_{j}(y)}=0.$

This is equivalent to

$\sum\limits_{i,j}{\lambda_{ij}\nu_{i}\nu^{\prime}_{j}e^{\prime}_{i}\otimes
f^{\prime}_{j}}=0.$

where $\nu_{i}$ and $\nu^{\prime}_{j}$ are coefficients of $x$ and $y$ in
bases $\\{e_{i}\\}_{i}$ and $\\{f_{j}\\}_{j}$. Since this is true for any
$x,y$ and that $\\{e^{\prime}_{i}\otimes^{\prime}f^{\prime}_{j}\\}_{ij}$ is a
basis of $\mathcal{E}^{\prime}\otimes^{\prime}\mathcal{F}^{\prime}$, all
lambdas have to go to zero, thus proving theorem 1.

## Acknowledgment

The author would like to thank Pierre Comon and Rodrigo Cabral Farias for
useful discussions.

## References

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# Volumes of spheres and special values of zeta functions of $\mathbb{Z}$ and
$\mathbb{Z}/n\mathbb{Z}$

Anders Karlsson111The first author was supported in part by the Swedish
Research Council grant 104651320 and the Swiss NSF grant 200020-200400. and
Massimiliano Pallich

(September 6, 2022)

###### Abstract

The volume of the unit sphere in every dimension is given a new interpretation
as a product of special values of the zeta function of $\mathbb{Z}$, akin to
volume formulas of Minkowski and Siegel in the theory of arithmetic groups. A
product formula is found for this zeta function that specializes to Catalan
numbers. Moreover, certain closed-form expressions for various other zeta
values are deduced, in particular leading to an alternative perspective on
Euler’s values for the Riemann zeta function.

##  1 Introduction

The determination of circumference, area, and volume of spheres is one of the
oldest topics in geometry. A more sophisticated volume formula, due to
Minkowski, is the following:

$\mathrm{vol(}\mathrm{SL}_{n}(\mathbb{R})/\mathrm{SL}_{n}(\mathbb{Z}))=\zeta(2)\zeta(3)...\zeta(n),$

where $\zeta(s)$ is the Riemann zeta function, and with a suitable coherent
choice of normalization of the Haar measure. This is not an incidental fact,
instead it is part of a more general phenomenon discovered and developed by
Siegel, Weil, Langlands, Harder, and others. It takes the form

$\mathrm{vol}(G/\Gamma)=c^{[K:\mathbb{Q}]}\prod_{i=1}^{l}\zeta_{K}(-m_{i}),$

and without going into details about this formula and when it holds (see [H71,
P01] for more information), let us just highlight one further example:

$\mathrm{vol(}\mathrm{Sp}_{n}(\mathbb{R})/\mathrm{Sp}_{n}(\mathbb{Z}))=\zeta(2)\zeta(4)...\zeta(2n).$

The passage from negative integers to positive ones is done by means of the
fundamental functional equation. In the present paper we will in particular
provide a new zeta value interpretation of the $(n-1)$-dimensional volume of
spheres, which are the homogeneous spaces
$S^{n-1}\cong\mathrm{SO}(n)/\mathrm{SO}(n-1)$.

As is well known, the Riemann zeta function is essentially the spectral zeta
function of the circle $\mathbb{R}/\mathbb{Z}$, more precisely

$\zeta(s)=\frac{1}{2}(2\pi)^{s}\zeta_{\mathbb{R}/\mathbb{Z}}(s/2)=\frac{(2\pi)^{s}}{2}\frac{1}{\Gamma(s)}\int_{0}^{\infty}\frac{1}{\sqrt{4\pi
t}}\sum_{k\neq 0}e^{-k^{2}/4t}t^{s/2}\frac{dt}{t},$

for $\mathrm{Re}(s)<1$ and then extended by meromorphic continuation. From
Fourier analysis we know that the circle and the integers are dual groups. The
function $e^{-2t}I_{0}(2t),$ with a Bessel function appearing, is the
$\mathbb{Z}$-analog of the theta series for the circle, found inside the
Mellin transform expression above. Therefore, entirely analogously to
$\mathbb{R}/\mathbb{Z}$ as is explained in [FK17], one can define the spectral
zeta function of $\mathbb{Z}$ as

