††thanks: The first two authors contributed equally in this research.

# Kernel Method based on Non-Linear Coherent State

Prayag Tiwari 1<EMAIL_ADDRESS>Shahram Dehdashti 2
<EMAIL_ADDRESS>Abdul Karim Obeid 2 Massimo Melucci 1 Peter Bruza
2<EMAIL_ADDRESS>1 Department of Information Engineering, University of
Padova, Padova, 35131, Italy.  2School of Information Systems, Queensland
University of Technology, Brisbane 4000, Australia.

###### Abstract

In this paper, by mapping datasets to a set of non-linear coherent states, the
process of encoding inputs in quantum states as a non-linear feature map is
re-interpreted. As a result of this fact that the Radial Basis Function is
recovered when data is mapped to a complex Hilbert state represented by
coherent states, non-linear coherent states can be considered as natural
generalisation of associated kernels. By considering the non-linear coherent
states of a quantum oscillator with variable mass, we propose a kernel
function based on generalized hypergeometric functions, as orthogonal
polynomial functions. The suggested kernel is implemented with support vector
machine on two well known datasets (make_circles, and make_moons) and
outperforms the baselines, even in the presence of high noise. In addition, we
study impact of geometrical properties of feature space, obtaining by non-
linear coherent states, on the SVM classification task, by using considering
the Fubini-Study metric of associated coherent states.

##  I Introduction

Quantum machine learning is a rapidly growing field of investigation. It can
be argued that developments are being driven from two directions. Firstly,
quantum computers offer the promise of massive improvement in the speed of
computational processing Aaronson and Arkhipov (2013); Tillmann et al. (2013);
Brod et al. (2019). Secondly, the mathematical framework of quantum mechanics
is increasingly been seen as a suitable framework for designing algorithms
that aren’t constrained by Boolean algebra and logic Pitowsky (1994); Vourdas
(2019).The reasons that support the latter claim are many, e.g., the linearity
of the Schrödinger equation Schrödinger (1987); Tsutsumi (1987), which leads
to the definition of superposed states in complex Hilbert spaces with an
‘interference’ term affecting probabilities. Consider modelling the dependence
that measurement outcomes have on the preparation of states: duly named
‘contextual scenarios’ Kochen and Specker (1975). There is also the novelty of
correlations observed through entanglement, discord, etc in quantum mechanics
that go beyond classically correlated structures Zurek (2000); Girolami et al.
(2014), as well as quasi-distributions, which occur in phase space, so-called
Wigner distributions Simon et al. (1987); Lorce and Pasquini (2011), and
achieve negativity - this is not possible in Kolmogorovian probability theory.
All of the preceding introduce the potential for access to an information
space greater than that of classical alternatives Goh et al. (2018); Pusey et
al. (2019); Dehdashti et al. (2020a); Uprety et al. (2020). This is
encouraging for scientists wishing to apply quantum formalism within machine
learning (ML) as a generalisation of probability theory.
In ML, kernel methods Shawe-Taylor et al. (2004); Zelenko et al. (2003);
Soentpiet (1999); Hofmann et al. (2008); Evgeniou et al. (2005); Campbell
(2002) are a class of categorization algorithms. Used within a wide range of
methods and algorithms, they include the support vector machine (SVM) Amari
and Wu (1999); Wang (2005); Noble (2006), kernel operators with principal
components analysis (PCA) Schölkopf et al. (1997), spectral clustering Dhillon
et al. (2004), canonical correlation analysis Akaho (2006), linear adaptive
filters Liu et al. (2009), and ridge regression An et al. (2007). Indeed,
kernel methods are proving to assist also in deep neural networks, for which
there are many recently published works Cho and Saul (2009); Belkin et al.
(2018). There exist vast prospects of kernel methods in ML due to the non-
linear nature of the underlying data. Within application settings, the data
are usually non-separable, for which the requirement of the kernel then
becomes transformations (of the data) into higher dimensions where it may be
(linearly) separable as can be seen in Figure 1. One of the well-known kernel
functions in ML, the Radial Basis function (RBF), is defined by
$K(x,x^{\prime})=\exp\left[-|x-x^{\prime}|^{2}/2\sigma^{2}\right]$ where $x$
and $x^{\prime}$ are two sample elements, and $\sigma$ controls the decision
boundary Musavi et al. (1992); Buhmann (2000); Orr et al. (1996). It is
worthwhile to mention that the RBF can be understood as an inner product of
the linear coherent state, see Kübler et al. (2019), which is defined as an
eigenstate of the annihilation operator of a harmonic oscillator. The idea of
using a quantum mechanics formalism in kernel methods was suggested by Schuld
and Killoran, who introduced squeezed kernels in feature spaces Schuld and
Killoran (2019). In fact, they defined the feature space as a set of squeezed
states so that the kernel is obtained by inner products of squeezed states
Datko (1970); Gleason (1957).
In this paper, we express non-linear coherent states de Matos Filho and Vogel
(1996); Man’ko et al. (1997); Mancini (1997); Roy and Roy (2000); Sivakumar
(2000) as a quantum feature space, such that kernel functions are defined as
their inner products. We show that the mathematical structure of non-linear
coherent states provides infinite kernels. As an example of non-linear feature
space, we investigate coherent states constructed by wave-functions of a
quantum oscillator with variable mass. Generalized hypergeometric functions,
as orthogonal polynomials, are identified as provide the associated kernel.
Our proposed KMNCS has been demonstrated in an SVM classifier, along with the
RBF and squeezed kernel as a baseline on two-well-known datasets (make_moons,
make_circles). KMNCS is shown to outperform the baselines (squeezed and RBF
kernels) even as we increase the noise in the dataset (which increases
difficulty of generalisation). In addition, we study the geometrical
properties of the feature space, by obtaining the Fubini-Study metric of non-
linear coherent states. We show that the feature space of a non-linear
coherent state of a oscillator with variable mass is a surface with negative
curvature, which opens a new line of investigation of how the feature space’s
curvature affects the accuracy SVM classification.
The rest of the paper is organised as follows: in Section II, we briefly
review the Kernel Method. Section III introduces the previously mentioned
coherent states. Section IV provides an overview on non-linear coherent states
and introduces the coherent states of a quantum oscillator with variable mass.
This allows the main contribution of the paper to be defined: a kernel
function based on generalized hypergeometric functions. Section V details the
experimental design which allows the proposed kernel function to be
empirically evaluated against two baseline: RBF and squeezed kernel. Section
VI discusses the results of the empirical evaluation. Also, this section is
examines the geometrical properties of a non-linear coherent state. Finally,
Section VII concludes the article.

Figure 1: Kernel visualization from low dimension input space to high
dimension feature space.

##  II Kernel Method

Traditional ML begins with a dataset of inputs
$\mathcal{D}=\\{x_{1},\cdots,x_{M}\\}$ drawn from a set $\mathcal{X}$. The
goal is to produce a predictive model that allows patterns to be discovered in
yet to be observed data. Kernel methods use the inner product
$K(x,x^{\prime})$ between any given two inputs $x,x^{\prime}\in\mathcal{X}$,
as a distance measure to build predictive models that assist with representing
characteristics of a data distribution.

######  Definition II.1.

Let $\mathcal{X}$ be a nonempty set, called the index set, and $\mathcal{H}$
by a Hilbert space of functions $\phi:\mathcal{X}\rightarrow\mathbb{R}$. Then
$\mathcal{H}$ is called a reproducing kernel Hilbert space endowed with the
dot product, $\bra{\cdot}\ket{\cdot}$, if there exists a function
$k:\mathcal{X}\times\mathcal{X\rightarrow\mathcal{R}}$ with the two following
properties: (i) $k$ has the reproducing property, i.e.,
$\bra{\phi}\ket{k(x,\cdot)}=\phi(x)$ for all $\phi\in\mathcal{H}$; (ii) $k$
spans $\mathcal{H}$ Scholkopf and Smola (2001).