$\zeta_{\mathbb{Z}}(s)=\frac{1}{\Gamma(s)}\int_{0}^{\infty}e^{-2t}I_{0}(2t)t^{s}\frac{dt}{t},$

for $0<\mathrm{Re(s)<1/2}$ and then extended by meromorphic continuation as
will soon become clear. We then define in parallel to the above the function

$Z(s)=\frac{1}{2}2\pi 2^{s}\zeta_{\mathbb{Z}}(s/2),$

which shares with $\zeta(s)$ the property of having a functional equation of
the type of a $s\longleftrightarrow 1-s$ symmetry ([FK17]). We observe the
following formula very reminiscent of the above volume expressions:

######  Theorem 1.

For $n>0$, the $n$-dimensional volume of the unit sphere in $\mathbb{R}^{n+1}$
is

$\mathrm{vol}(S^{n})=2\cdot Z(0)\cdot Z(-1)...\cdot Z(-n+1).$

It turns out, by [FK17], and then by Dubout’s formula in [D19], that
$\zeta_{\mathbb{Z}}(s)$ is an analytic continuation essentially of the Catalan
numbers

$\zeta_{\mathbb{Z}}(s)=\frac{1}{4^{s}\sqrt{\pi}}\frac{\Gamma(1/2-s)}{\Gamma(1-s)}=\left(\begin{array}[]{c}-2s\\\
-s\end{array}\right).$

The appearance of the gamma function begins explaining the volume formula.
Indeed, with the values $Z(0)=\pi$, $Z(-1)=2$ and $Z(-2)=\pi/2$ one sees that
the volume formula is true for the first cases. One could alternatively say
that the Catalan numbers

$C_{m}=\frac{1}{m+1}\left(\begin{array}[]{c}2m\\\ m\end{array}\right),$

but now at the nonstandard indices $m=k/2$, appear in the classical expression
of the surface area of spheres.

The spectral zeta function of $\mathbb{Z}$ has the following beautiful product
formula:

######  Theorem 2.

It holds that

$\zeta_{\mathbb{Z}}(s)=\prod_{k=1}^{\infty}\frac{(k-s)^{2}}{k(k-2s)}$

interpreted suitably when $s$ is a positive integer or half-integer.

From this expression, the simple zeros and poles of $\zeta_{\mathbb{Z}}(s)$ at
the positive integers and half-integers respectively, are clearly visible. As
will be shown below, it specializes to the standard product expression for
Catalan numbers:

$C_{m}=\prod_{k=2}^{m}\frac{m+k}{k}.$

In the following we shall also provide an exposition of special values of
$\zeta_{\mathbb{Z}}(s)$ and $\zeta(s)$, including correcting certain minor
inaccuracies in [FK17], as well as studying the related and analogously
defined spectral zeta function of $\mathbb{Z}/n\mathbb{Z}$,

$\zeta_{\mathbb{Z}/n\mathbb{Z}}(s)=\frac{1}{4^{s}}\sum_{k=1}^{n-1}\frac{1}{\sin^{2s}(\pi
k/n)}.$

In particular, we will provide an elementary calculation of
$\zeta_{\mathbb{Z}/n\mathbb{Z}}(1/2-m)$, and indicate how then to pass to
Euler’s $\zeta(1-2m)$ values from the asymptotics expansions in [Si04, FK17,
MV22]. After that via the symmetry $s$ vs $1-s$ one gets $\zeta(2m)$, which
leads to $\zeta_{\mathbb{Z}/n\mathbb{Z}}(m).$

In this way we relate $\zeta_{\mathbb{Z}/n\mathbb{Z}}(2m/2)$ to
$\zeta_{\mathbb{Z}/n\mathbb{Z}}((1-2m)/2)$. Note that asymptotical symmetries
of the type $s$ vs $1-s$ for $\zeta_{\mathbb{Z}/n\mathbb{Z}}(s/2)$ and related
finite sums is not a trivial matter, in fact a certain version of it is
equivalent to various Riemann hypotheses, for $\zeta(s)$ as shown in [FK17],
for certain Dirichlet $L$-functions as proven in [F16], and for the Dedekind
zeta function of the Gaussian rationals as established in [MV22].

Acknowledgement: It is a pleasure to thank Fabien Friedli for several useful
remarks.

##  2 Proof of the volume and product formulas

Proof of the volume formula. The well-known volume (hypersurface area) of
spheres is

$\mathrm{vol}(S^{n})=\frac{2\pi^{(n+1)/2}}{\Gamma((n+1)/2)},$

generating the numbers $2,2\pi,4\pi,2\pi^{2}$ etc.

This leads to the following calculation

$\frac{\mathrm{vol}(S^{n})}{\mathrm{vol}(S^{n-1})}=\frac{2\pi^{(n+1)/2}}{\Gamma((n+1)/2)}\frac{\Gamma(n/2)}{2\pi^{n/2}}=\frac{\sqrt{\pi}\Gamma(n/2)}{\Gamma(n/2+1/2)}.$

This we can rewrite as follows

$\frac{\mathrm{vol}(S^{n})}{\mathrm{vol}(S^{n-1})}=\frac{\sqrt{\pi}\Gamma(1/2-(1/2-n/2)}{\Gamma(1-(1/2-n/2))}=\pi
4^{1/2-n/2}\zeta_{\mathbb{Z}}((1-n)/2)=Z(-n+1).$

Since $\mathrm{vol}(S^{0})=2$ we inductively arrive at the proof of the
formula stated in Theorem 1.

Proof of the product formula. Recall that the Euler’s beta function has the
following product formula [AAR99, Formula 1.1.26]:

$B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}=\frac{x+y}{xy}\prod_{k=1}^{\infty}\frac{(1+(x+y)/k)}{(1+x/k)(1+y/k)}$

with poles for $x$ or $y$ equal to $0$ or a negative integer, and analytic
elsewhere. On the other hand, Dubout’s expression gives

$\zeta_{\mathbb{Z}}(s)=\frac{\Gamma(1-2s)}{\Gamma(1-s)\Gamma(1-s)}=\frac{\Gamma(2-2s)}{(1-2s)\Gamma(1-s)\Gamma(1-s)}=\frac{(1-s)^{2}}{(1-2s)(2-2s)}\prod_{k=1}^{\infty}\frac{(1+(1-s)/k)^{2}}{(1+(2-2s)/k)}$

$=\frac{(1-s)^{2}}{(1-2s)(2-2s)}\prod_{k=1}^{\infty}\frac{(k+1-s)^{2}}{k(k+2-2s)}=\prod_{k=1}^{\infty}\frac{(k-s)^{2}}{k(k-2s)},$

as was to be proved.

To see that it gives back a correct expression in the case of $s=-m$ a
negative integers (the Catalan number case) note that except for the small
values, the others integer appear twice in the numerator as well as in the
denominator:

$C_{m}=\frac{1}{m+1}\zeta_{\mathbb{Z}}(-m)=\frac{1}{m+1}\prod_{k=1}^{\infty}\frac{(k+m)^{2}}{k(k+2m)}$
$=\frac{1}{m+1}\frac{(m+1)^{2}(m+2)^{2}...}{1\cdot
2...(1+2m)(2+2m)...}=\prod_{k=2}^{m}\frac{m+k}{k},$

which is an expression whose validity is immediate from the definition of
$C_{m}$.