Note that in particular
$\bra{k(x,\cdot)}\ket{k(x^{\prime},\cdot)}=k(x,x^{\prime})=k(x^{\prime},x)$
guarantees symmetry of the arguments of $k$.

######  Theorem II.1.

Let $\phi:\mathcal{X}\rightarrow\mathcal{F}$ be a feature map. Kernel function
can be defined as the inner product of two inputs mapped to some feature space
by, $\mathbf{k}(x,x^{\prime})=\langle\phi(x),\phi(x^{\prime})\rangle$.

###### Proof.

The proof can be found in references Hofmann et al. (2008); Kübler et al.
(2019). ∎

######  Theorem II.2.

Consider a feature mapping is $\phi:\mathcal{X}\rightarrow\mathcal{F}$ over
some input set $\mathcal{X}$, which provide the basis to a complex kernel
$\mathbf{k}(x,x^{\prime})=\langle\phi(x),\phi(x^{\prime})\rangle$ that is
defined on $\mathcal{F}$. The associated reproducing kernel Hilbert space can
be written as
$\mathcal{R}_{\mathbf{k}}=\\{f:\mathcal{X}\rightarrow\mathbb{C}\\}$

$f(x)=\langle w,\phi(x^{\prime})\rangle,$ (1)

for all $w\in\mathcal{F}$ and $x\in\mathcal{X}$.

###### Proof.

The proof can be found in Hofmann et al. (2008) and
111http://www.gatsby.ucl.ac.uk/~gretton/coursefiles/lecture4_introToRKHS.pdf.
∎

##  III Coherent State

A harmonic oscillator in quantum physics is described by the following
Hamiltonian:

$\displaystyle H=\hbar\omega\left(\hat{n}+\frac{1}{2}\right)$ (2)

in which $\hbar$ is the Planck’s constant and $\omega$ is the angular
frequency of the oscillator; the number operator $\hat{n}$ is described by
annihilation and creation operators, i.e., $\hat{n}=\hat{a}^{\dagger}\hat{a}$.
The Schörodinger equation gives the discrete energy eigenvalue:

$\displaystyle H\ket{n}=E_{n}\ket{n},\ n=0,1,2,\cdots$ (3)

in which eigenvalue $E_{n}=\hbar\omega(n+1/2)$ is associated by eigenstate
$\ket{n}$. For simplicity, we consider $\hbar=1$ and $\omega=1$ in the rest of
the paper.
A coherent state is the specific quantum state of the quantum harmonic
oscillator, often described as a state which has dynamics most closely
resembling the oscillatory behavior of a classical harmonic oscillator, for
example see Ali et al. (2000).

######  Definition III.1.

A coherent state is defined as superposition of number state $\ket{n}$ as
following:

$\displaystyle\ket{\alpha}=e^{-|\alpha|^{2}/2}\sum_{n=0}^{\infty}\frac{\alpha^{n}}{\sqrt{n!}}\ket{n}$
(4)

which number states satisfy $\bra{n}\ket{m}=\delta_{n,m}$, where
$\delta_{n,m}$ is Kronecker delta.

Note that the inner product of two coherent states is given by:

$\displaystyle\bra{\alpha}\ket{\beta}=e^{|\alpha-\beta|^{2}/2}$ (5)

######  Lemma III.1.

A harmonic oscillator coherent state adheres the following:

  1. 1.

It is obtained by operation of the displacement operator
$D(\mathbf{\alpha})=\exp{\left(\alpha^{\ast}a^{\dagger}-\alpha a\right)}$ on a
reference state: $|\alpha\rangle=D(\alpha)|0\rangle$, in which $\ket{\alpha}$
is given by the relation (4).

  2. 2.

It is an eigenvector of the annihilation operator,
$a|\alpha\rangle=\alpha|\alpha\rangle$.

  3. 3.

It fulfills the minimum uncertainty relation, i.e.,
$\Delta(\mathbf{x})=\Delta(\mathbf{p})=1/\sqrt{2}$, where
$\Delta(\mathbf{x})=\sqrt{\langle\mathbf{x}^{2}\rangle-\langle\mathbf{x}\rangle^{2}}$,
where $\langle\mathbf{x}\rangle=\bra{\alpha}\mathbf{x}\ket{\alpha}$.

  4. 4.

It is over-complete,
$K(\alpha,\alpha^{\prime})=\bra{\alpha}\ket{\alpha^{\prime}}\neq\delta(\alpha-\alpha^{\prime})$,
i.e., the equation (5), despite the fact that they fulfil the resolution of
the identity, $\int
d\mu(\mathbf{x})|\mathbf{x}\rangle\langle\mathbf{x}|=\mathbb{I}$, which leads
to the following relation:

$\displaystyle\int
d\mu(\mathbf{x})\langle\phi|\mathbf{x}\rangle\langle\mathbf{x}|\psi\rangle=\langle\phi|\psi\rangle.$
(6)

###### Proof.

The proofs can be found in references Ali et al. (2000); Kübler et al. (2019).
∎

The latter, i.e., item $4$ of Lemma III.1, implies an arbitrary function can
be expressible as a linear combination of kernel functions in a “reproducing
Hilbert space” Combescure and Robert (2012). We should mention that the first
above-mentioned property leads to define a displacement-type coherent states
Ali et al. (2000), for generalized annihilation and creation operators.
Moreover a Gazeau-Klauder coherent state is defined by the second property and
fulfils the third property Ali et al. (2000). It should be noted that while
the latter, i.e., resolution of the identity, fulfils all types of coherent
states.

Recently Schuld and Killoran published a paper in which data is mapped into a
feature space established by squeezed states Schuld and Killoran (2019).
Squeezed states are states that saturate the Heisenberg uncertainty principle;
additionally, the quadrature variance of position and momentum depend on a
parameter, so-called squeezing parameter. The squeezing parameter causes the
uncertainty to be squeezed one of its quadrature components, while stretched
uncertainty for the other component, i.e.
$\Delta(\mathbf{x})=\exp{2\zeta}/\sqrt{2}$ and
$\Delta(\mathbf{p})=\exp{-2\zeta}/\sqrt{2}$, where $\zeta$ is the squeezing
parameter. Therefore, the squeezing parameter controls uncertainty via a
quadrature component, while the third and fourth properties of coherent states
are preserved. However, as we mentioned, one of the well-known kernel
functions in ML, the Radial Basis function (RBF), is defined by

$K(\mathbf{x},\mathbf{x}^{\prime};\sigma)=\exp{\left(-||\mathbf{x}-\mathbf{x}^{\prime}||^{2}/2\sigma^{2}\right)},$
(7)

where $\mathbf{x}$ and $\mathbf{x}^{\prime}$ are two sample elements, and
$\sigma$ is understood as a free parameter. However, drawing a comparison
between relations (5) and (7), inspires someone to interpret the RBF as an
inner product of two coherent states. This interpretation opens the door to
define new kernels and consequently improve the kernel method Kübler et al.
(2019).

##  IV Non-linear Coherent State

As mentioned before, some efforts have been devoted to study possible
generalization of the quantum harmonic oscillator algebra Man’ko et al.
(1997). A deformed algebra is a non-trivial generalization of a given algebra
through the introduction of one or more deformation parameters, such that, in
a certain limit of the parameters, the non-deformed algebra can be recovered.