##  3 Special values of $\zeta_{\mathbb{Z}}(s)$

Here is a summary and proof of special values for $\zeta_{\mathbb{Z}}$ and its
derivative. In particular, at the negative integers, $\zeta_{\mathbb{Z}}(-m)$
is rational even integral. This is not a priori obvious, but in view of the
Dubout formula for $\zeta_{\mathbb{Z}}$ this becomes clear since they are just
binomial coefficients at these points. This is the counterpart of theorems by
Hecke, Siegel, Klingen, and others for classical zeta functions.

The following is taken from [FK17, Pa22]. First, we recall that for integers
$n\geq 0$, we have

$\displaystyle\Gamma(n+1)=n!\quad\textrm{and}\quad\Gamma\left(\frac{1}{2}+n\right)=\frac{(2n)!}{4^{n}n!}\pi^{\frac{1}{2}},$

the latter being the Legendre duplication formula. Thus, by using these
formulas and the expressions above, we get for integers $n\geq 1$ that

$\displaystyle\zeta_{\mathbb{Z}}(0)=4^{0}\pi^{-\frac{1}{2}}\frac{\Gamma(\frac{1}{2})}{\Gamma(1)}=1,$
(1)
$\displaystyle\zeta_{\mathbb{Z}}(-n)=4^{n}\pi^{-\frac{1}{2}}\frac{\Gamma(\tfrac{1}{2}+n)}{\Gamma(1+n)}=\frac{(2n)!}{n!n!}=\binom{2n}{n},$
$\displaystyle\zeta_{\mathbb{Z}}(-n+\tfrac{1}{2})=4^{n}\pi^{-\frac{1}{2}}\frac{\Gamma(n)}{\Gamma(\tfrac{1}{2}+n)}=\frac{4^{2n}}{2\pi
n}\frac{n!n!}{(2n)!}=\frac{4^{2n}}{2\pi n}\binom{2n}{n}^{-1}.$

As said before the values at positive integers and positive half-integers are
precisely the zeros and poles.

Now, if we differentiate $\zeta_{\mathbb{Z}}(s)$, we get

$\displaystyle\zeta_{\mathbb{Z}}^{\prime}(s)=\pi^{-\frac{1}{2}}\left(\frac{\Gamma(\frac{1}{2}-s)}{4^{s}\Gamma(1-s)}\right)^{\prime}$
$\displaystyle=\pi^{-\frac{1}{2}}\frac{-\Gamma(\frac{1}{2}-s)\psi_{0}(\frac{1}{2}-s)-\Gamma(\frac{1}{2}-s)\big{(}\log(4)-\psi_{0}(1-s)\big{)}}{4^{s}\Gamma(1-s)}$
$\displaystyle=\zeta_{\mathbb{Z}}(s)\left(-\psi_{0}\left(\frac{1}{2}-s\right)-2\log(2)+\psi_{0}(1-s)\right),\phantom{\frac{}{}}$

since
$\Gamma(\frac{1}{2}-s)^{\prime}=-\Gamma(\frac{1}{2}-s)\psi_{0}(\frac{1}{2}-s)$
and $(4^{s}\Gamma(1-s))^{\prime}=4^{s}\Gamma(1-s)(\log(4)-\psi_{0}(1-s))$, and
where $\psi_{n}(s)$ is the polygamma function, which is defined by

$\displaystyle\psi_{n}(s)=\frac{d^{n}}{ds^{n}}\frac{\Gamma^{\prime}(s)}{\Gamma(s)}.$

We can therefore deduce the following special values of
$\zeta_{\mathbb{Z}}^{\prime}$.

######  Proposition 3.

It holds that

$\displaystyle\zeta_{\mathbb{Z}}^{\prime}(0)=0\quad\textrm{and}\quad\zeta_{\mathbb{Z}}^{\prime}(-\tfrac{1}{2})=\frac{8}{\pi}(1-2\log(2)),$

and

$\displaystyle\zeta_{\mathbb{Z}}^{\prime}(-n)=\binom{2n}{n}\left(\sum_{k=1}^{n}\frac{1}{k}-\frac{2}{2k-1}\right)$
$\displaystyle\text{for $n\geq 1$},$
$\displaystyle\zeta_{\mathbb{Z}}^{\prime}(-n+\tfrac{1}{2})=\frac{4^{2n}}{2\pi
n}\binom{2n}{n}^{-1}\left(-4\log(2)-\sum_{k=1}^{n-1}\frac{1}{k}+2\sum_{k=1}^{n}\frac{1}{2k-1}\right)$
$\displaystyle\text{for $n\geq 2$}.$

###### Proof.

According to 6.3.2 and 6.3.3 in [AS64], we have the following values

$\displaystyle\psi_{0}\left(\frac{1}{2}\right)=-\gamma-2\log(2)\quad\text{and}\quad\psi_{0}(1)=-\gamma.$
(2)

Furthermore, by formulas 6.3.2 and 6.3.4 in [AS64], we also have

$\displaystyle\psi_{0}\left(n+\frac{1}{2}\right)=-\gamma-2\log(2)+2\sum_{k=1}^{n}\frac{1}{2k-1}\qquad\text{for
$n\geq 1$},$ (3)
$\displaystyle\psi_{0}(n)=-\gamma+\sum_{k=1}^{n-1}\frac{1}{k}\qquad\qquad\qquad\qquad\qquad\quad\;\;\;\text{for
$n\geq 2$}.$ (4)

Hence, combining (2), (3), (4) and the special values already computed in (1)
concludes the proof of the proposition. ∎

Note that these values correct (confirmed by computer numerics) small errors
in [FK17, Proposition 6.1]. Indeed, the second formula in that reference
should have $-4\log 2$ instead of $-4\log 4$, and the case $n=1$ needs to be
interpreted correctly in view of the term $1/(n-1)$ appearing. Finally it is
stated in [FK17] that $\zeta_{\mathbb{Z}}^{\prime}(s)$ is zero at the positive
integers, this is not true (checked by numerics) and instead the values are
given here:

######  Proposition 4.

Let $n$ be a positive integer, then

$\displaystyle\zeta_{\mathbb{Z}}^{\prime}(n)=\frac{1}{n}\binom{2n}{n}^{-1}.$

###### Proof.

By the reflection formula 6.3.7 in [AS64], we have

$\displaystyle\psi_{0}(1-z)=\psi_{0}(z)+\pi\cot(\pi z).$

Hence, applying this last formula, as well as (3) and (4), gives

$\displaystyle\zeta_{\mathbb{Z}}^{\prime}(n)$
$\displaystyle=\binom{-2n}{-n}\left(-\psi_{0}\left(\frac{1}{2}-s\right)-2\log(2)+\psi_{0}(1-n)\right)$
$\displaystyle=\binom{-2n}{-n}\left(\sum_{k=1}^{n-1}\frac{1}{k}-2\sum_{k=1}^{n}\frac{1}{2k-1}+\pi\cot(\pi
n)\right).$

We observe that for any positive integer $n$, we have $\binom{-2n}{-n}=0$.
Therefore the two sums are eliminated by the multiplication with the binomial.
However we also note that we have $\cot(\pi n)=\pm\infty$. Hence

$\displaystyle\zeta_{\mathbb{Z}}^{\prime}(n)=\binom{-2n}{-n}\pi\cot(n\pi)=\frac{\Gamma(1-2n)}{\Gamma(1-n)^{2}}\pi\cot(n\pi)=\frac{1}{n}\binom{2n}{n}^{-1},$

where the last equality is obtained first by applying the reflection formula
$\Gamma(1-z)\Gamma(z)=\pi\sin^{-1}(\pi z)$, then by applying the recurrence
formula $\Gamma(z+1)=z\Gamma(z)$ and finally by using the double-angle
formulas for sine. ∎