A particular deformation of the W-H algebra led to the notion of $f$-deformed
oscillator Man’ko et al. (1997). An $f$-deformed oscillator is a non-harmonic
system where its dynamical variables (creation and annihilation operators) are
constructed from a non-canonical transformation through

$\displaystyle\hat{A}=\hat{a}f(\hat{n})=f(\hat{n}+1)\hat{a},$ (8)
$\displaystyle\hat{A}^{{\dagger}}=f^{{\dagger}}(\hat{n})\hat{a}^{{\dagger}}=\hat{a}^{{\dagger}}f^{{\dagger}}(\hat{n}+1).$
(9)

where $f(\hat{n})$ is called deformation function by which non-linear
properties of this system are governed. An $f$-deformed oscillator is
characterized by a Hamiltonian of the harmonic oscillator form,

$\displaystyle\hat{H}=\frac{\omega}{2}(\hat{A}\hat{A}^{{\dagger}}+\hat{A}^{{\dagger}}\hat{A}),$
(10)

where $\hat{A}$ and $\hat{A}^{{\dagger}}$ are given in equations (8) and (9).
In this Hamiltonian, $\omega$ is frequency of harmonic oscillator and
$\hslash=m=1$. The deformed operators satisfy the following commutation
relation

$[\hat{A},\hat{A}^{{\dagger}}]=(\hat{n}+1)f^{2}(\hat{n}+1)-\hat{n}f^{2}(\hat{n}).$
(11)

Relations (8) and (9) give eigenvalues of the Hamiltonian (10) as follows:

$E_{n}=\frac{\omega}{2}((n+1)f^{2}(n+1)+nf^{2}(n)).$ (12)

It is worth to mention that by approaching deformation function into $1$,
i.e., $f(n)\rightarrow 1$, the non-deformation energy eigenvalues,
$E_{n}=\omega(n+\frac{1}{2})$, and the non-deformed commutation relation
$[\hat{a},\hat{a}^{{\dagger}}]=1$ are recovered. However, similar to the
harmonic oscillator, it is possible to construct coherent states for the
$f$-deformed oscillator. The non-linear transformation of the creation and
annihilation operators leads naturally to the notion of non-linear coherent
states or $f$-coherent states Dehdashti et al. (2015a, 2013a, 2013b, b).

######  Definition IV.1.

Non-linear coherent states are defined as the right-hand eigenstates of the
deformed annihilation operator $\hat{A}$ as follows Ali et al. (2000):

$\hat{A}|\alpha\rangle_{f}=\alpha|\alpha\rangle_{f}.$ (13)

From equation (13) one can obtain an explicit form of the non-linear coherent
states in the number state representation as,

$|\alpha\rangle_{f}=\frac{1}{\mathcal{N}}\sum_{n}^{M}\frac{\alpha^{n}}{\sqrt{n![f(n)]!}}|n\rangle,$
(14)

where $M$ can be finite, or infinite (corresponding to finite or infinite
dimensional Hilbert space), $\alpha$ is a complex number and
$[f(n)]!=\prod_{i=0}^{n}f(i)$, with $[f(0)]!=1$; the normalization factor
$\mathcal{N}$ is given by

$\displaystyle\mathcal{N}=\left(\sum_{n=0}^{M}\frac{|\alpha|^{2n}}{n![f(n)]!}\right)^{-1/2}.$
(15)

Therefore, based on the definition II.1, we can define a kernel as the
following:

######  Definition IV.2.

By mapping multi-dimensional input set
$\mathbf{x}=(x_{1},\cdots,x_{N})^{T}\in\mathbb{R}^{N}$ into non-linear
coherent states, defined by the relation (14), so that they are fulfilled with
resolution of the identity, a feature space is defined as

$\phi:(x_{1},\cdots,x_{N})\rightarrow|x_{1}\rangle_{f}\otimes|x_{2}\rangle_{f}\otimes\cdots\otimes|x_{N}\rangle_{f}.$

In addition, the associated kernel is obtained by the inner product as the
following:

$\displaystyle K(\mathbf{x},\mathbf{x^{\prime}})=\prod_{i=1}^{N}\
{}_{f}\langle x_{i}|x_{i}^{\prime}\rangle_{f},$ (16)

in which

${}_{f}\langle
x_{i}|x_{i}^{\prime}\rangle_{f}=\frac{1}{\mathcal{N}^{2}}\sum_{n=0}^{M}\frac{(x_{i}x_{i}^{\prime})^{m}}{m![f(m)]!}.$
(17)

###  IV.1 Non-Linear coherent state of an oscillator with variable mass

Figure 2: Schematic shape of the kernel function (22) for $k=-0.001$ and
$k=-0.5$ in plots (a) and (b), respectively. The input $x$ is fixed at $(0,0)$
and $x^{\prime}$ is varied.

The quantum version of a non-linear oscillator Hamiltonian with variable mass
is given by Tchoffo et al. (2019)

$H=\frac{1}{2\lambda}\left[-(1+\delta^{2}x^{2})\frac{d^{2}}{dx^{2}}-2\delta^{2}x\frac{d}{dx}+\frac{\lambda^{2}x^{2}}{1+\delta^{2}x^{2}}\right]$
(18)

in which where $\lambda$ is a real parameter and $\delta$ is constant that
measures the force of the nonlinearity of the oscillator. By using ladder
operators, one can define a non-linear coherent state as the following:

$\displaystyle\ket{x}=\frac{1}{\mathcal{N}(x)}\sum_{n=0}^{\infty}\frac{x^{n}}{\rho_{n}}\ket{n}$
(19)

in which $\rho_{n}=n!(-k)^{n}(2-1/k)_{n}$, which $(u)_{n}=u(u-1)\cdots(u+n-1)$
represents the Pochhammer symbol, $k=\delta^{2}/2\lambda$ and normalization
factor $\mathcal{N}(x)$ is given by

$\displaystyle\mathcal{N}^{2}(x)$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\left(\frac{(1/k)^{n}}{n!(2-1/k)_{n}}\right)^{2}|x|^{2n}$
(20) $\displaystyle=$ ${}_{0}F_{3}(;1,2-1/k,2-1/k;|x/k|^{2})$

which ${}_{0}F_{3}(;1,2-1/k,2-1/k;|x/k|^{2})$ is a generalized hyper-geometric
function.
By considering a multi-dimensional input set in a data set of vectors
$\mathbf{x}=(x_{1},\cdots,x_{N})^{T}\in\mathbb{R}^{N}$, one can define the
joint state of $N$ deformed coherent states,

$\phi:(x_{1},\cdots,x_{N})\rightarrow|x_{1},k\rangle\otimes|x_{2},k\rangle\otimes\cdots\otimes|x_{N},k\rangle.$

Therefore, the kernel is defined as the following:

$\displaystyle K(\mathbf{x},\mathbf{x^{\prime}})=\prod_{i=1}^{N}\langle
x_{i};k|x_{i}^{\prime};k\rangle,$ (21)

in which

$\displaystyle\langle
x_{i};k|x_{i}^{\prime};k\rangle=\frac{{}_{0}F_{3}(;1,2-1/k,2-1/k;x_{i}x_{i}^{\prime}/k^{2})}{\mathcal{N}(x)\mathcal{N}(x^{\prime})}$
(22)

Figure 2 schematically illustrates the kernel function (22), for $k=-0.001$
and $k=-0.05$.

###  IV.2 Geometrical properties of associated Hilbert Space constructed by
Non-linear coherent state

For understanding the rule of nonlinear parameter $k$, we will study the
geometrical properties of above-mentioned feature spaces. We can define the
line element of the feature space by using the Fubini-Study metric Bengtsson
and Życzkowski (2017).

######  Definition IV.3.

A suitable metric between two Hilbert space vectors, e.g, $\ket{\psi}$ and
$\ket{\phi}$, is defined by

$\displaystyle
d(\ket{\psi},\ket{\phi})=\min||\ket{\psi}-e^{i\alpha}\ket{\phi}||,\
0\leq\alpha\leq 2\pi.$ (23)

The infinitesimal form of this metric is given by the Fubini-Study metric:

$\displaystyle ds^{2}=||d\ket{x}||^{2}-||\bra{x}d\ket{x}||^{2}.$ (24)

The following gives the Fubini-Study metric of a non-Linear coherent state
(19).

######  Theorem IV.1.

The Fubini-Study metric of a non-linear coherent state (19) is a surface with
non-zero curvature with the following metric:

$\displaystyle ds^{2}=\Omega(r)\left(dr^{2}+r^{2}d\phi^{2}\right),$ (25)

in which

$\displaystyle\Omega(r)=\frac{1}{2}\left[\frac{\partial^{2}_{r}\mathcal{N}(r)}{\mathcal{N}(r)}+\frac{\partial_{r}\mathcal{N}(r)}{r\mathcal{N}(r)}-\left(\frac{\partial_{r}\mathcal{N}(r)}{\mathcal{N}(r)}\right)^{2}\right].$
(26)

with $\partial_{r}\equiv\frac{\partial}{\partial r}$.