##  4 Special values of $\zeta_{\mathbb{Z}/n\mathbb{Z}}(s)$ and $\zeta(s)$

Sums of powers of the sine function are special cases of important sums in
works of Dedekind, Verlinde, Dowker and others in number theory and physics,
see [Do92, Z96, Do15, K20]. From our perspective, and also partly from
Dowker’s, they are special values of spectral zeta functions of discrete
circles. It is therefore of interest to recall the following values, see for
example [Me14]. Let $n$ and $m$ be two positive integer, then we have

$\displaystyle\zeta_{\mathbb{Z}/n\mathbb{Z}}(-m)=2^{2m}\sum_{k=1}^{n-1}\sin^{2m}\left(\frac{\pi
k}{n}\right)=n\sum_{k=-\lfloor\frac{m}{n}\rfloor}^{\lfloor\frac{m}{n}\rfloor}(-1)^{kn}\binom{2m}{m+kn}.$

In the special case $m<n$, we have

$\displaystyle\zeta_{\mathbb{Z}/n\mathbb{Z}}(-m)=2^{2m}\sum_{k=1}^{n-1}\sin^{2m}\left(\frac{\pi
k}{n}\right)=n\binom{2m}{m}.$

Comparing the last formula to the one of $\zeta_{\mathbb{Z}}(-m)$ above (1),
one sees a manifestation of all the trivial zeros of the Riemann zeta function
at the negative even integers, in view of the asymptotics in [Si04, FK17,
MV22]

$\sum_{k=1}^{n-1}\frac{1}{\sin^{s}(k\pi/n)}=\frac{1}{\sqrt{\pi}}\frac{\Gamma(1/2-s/2)}{\Gamma(1-s/2)}n+2\pi^{-s}\zeta(s)n^{s}+\frac{s}{3}\pi^{2-s}\zeta(s-2)n^{s-2}+...$

as $n\rightarrow\infty.$ This asymptotic relation demonstrates the intimate
connection between the three zeta functions appearing in this paper,
$\zeta(s)$, $\zeta_{\mathbb{Z}}(s)$ and $\zeta_{\mathbb{Z}/n\mathbb{Z}}(s)$.

Friedli provided us with the proof of the following formula, empirically found
in [Pa22]:

######  Proposition 5.

Let $n$ and $m$ be positive integer, then

$\displaystyle\zeta_{\mathbb{Z}/n\mathbb{Z}}\left(-\frac{1}{2}-m\right)=2^{2m+1}\sum_{k=1}^{n-1}\sin^{2m+1}\left(\frac{\pi
k}{n}\right)=2\sum_{j=0}^{m}(-1)^{m-j}\binom{2m+1}{j}\cot\left(\frac{2m+1-2j}{2n}\pi\right).$

###### Proof.

First we recall

$\sin^{2m+1}(k\pi/n)=(2i)^{-2m-1}\left(e^{i\pi k/n}-e^{-i\pi
k/n}\right)^{2m+1}$
$=(2i)^{-2m-1}\sum_{j=0}^{2m+1}(-1)^{j}\left(\begin{array}[]{c}2m+1\\\
j\end{array}\right)e^{i\pi k(2j-2m-1)/n}.$

This gives switching the finite sums

$\sum_{k=1}^{n-1}\sin^{2m+1}(k\pi/n)=(2i)^{-2m-1}\sum_{j=0}^{2m+1}(-1)^{j}\left(\begin{array}[]{c}2m+1\\\
j\end{array}\right)\sum_{k=1}^{n-1}e^{i\pi k(2j-2m-1)/n}.$

The interior sum is of geometric type and can therefore be summed and equals

$-i\cdot\cot\left(\pi(2j-2m-1)/2n\right).$

Observe that there is a symmetry $j$ vs $2m+1-j$

$\cot\left(\pi(2(2m+1-j)-2m-1)/2n\right)=-\cot\left(\pi(2j-2m-1)/2n\right).$

Using the same symmetry for the binomial coefficients one arrives at

$\displaystyle\sum_{k=1}^{n-1}\sin^{2m+1}\left(\frac{\pi
k}{n}\right)=\frac{1}{2^{2m}}\sum_{j=0}^{m}(-1)^{m-j}\binom{2m+1}{j}\cot\left(\left(\frac{2m+1-2j}{2n}\right)\pi\right).$

∎

Thanks to the above asymptotics and the well-known series expansion of the
cotangent function

$\cot(z)=\sum_{n=0}^{\infty}\frac{(-1)^{n}2^{2n}B_{2n}}{(2n)!}z^{2n-1},$

one can deduce Euler’s celebrated formulas that for positive integers $m$,

$\zeta(-m)=(-1)^{m}\frac{B_{m+1}}{m+1},$

in particular $\zeta(s)$ vanishes at the negative even integers, and
$\zeta(0)=-1/2$. Indeed, setting $s=0$ in the asymptotics one has

$n-1=\frac{1}{\sqrt{\pi}}\frac{\Gamma(1/2)}{\Gamma(1)}n+2\pi^{0}\zeta(0)n^{0}+...$

giving $\zeta(0)=-1/2.$ Specializing to $s=-1$ produces

$\sum_{k=1}^{n-1}\sin(k\pi/n)=\frac{1}{\sqrt{\pi}}\frac{\Gamma(1)}{\Gamma(3/2)}n+2\pi\zeta(-1)n^{-1}+\frac{-1}{3}\pi^{3}\zeta(-3)n^{-3}+...$

and on the other hand from Proposition 5,

$\sum_{k=1}^{n-1}\sin(k\pi/n)=\cot(\pi/2n)=\frac{2n}{\pi}B_{0}+\frac{\pi}{2n}(-2)B_{2}+\frac{\pi^{3}}{8n^{3}}\frac{16B_{4}}{24}+O(1/n^{5}).$

This gives that

$\zeta(-1)=-\frac{B_{2}}{2},\textrm{ $\zeta(-3)=-\frac{B_{4}}{4},...$}$

In view of the functional equation of $\zeta(s)$ one can then get the values
of $\zeta(2m)$ such as $\pi^{2}/6$, $\pi^{4}/90$ etc. which appear in the
volume formulas in the introduction. This, in turn, again via the above stated
asymptotics (the asymptotics for these special $s$ here is just a formula due
to the trivial zeros $\zeta(-2m)=0$) gives the values in closed form of

$\zeta_{\mathbb{Z}/n\mathbb{Z}}(m)=\frac{1}{4^{s}}\sum_{k=1}^{n-1}\frac{1}{\sin^{2m}(\pi
k/n)}.$

For example,

$\zeta_{\mathbb{Z}/n\mathbb{Z}}(1)=0n+\frac{2}{4\pi^{2}}\frac{\pi^{2}}{6}n^{2}+\frac{2}{12}\pi^{2-2}\left(-\frac{1}{2}\right)+0=\frac{1}{12}(n^{2}-1)$

and

$\zeta_{\mathbb{Z}/n\mathbb{Z}}(2)=\frac{1}{720}\left(n^{4}+10n^{2}+11\right).$

An easier approach to these last two formulas can be found in [Do92, Z96]. The
values of $\zeta_{\mathbb{Z}/n\mathbb{Z}}$ at the positive half-integral
points do not have such polynomial expressions, which is related to the
elusive nature of the zeta values $\zeta(2m+1)$, $m>0.$

Finally, let us remark that the Riemann Hypothesis can be reformulated solely
in terms of a hypothetical functional symmetry of the standard type
$s\longleftrightarrow 1-s$ for $\zeta_{\mathbb{Z}}(s/2)$ and
$\zeta_{\mathbb{Z}/n\mathbb{Z}}(s/2)$. This is not an incidental fact, as it
was shown to extend in [F16] and [MV22] to some cases of a generalized Riemann
Hypothesis.

## References

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Section de mathématiques, Université de Genève, Case postale 64, 1211 Genève,
Switzerland; Mathematics department, Uppsala University, Box 256, 751 05
Uppsala, Sweden.