###### Proof.

We consider the $x=re^{i\phi}$, using the definition (24), directly leads to
the metric (25).
By using the definition of Christoffel symbol,

$\displaystyle\Gamma_{ab}^{c}=\frac{1}{2}g^{cd}\left(\partial_{a}g_{bd}+\partial_{b}g_{ad}-\partial_{d}g_{ab}\right),\
a,b,c=r,\phi$ (27)

with the Einstein summation rule, the non-zero Christoffel symbols are given
by

$\displaystyle\Gamma_{rr}^{r}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\partial_{r}\ln\Omega(r),$ (28)
$\displaystyle\Gamma_{\phi\phi}^{r}$ $\displaystyle=$
$\displaystyle-\left(r+\frac{r^{2}}{2}\partial_{r}\ln\Omega(r)\right),$ (29)
$\displaystyle\Gamma_{\phi r}^{\phi}$ $\displaystyle=$
$\displaystyle\frac{1}{r}+\frac{1}{2}\partial_{r}\ln\Omega(r).$ (30)

Hence, the non-zero Ricci tensors are given by,

$\displaystyle R_{rr}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(\partial_{r}^{2}\ln\Omega(r)+\frac{1}{r}\partial_{r}\ln\Omega(r)\right)$
(31) $\displaystyle R_{\phi\phi}$ $\displaystyle=$
$\displaystyle-\frac{r}{2}\left(\partial_{r}\ln\Omega(r)+r\partial_{r}^{2}\ln\Omega(r)\right)$
(32)

Hence, the Ricci scalar, $R=g^{ab}R_{ab}$, is obtained as

$\displaystyle
R=-\Omega(r)^{-1}\left(\partial_{r}^{2}\ln\Omega(r)+\frac{1}{r}\partial_{r}\ln\Omega(r)\right)$
(33)

∎

Figure 3: Ricci scalar of the Fubini-Study metric of a feature space
constructed by a oscillator with variable mass as function of $r$, for
different values of $k$.

It is worthwhile to note that the metric (25), which describes the feature
space, is a surface that is conformal with flat space, that is conformally
preserves angles, while lengths can be changed. In fact, this metric
illustrates a two-dimensional curved space, depending on conformal function
$\Omega(r)$. By considering the non-linear coherent state (19) and normalized
coefficient (5), Fig. 3 represents the Ricci scalar for different values of
$k$. This figure indicates that the feature space is a surface with negative
curvature. Decreasing the value $k$ causes the Ricci curvature to increase. In
other words, increasing value of $k$ causes the feature space and the
associated kernel to approach a flat space and RBF kernel, respectively.
Despite the fact that the feature space constructed on a non-linear coherent
state is a non-zero curved space, RBF, which is constructed from a linear
coherent state, is a kernel on flat space, with the zero Ricci scalar.

##  V Experimental Design

In the following, we empirically evaluate the KMNCS, i.e. the relation (21),
RBF kernel, i.e., the relation (7), and the Squeezed kernel, given by

$\displaystyle K_{Sq}=\prod_{i=1}\left[\frac{\sech c\sech
c^{\prime}}{1-e^{i(x-x^{\prime})}\tanh c\tanh c^{\prime}}\right]^{1/2}.$ (34)

SVM uses a kernel function to define a decision boundary for separating the
data points. Generally, the hyper parameter $C$ in the Gaussian kernel is used
to optimize the performance of SVM, as a cost function connected with mis-
classifications of data points in feature space of the training set. We kept
the hyperparameter $C=1$ as the optimal value.
In order to provide a comprehensive picture of the performance of the kernels,
i.e., kernels (7), (34) and (21), noise was systematically applied to the
input data. Applying noise implies adding it to the target variable. For
example, if the noise parameter is say $0.2$, a standard deviation of $0.2$
would be observed (originating from Gaussian noise) in the output. When data
points become inseparable due to noise, it becomes more challenging for the
classifier to accurately classify the data points.
We have used synthetic datasets (simple toy datasets) which are commonly used
to check the performance of kernels. The two datasets
($make\\_moons$222https://scikit-
learn.org/stable/modules/generated/sklearn.datasets.make_moons.html#sklearn.datasets.make_moons
and $make\\_circles$333https://scikit-
learn.org/stable/modules/generated/sklearn.datasets.make_circles.html#sklearn.datasets.make_circles)
are taken from sklearn. There is some flexibility in regard to each dataset.
Random noise can be introduced by adjusting different parameters for each
dataset: the Moons dataset generates two half circles with the noise
parameters affecting ’interleave’, the Circles dataset generates concentric
circles also affected by ’interleave’ via the noise parameter. The decision
region for class 1 is color-coded ’red’, and ’blue’ for class 0. We have also
recorded differing values of $flip\\_y$ which is an inbuilt parameter in
$make\\_classification$ 444https://scikit-
learn.org/stable/modules/generated/sklearn.datasets.make_classification.html#sklearn.datasets.make_classification
( Generate a random n-class classification problem) also from sklearn. A large
$flip\\_y$ supplements the noise effects, making accurate classification even
more challenging. The $factor$ and $random\\_state$ in the setting have also
been recorded. We have divided the data into $60\%$ training set and $40\%$
test set. We have also used 5 fold cross validation during training in order
to avoid overfitting problems. We have examined KMNCS using different values
of $k$ in order to evaluate the classification performance and to understand
how the decision boundaries are formed.

Figure 4: Kernel visualization (Squeezed, RBF kernels and KMNCS) for
$k=-0.001$, while the noise is $0.4$ on both the make_moons and make_circles
datasets. Figure 5: Kernel visualization (Squeezed, RBF, and KMNCS) for
$k=-0.1$, while noise is $0.5$ on the make_moons dataset, and $0.7$ on the
make_circles dataset. Figure 6: Kernel visualization (Squeezed, RBF, and
KMNCS) for $k=-0.01$, while noise is $0.2$ on the make_moons dataset, and
$0.1$ on make_circles dataset.

##  VI Results

The sklearn package includes a “fit” method that is used for informing the
model by applying a training set. To compute the score by cross-validation of
SVM, “cross_val_score” is used also from sklearn with a “5”-fold cross-
validation.
As we can see from Table 1, KMNCS has been tested on several values of $k$. If
we check the value of $k=-0.001$ with very low noise, i.e. $=0.1$, all the
three kernels provide almost the same classification accuracy as can be seen
in Figure 6 (Here the input data is separable and it is an easy task for the
classifier to classify them). Interestingly, KMNCS has better performance than
both baselines for both datasets when $k=-0.001$ and noise is $0.4$. This
performance in the presence of such noise demonstrates effectiveness of KMNCS.
The decision boundary is depicted in Figure 4.

When we increase the noise effect in the data, then classification becomes
more challenging i.e., when noise is $0.5$ then KMNCS outperforms the squeezed
and RBF kernel as can be seen in Figure 5. Note that it is possible to
increase the accuracy of the classifier if we select $k=-0.01,-0.1$ with the
same level of noise, that is $0.5$, as it can be seen in the Table 1. The
KMNCS performance is deemed better than the baselines as data is inseparable
(due to $50\%$ noise in the target variable). Again, where noise is $0.7$ and
$k=-0.1$, then KMNCS provides better performance than the two baseline
classifiers due to clean decision boundaries where data is separable, as can
be seen in Figure 5.

We also tested on $3000$ samples as can be seen in Table 3. KMNCS outperforms
both baselines which suggests it is effective on larger samples by producing
clean decision boundaries.
There is a trade-off with decision boundaries and accuracy. It is possible to
achieve even higher accuracy if we continue to modify $k$, however this may
obscure the decision boundaries ( despite high accuracy scores), which could
be related to bias and variance issues. We also computed the score without the
cross validation set with the results presented in Table 2.

Table 1: Accuracy on Squeezed kernel, RBF, and KMNCS on the make_moon and make_circles datasets with the following parameters: n_samples=200, random_state=50, y_flip=0.2, factor=0.5, n_informative=2, 5-fold cross validation. Noise effects are introduced in the data to increase difficulty of accurate classification. | Parameters | moons | circles
---|---|---|---
Squeezed Kernel | Noise=0.1 | 0.90 | 0.95
| Noise=0.2 | 0.85 | 0.86
| Noise=0.3 | 0.82 | 0.80
| Noise=0.4 | 0.79 | 0.71
| Noise=0.5 | 0.74 | 0.68
| Noise=0.7 | 0.62 | 0.61
RBF | Noise=0.1 | 0.99 | 1.0
| Noise=0.2 | 0.95 | 0.90
| Noise=0.3 | 0.89 | 0.80
| Noise=0.4 | 0.82 | 0.69
| Noise=0.5 | 0.72 | 0.64
| Noise=0.7 | 0.65 | 0.62
KMNCS | Noise=0.1, k=-0.001 | 0.97 | 1.0
| Noise=0.2, k=-0.001 | 0.95 | 0.88
| Noise=0.2, k=-0.01 | 0.95 | 0.89
| Noise=0.2, k=-0.1 | 0.94 | 0.88
| Noise=0.3, k=-0.001 | 0.90 | 0.79
| Noise=0.3, k=-0.01 | 0.90 | 0.79
| Noise=0.4, k=-0.001 | 0.84 | 0.72
| Noise=0.4, k=-0.01 | 0.84 | 0.72
| Noise=0.5, k=-0.001 | 0.75 | 0.68
| Noise=0.5, k=-0.01 | 0.76 | 0.69
| Noise=0.5, k=-0.1 | 0.76 | 0.70
| Noise=0.7, k=-0.001 | 0.64 | 0.62
| Noise=0.7, k=-0.01 | 0.62 | 0.62
| Noise=0.7, k=-0.1 | 0.65 | 0.66
Table 2: Accuracy on same parameters from Table 1 without cross validation | Parameters | moons | circles
---|---|---|---
Squeezed Kernel | Noise=0.2 | 0.90 | 0.94
| Noise=0.5 | 0.75 | 0.70
| Noise=0.7 | 0.68 | 0.56
RBF | Noise=0.2 | 0.96 | 0.90
| Noise=0.5 | 0.76 | 0.64
| Noise=0.7 | 0.70 | 0.54
KMNCS | Noise=0.2, k=-0.001 | 0.99 | 0.93
| Noise=0.5, k=-0.001 | 0.75 | 0.66
| Noise=0.7, k=-0.001 | 0.71 | 0.54
Table 3: Accuracy on 3000 samples with same parameters from Table 1 as well as 5-fold cross validation | Parameters | moons | circles
---|---|---|---
Squeezed Kernel | Noise=0.5 | 0.81 | 0.65
| Noise=0.8 | 0.72 | 0.55
| Noise=1.0 | 0.68 | 0.52
RBF | Noise=0.5 | 0.81 | 0.64
| Noise=0.8 | 0.73 | 0.54
| Noise=1.0 | 0.68 | 0.51
KMNCS | Noise=0.5, k=-0.1 | 0.82 | 0.65
| Noise=0.5, k=-0.01 | 0.81 | 0.65
| Noise=0.8, k=-0.1 | 0.73 | 0.56
| Noise=1.0, k=-0.1 | 0.69 | 0.53

##  VII Discussion

With respect to the general behaviour of KMNCS, it can be said that the kernel
is conducive to forming large decision boundaries, which is a major point of
contrast when considering the RBF kernel, as evident in Figure 4,5. As
previously mentioned, the hyper-parameter $C$ can be used to optimise an SVM
classifier, as a cost function associated with mis-classification of elements
in feature spaces of the ’training’ set. It implies the maximisation of $C$
tightens decision boundaries (so called ’hard’ margins), and was introduced by
Boser et al. Boser et al. (1992). In later work, ’hard’ margins were found to
fail on even slightly inseparable data. As a solution, Schiilkop et al. (1995)
introduced a meaningful technique for the minimisation of $C$ that was found
to enlarge the space covered by decision boundaries (’soft’ margins). While
this has largely benefited the field of SVM in contention with other
classification techniques, the trade-off is that significantly ’soft’ margins
fail to classify data entirely - formally known as over-generalisation, large
decision boundaries (as produced by the Squeezed kernel) become non-
representative of the data. As a result, the objective is to minimise the
hyper-parameter $C$, while maintaining the highest possible classification.
Relating this back to the findings of this paper, the topological structure of
the KMNCS is reminiscent of Ref. Boser et al. (1992), surprisingly in cases of
sparse data ( in this case the sparse $make\\_moons$ dataset), without concern
for $C$.

##  VIII Conclusions and Remarks

In this paper, we mapped datasets into non-linear coherent states, as a non-
linear feature space, constructed by a complex Hilbert space. We showed that
the RBF kernel is recovered when data is mapped to a complex Hilbert space
represented by coherent states. Therefore, non-linear coherent states can be
considered as natural generalized candidates for formalizing kernels. In
addition, by considering the non-linear coherent states of a quantum
oscillator with variable mass, we proposed a kernel function based on
generalized hypergeometric functions. This idea suggests a method for
obtaining a generalized kernel function which can be expressed by orthogonal
polynomial functions on the one hand, and opens a new door for using quantum
formalism to specify quantum algorithms in continuous variable quantum
computing, on the other. In addition, we studied the geometrical properties of
the surface in which the kernel lives. We indicated that the feature space of
a non-linear coherent state is a non-zero curved space, despite the fact that
the RBF kernel lives on feature space which is flat. This method can
potentially open a door for studying the impact of general curved space on the
machine learning methods more generally, and the problem of classification
more specifically.
More generally, this research has demonstrated how quantum approaches to
machine learning may prove beneficial. In practical usage, machine learning
applications of quantum theory have begun involving developments of physical
circuitry Havlíček et al. (2019). These contributions have begun to realise
the quantum processing components required to build quantum computational
devices designed solely for feature classification. It is inspiring to think
that eventually, SVM classification may ( with the assistance of quantum
theory) be computed substantially faster than ever before.

###### Acknowledgements.

This reserach has received funding from the European Union’s Horizon 2020
research and innovation programme under the Marie Sklodowska-Curie grant
agreement No 721321. It was additionally supported by the Asian Office of
Aerospace Research and Development (AOARD) grant: FA2386-17-1-4016.

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# Additive transform of an arithmetic function : Part I

Es-said En-naoui
University Sultan Moulay Slimane
Morocco
<EMAIL_ADDRESS>

###### Abstract

The Additive Transform of an arithmetic function represents a novel approach
to examining the interplay between multiplicative arithmetic function and
additive functions. This transform concept introduces a method to
systematically generate new arithmetic functions by combining the values of an
existing function under an additive operation. The resulting framework not
only extends our understanding of classical arithmetic functions but also
provides a versatile tool for exploring additive relationships within the
realm of number theory. In this article, we present the fundamental principles
of the Additive Transform and illustrate its application through various
examples, shedding light on its potential implications for diverse
mathematical domains.
For all positive integer $n$. a motivation for the present study is to give a
new concept named the Additive transform of an arithmetic function $f$ when
$f$ equals some special arithmetic functions, that new concept can help us to
prove many results like :

$\big{(}\mu*f.Id\big{)}(n)=\varphi(n)f(n)+\varphi(n)\sum\limits_{p^{\alpha}||n}\frac{f(p^{\alpha})-f(p^{\alpha-1})}{p-1}$
(1)

where $f$ is an additive function .

##  1 Introduction

First of all, to cultivate analytic number theory one must acquire a
considerable skill for operating with arithmetic functions. We begin with a
few elementary considerations.

######  Definition 1.1 (arithmetic function).

An arithmetic function is a function $f:\mathbb{N}\longrightarrow\mathbb{C}$
with domain of definition the set of natural numbers $\mathbb{N}$ and range a
subset of the set of complex numbers $\mathbb{C}$.

######  Definition 1.2 (multiplicative function).

A function $f$ is called an multiplicative function if and only if :

$f(nm)=f(n)f(m)$ (2)

for every pair of coprime integers $n$,$m$. In case (2) is satisfied for every
pair of integers $n$ and $m$ , which are not necessarily coprime, then the
function $f$ is called completely multiplicative.

Clearly , if $f$ are a multicative function , then
$f(n)=f(p_{1}^{\alpha_{1}})\ldots f(p_{s}^{\alpha_{s}})$, for any positive
integer $n$ such that $n=p_{1}^{\alpha_{1}}\ldots p_{s}^{\alpha_{s}}$ , and if
$f$ is completely multiplicative , so we have :
$f(n)=f(p_{1})^{\alpha_{1}}\ldots f(p_{s})^{\alpha_{s}}$.

The functions defined above are widely studied in the literature, see, e.g.,
[Ap, 11, 13, 14, 15, 16, 17].

######  Definition 1.3 (additive function).

A function $f$ is called an additive function if and only if :

$f(nm)=f(n)+f(m)$ (3)

for every pair of coprime integers $n$,$m$. In case (3) is satisfied for every
pair of integers $n$ and $m$ , which are not necessarily coprime, then the
function $f$ is called completely additive.

Clearly , if $f$ are a additive function , then
$f(n)=f(p_{1}^{\alpha_{1}})+\ldots+f(p_{s}^{\alpha_{s}})$, for any positive
integer $n$ such that $n=p_{1}^{\alpha_{1}}\ldots p_{s}^{\alpha_{s}}$ , and if
$f$ is completely additive , so we have :
$f(n)=\alpha_{1}f(p_{1})+\ldots+\alpha_{s}f(p_{s})$.

######  Definition 1.4 (L-additive function).

We say that an arithmetic function $f$ is Leibniz-additive (or, L-additive, in
short) if there is a completely multiplicative function $h_{f}$ such that

$f(mn)=f(m)h_{f}(n)+f(n)h_{f}(m)$ (4)

for all positive integers $m$ and $n$.

##  2 Additive transform of an arithmetic function .

Let us begin by defining the new concept related to thearithmetic function,
this concept called Additive transform of arithmetic function.

######  Definition 2.1.

Let $f$ an function arithmetic.
The Additive transform of an arithmetic function $f$ is an application
$\Phi_{f}:\mathbb{N}\longrightarrow\mathbb{C}$ verify that condition :

  1. 1.

$\forall\;p\;\in\mathbb{P}\;,\;\forall\;\alpha\;\in\mathbb{N}^{*}\;\;,\;\;\Phi_{f}(p^{\alpha})=f(p^{\alpha}).$

  2. 2.

$\Phi_{f}(nm)=n\Phi_{f}(m)+m\Phi_{f}(n)$ for relatively prime $m$ and $n$.
(Leibniz formula).

######  Definition 2.2.

Let $f$ an arithmetic function.
The Additive transform of an arithmetic function $f$ is a function
$\Phi_{f}:\mathbb{N}\longrightarrow\mathbb{C}.$,It is :

  1. 1.

Additive transform if $\Phi_{f}(nm)=n\Phi_{f}(m)+m\Phi_{f}(n)$ for relatively
prime $m$ and $n$.

  2. 2.

Completely additive transform if $\Phi_{f}(nm)=n\Phi_{f}(m)+m\Phi_{f}(n)$ for
any $m$ and $n$.

######  Remark 2.1.

Let $f$ an arithmetic function , if the additive transform of $f$ is
completely additive transform , then $\Phi_{f}$ is L-additive with
$h_{\Phi_{f}}(n)=n$ .

######  Theorem 2.1.

Let $n$ a positive integer such that $n=p_{1}^{\alpha_{1}}\ldots
p_{s}^{\alpha_{s}}$ and let $f$ an arithmetic function , So we have :

$\Phi_{f}(n)=n\sum\limits_{i=1}^{s}\frac{f\big{(}p_{i}^{\alpha_{i}}\big{)}}{p_{i}^{\alpha_{i}}}.$
(5)

###### Proof.

Let $n$ a positive integer such that $n=p_{1}^{\alpha_{1}}\ldots
p_{s}^{\alpha_{s}}$ and let $f$ an arithmetic function ,bu definition
$(\ref{def-2-1})$ Then :

$\displaystyle\Phi_{f}(n)$ $\displaystyle=\Phi_{f}(p_{1}^{\alpha_{1}}\ldots
p_{s}^{\alpha_{s}})=p_{1}^{\alpha_{1}}\Phi_{f}(p_{2}^{\alpha_{2}}\ldots
p_{s}^{\alpha_{s}})+p_{2}^{\alpha_{2}}\ldots
p_{s}^{\alpha_{s}}\Phi_{f}(p_{1}^{\alpha_{1}})$
$\displaystyle=n\frac{f(p_{1}^{\alpha_{1}})}{p_{1}^{\alpha_{1}}}+p_{1}^{\alpha_{1}}\bigg{(}p_{2}^{\alpha_{2}}\Phi_{f}(p_{3}^{\alpha_{3}}\ldots
p_{s}^{\alpha_{s}})+p_{3}^{\alpha_{3}}\ldots
p_{s}^{\alpha_{s}}\Phi_{f}(p_{2}^{\alpha_{2}})\bigg{)}$
$\displaystyle=n\frac{f(p_{1}^{\alpha_{1}})}{p_{1}^{\alpha_{1}}}+n\frac{f(p_{2}^{\alpha_{2}})}{p_{2}^{\alpha_{2}}}+p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}\Phi_{f}(p_{3}^{\alpha_{3}}\ldots
p_{s}^{\alpha_{s}})$ $\displaystyle\vdots$
$\displaystyle=n\frac{f(p_{1}^{\alpha_{1}})}{p_{1}^{\alpha_{1}}}+n\frac{f(p_{2}^{\alpha_{2}})}{p_{2}^{\alpha_{2}}}+\ldots+n\frac{f(p_{s}^{\alpha_{s}})}{p_{s}^{\alpha_{s}}}$
$\displaystyle=n\bigg{(}\frac{f(p_{1}^{\alpha_{1}})}{p_{1}^{\alpha_{1}}}+\ldots+\frac{f(p_{s}^{\alpha_{s}})}{p_{s}^{\alpha_{s}}}\bigg{)}$
$\displaystyle=n\sum\limits_{i=1}^{s}\frac{f\big{(}p_{i}^{\alpha_{i}}\big{)}}{p_{i}^{\alpha_{i}}}$

∎

######  Definition 2.3.

Let $n$ a positive integer , and $p\in\;\mathbb{P}$ such that
$\alpha=v_{p}(n)\neq 0$ , So one can define the arithmetic partial derivative
of a function arithmetic $f$ via :

$\frac{\partial f}{\partial p}(n)=\frac{f(p^{\alpha})}{p^{\alpha}}n$ (6)

Then we have this result :

$\Phi_{f}(n)=\sum\limits_{p^{\alpha}||n}\frac{\partial f}{\partial p}(n)$ (7)

######  Proposition 2.1.

Let $n$ a positive integer such that and let $f$ an arithmetic function
multiplicative , So we have :

$\Phi_{f}(n)\geq
n\omega(n)\bigg{(}\frac{f(n)}{n}\bigg{)}^{\frac{1}{\omega(n)}}.$ (8)

###### Proof.

Let $n$ a positive integer such that $n=p_{1}^{\alpha_{1}}\ldots
p_{s}^{\alpha_{s}}$ and let $f$ an arithmetic function .
for every $x_{1},x_{2},\ldots,x_{s}\in\mathbb{R}^{+*}$, by The AM-GM
inequality we have :

$\frac{x_{1}+x_{2}+\ldots+x_{n}}{n}\geq\sqrt[n]{x_{1}\cdot
x_{2}\cdot\ldots\cdot x_{n}}$

So if we put $x_{i}=\frac{f(p_{i}^{\alpha_{i}})}{p_{i}^{\alpha_{i}}}$ we find
that:

$\displaystyle\frac{1}{s}\sum\limits_{i=1}^{s}\frac{f(p_{i}^{\alpha_{i}})}{p_{i}^{\alpha_{i}}}\geq\sqrt[s]{\prod\limits_{i=1}^{s}\frac{f(p_{i}^{\alpha_{i}})}{p_{i}^{\alpha_{i}}}}\Longrightarrow
n\sum\limits_{i=1}^{s}\frac{f(p_{i}^{\alpha_{i}})}{p_{i}^{\alpha_{i}}}\geq
ns\sqrt[s]{\frac{\prod\limits_{i=1}^{s}f(p_{i}^{\alpha_{i}})}{\prod\limits_{i=1}^{s}p_{i}^{\alpha_{i}}}}$

Since $\prod\limits_{i=1}^{s}f(p_{i}^{\alpha_{i}})=f(n)$ and
$\prod\limits_{i=1}^{s}p_{i}^{\alpha_{i}}=n$ , Then we have :

$\Phi_{f}(n)\geq ns\sqrt[s]{\frac{f(n)}{n}}$

∎

######  Proposition 2.2.

Let $n$ a positive integer such that and let $f$ an arithmetic function
additive , then we have :

$\Phi_{f}(n)\leq\frac{1}{2}nf(n).$ (9)

###### Proof.

Let $n$ a positive integer such that $n=p_{1}^{\alpha_{1}}\ldots
p_{s}^{\alpha_{s}}$ and let $f$ an arithmetic function , then we have .

$p^{\alpha}\geq
2\Longrightarrow\frac{1}{p^{\alpha}}\leq\frac{1}{2}\Longrightarrow\frac{f(p^{\alpha})}{p^{\alpha}}\leq\frac{f(p^{\alpha})}{2}\Longrightarrow\sum\limits_{p^{\alpha}||n}\frac{f(p^{\alpha})}{p^{\alpha}}\leq\frac{1}{2}\sum\limits_{p^{\alpha}||n}f(p^{\alpha})$

∎

######  Proposition 2.3.

Let $n$ a positive integer .
For every multiplicative function $f$ , there are exist a function additive
$g$ such that : $\Phi_{f}(n)=\Phi_{g}(n)$ , with :

$g(n)=\sum\limits_{p^{\alpha}||n}f(p^{\alpha})$

###### Proof.

Let $f$ be a multiplicative function, and define
$g(n)=\sum_{p^{\alpha}\parallel n}f(p^{\alpha})$. Then, for any prime power
$p^{\alpha}$, we have $g(p^{\alpha})=f(p^{\alpha})$, implying
$\Phi_{f}(n)=\Phi_{g}(n)$. ∎

######  Proposition 2.4.

Given two multiplicative arithmetic functions $f$ and $g$ ,then for every
positive integer $n$ we have :

$f(n)=g(n)\Longleftrightarrow\Phi_{f}(n)=\Phi_{g}(n)$ (10)

After we have seen that many fundamental properties of the additive transform
of an arithmetic function. We complete this article by changing our point of
view slightly and demonstrate that additive transform can also be studied in
terms of the Dirichlet convolutions .

Let $f$ and $g$ be arithmetic functions. Their Dirichlet convolution is

$(f\ast g)(n)=\sum_{\begin{subarray}{c}a,b=1\\\
ab=n\end{subarray}}^{n}f(a)g(b)=\sum_{\begin{subarray}{c}d|n\end{subarray}}^{n}f(d)g\left(\frac{n}{d}\right).$

where the sum extends over all positive divisors $d$ of $n$ , or equivalently
over all distinct pairs $(a,b)$ of positive integers whose product is $n$.
In particular, we have $(f*g)(1)=f(1)g(1)$ ,$(f*g)(p)=f(1)g(p)+f(p)g(1)$ for
any prime $p$ and for any power prime $p^{m}$ we have :

$(f*g)(p^{m})=\sum\limits_{j=0}^{m}f(p^{j})g(p^{m-j})$ (11)

This product occurs naturally in the study of Dirichlet series such as the
Riemann zeta function. It describes the multiplication of two Dirichlet series
in terms of their coefficients:

$\bigg{(}\sum\limits_{n\geq
1}\frac{\big{(}f*g\big{)}(n)}{n^{s}}\bigg{)}=\bigg{(}\sum\limits_{n\geq
1}\frac{f(n)}{n^{s}}\bigg{)}\bigg{(}\sum\limits_{n\geq
1}\frac{g(n)}{n^{s}}\bigg{)}$ (12)

with Riemann zeta function or is defined by :

$\zeta(s)=\sum\limits_{n\geq 1}\frac{1}{n^{s}}$

These functions are widely studied in the literature (see, e.g., [3, 6, 7]).

######  Proposition 2.5.

Given two arithmetic functions $f$ and $g$, and lets $n$ and $m$ two positive
integer such that $gcd(n,m)=1$ , if $f$ is multiplicative function ,Then we
have :

$(f*\Phi_{g})(nm)=(Id*f)(n).(f*\Phi_{g})(m)+(Id*f)(m).(f*\Phi_{g})(n)$ (13)

###### Proof.

Lets $n$ and $m$ two positive integers such that $gcd(n,m)=1$, and lets $f$
and $g$ two arithmetics function , if $f$ is multuplicative then we have :

$\displaystyle(f*\Phi_{g})(nm)$
$\displaystyle=\sum\limits_{d|nm}f\big{(}\frac{nm}{d}\big{)}\Phi_{g}(d)=\sum\limits_{\underset{d_{2}|m}{d_{1}|n}}f(\frac{nm}{d_{1}d_{2}})\Phi_{g}(d_{1}d_{2})$
$\displaystyle=\sum\limits_{\underset{d_{2}|m}{d_{1}|n}}f(\frac{n}{d_{1}})f(\frac{m}{d_{2}})\bigg{(}d_{1}\Phi_{g}(d_{2})+d_{2}\Phi_{g}(d_{1})\bigg{)}$
$\displaystyle=\sum\limits_{\underset{d_{2}|m}{d_{1}|n}}\bigg{(}d_{1}f(\frac{n}{d_{1}})f(\frac{m}{d_{2}})\Phi_{g}(d_{2})+d_{2}f(\frac{m}{d_{2}})f(\frac{n}{d_{1}})\Phi_{g}(d_{1})\bigg{)}$
$\displaystyle=\bigg{(}\sum\limits_{d_{1}|n}d_{1}f(\frac{n}{d_{1}})\bigg{)}\bigg{(}\sum\limits_{d_{2}|m}f(\frac{m}{d_{2}})\Phi_{g}(d_{2})\bigg{)}+\bigg{(}\sum\limits_{d_{2}|m}d_{2}f(\frac{m}{d_{2}})\bigg{)}\bigg{(}\sum\limits_{d_{1}|n}f(\frac{n}{d_{1}})\Phi_{g}(d_{1})\bigg{)}$
$\displaystyle=\big{(}Id*f\big{)}(n).\big{(}f*\Phi_{g}\big{)}(m)+\big{(}Id*f\big{)}(m).\big{(}f*\Phi_{g}\big{)}(n)$

∎

######  Corollary 2.1.

Let $n$ a positive integer , for every multiplicative function we have :

$\big{(}f\ast\Phi_{g}\big{)}(n)=(Id*f)(n)\sum\limits_{p^{\alpha}||n}\frac{\big{(}f\ast\Phi_{g}\big{)}(p^{\alpha})}{(Id*f)(p^{\alpha})}$
(14)

###### Proof.

Let $n$ a positive integer such that $n=p_{1}^{\alpha_{1}}\ldots
p_{s}^{\alpha_{s}}$ and lets $f$ and $g$ two arithmetics functions , if $f$ is
multiplicative function Then by using the proposition $(\ref{prop-2-5})$ we
have :

$\displaystyle\big{(}f*\Phi_{g}\big{)}(n)$
$\displaystyle=\big{(}f*\Phi_{g}\big{)}(p_{1}^{\alpha_{1}}\ldots
p_{s}^{\alpha_{s}})$
$\displaystyle=\big{(}Id*f\big{)}(p_{2}^{\alpha_{2}}\ldots
p_{s}^{\alpha_{s}}).\big{(}f*\Phi_{g}\big{)}(p^{\alpha_{1}}_{1})+\big{(}Id*f\big{)}(p^{\alpha_{1}}_{1}).\big{(}f*\Phi_{g}\big{)}(p_{2}^{\alpha_{2}}\ldots
p_{s}^{\alpha_{s}})$
$\displaystyle=\big{(}Id*f\big{)}(n).\frac{\big{(}f*\Phi_{g}\big{)}(p^{\alpha_{1}}_{1})}{\big{(}Id*f\big{)}(p_{1}^{\alpha_{1}})}+\big{(}Id*f\big{)}(p^{\alpha_{1}}_{1}).\bigg{[}\big{(}Id*f\big{)}(p_{3}^{\alpha_{3}}\ldots
p_{s}^{\alpha_{s}}).\big{(}f*\Phi_{g}\big{)}(p^{\alpha_{2}}_{2})+$
$\displaystyle+\big{(}Id*f\big{)}(p^{\alpha_{2}}_{2}).\big{(}f*\Phi_{g}\big{)}(p_{3}^{\alpha_{3}}\ldots
p_{s}^{\alpha_{s}})\bigg{]}$
$\displaystyle=\big{(}Id*f\big{)}(n)\frac{\big{(}f*\Phi_{g}\big{)}(p^{\alpha_{1}}_{1})}{\big{(}Id*f\big{)}(p_{1}^{\alpha_{1}})}+\big{(}Id*f\big{)}(n)\frac{\big{(}f*\Phi_{g}\big{)}(p^{\alpha_{2}}_{2})}{\big{(}Id*f\big{)}(p_{2}^{\alpha_{2}})}+$
$\displaystyle+\big{(}Id*f\big{)}(p^{\alpha_{1}}_{1})\big{(}Id*f\big{)}(p^{\alpha_{2}}_{2})\big{(}f*\Phi_{g}\big{)}(p_{3}^{\alpha_{3}}\ldots
p_{s}^{\alpha_{s}})$ $\displaystyle\vdots$
$\displaystyle=\big{(}Id*f\big{)}(n)\frac{\big{(}f*\Phi_{g}\big{)}(p^{\alpha_{1}}_{1})}{\big{(}Id*f\big{)}(p_{1}^{\alpha_{1}})}+\ldots+\big{(}Id*f\big{)}(n)\frac{\big{(}f*\Phi_{g}\big{)}(p^{\alpha_{s}}_{s})}{\big{(}Id*f\big{)}(p_{s}^{\alpha_{s}})}$
$\displaystyle=\big{(}Id*f\big{)}(n)\bigg{[}\frac{\big{(}f*\Phi_{g}\big{)}(p^{\alpha_{1}}_{1})}{\big{(}Id*f\big{)}(p_{1}^{\alpha_{1}})}+\ldots+\frac{\big{(}f*\Phi_{g}\big{)}(p^{\alpha_{s}}_{s})}{\big{(}Id*f\big{)}(p_{s}^{\alpha_{s}})}\bigg{]}$
$\displaystyle=\big{(}Id*f\big{)}(n)\sum\limits_{i=1}^{s}\frac{\big{(}f*\Phi_{g}\big{)}(p^{\alpha_{i}}_{i})}{\big{(}Id*f\big{)}(p^{\alpha_{i}}_{i})}$

∎

######  Lemma 2.1.

Let $p$ be a number prime ,and let $f$ an arithmetic function , then we have :

$\forall\;\alpha\;\in\mathbb{N}^{*}\;\;\;,\;\big{(}\mu*\Phi_{f}\big{)}(p^{\alpha})=\left\\{\begin{array}[]{lll}f(p^{\alpha})-f(p^{\alpha-1})&if&\alpha>1\\\
f(p)&if&\alpha=1\par\end{array}\right.$ (15)

###### Proof.

Let $p$ be a number prime ,and $\alpha\;\in\mathbb{N}^{*}$ , if $\alpha=1$
then :

$\big{(}\mu*\Phi_{f}\big{)}(p^{1})=\mu(1)\Phi_{f}(p)+\mu(p)\Phi_{f}(1)=f(p)$

If $\alpha>1$ then we have :

$\big{(}\mu*\Phi_{f}\big{)}(p^{\alpha})=\sum\limits_{j=0}^{\alpha-1}\mu(p^{j})\Phi_{f}(p^{\alpha-j})=\sum\limits_{j=0}^{\alpha-1}\mu(p^{j})f(p^{\alpha-j})=\mu(p^{0})f(p^{\alpha})+\ldots+\mu(p^{\alpha-1})f(p)$

Since for every integer $m\geq 2$ we know that $\mu(p^{m})=0$ , then :

$\big{(}\mu*\Phi_{f}\big{)}(p^{\alpha})=\mu(1)f(p^{\alpha})+\mu(p)f(p^{\alpha-1})=f(p^{\alpha})-f(p^{\alpha-1})$

∎

######  Proposition 2.6.

Let $n$ a positive integer such that $n=p_{1}^{\alpha_{1}}\ldots
p_{s}^{\alpha_{s}}$ and let $f$ an arithmetic function , we have :

$\big{(}\mu*\Phi_{f}\big{)}(n)=\left\\{\begin{array}[]{lll}\varphi(n)\sum\limits_{p^{\alpha}||n}\frac{f(p^{\alpha})-f(p^{\alpha-1})}{p^{\alpha}-p_{i}^{\alpha-1}}&if&n\text{
is a perfect square}\\\
\varphi(n)\sum\limits_{p^{\alpha}||n}\frac{f(p)}{p-1}&if&n\text{ is a
squarefree}\end{array}\right.$ (16)

###### Proof.

Let $n$ a positive integer ,Since $\mu*Id=\varphi$ , then substitiling $f=\mu$
into corollary $(\ref{cor-2-1})$ to found :

$\big{(}\mu\ast\Phi_{f}\big{)}(n)=(Id*\mu)(n)\sum\limits_{p^{\alpha}||n}\frac{\big{(}\mu\ast\Phi_{f}\big{)}(p^{\alpha})}{(Id*\mu)(p^{\alpha})}=\varphi(n)\sum\limits_{p^{\alpha}||n}\frac{\big{(}\mu\ast\Phi_{f}\big{)}(p^{\alpha})}{\varphi(p^{\alpha})}$

Since $\varphi(p^{\alpha})=p^{\alpha}-p^{\alpha-1}$ and by usine Lemma
$(\ref{lem-2-2})$ we have :

$\big{(}\mu*\Phi_{f}\big{)}(n)=\left\\{\begin{array}[]{lll}\varphi(n)\sum\limits_{p^{\alpha}||n}\frac{f(p^{\alpha})-f(p^{\alpha-1})}{p^{\alpha}-p^{\alpha-1}}&if&n\text{
is a perfect square}\\\
\varphi(n)\sum\limits_{p^{\alpha}||n}\frac{f(p)}{p-1}&if&n\text{ is a
squarefree}\end{array}\right.$

∎

######  Remark 2.2.

if $f$ is completely additive then we have :

$\big{(}\mu*\Phi_{f}\big{)}(n)=\sum\limits_{p^{\alpha}||n}\frac{f(p)}{p^{\alpha}-p_{i}^{\alpha-1}}$
(17)

######  Proposition 2.7.

Let $n$ a positive integer ,if $f$ is an arithmetic function additive then we
have :

$\Phi_{Id.f}(n)=nf(n)$ (18)

###### Proof.

Let $n$ a positive integer ,and $f$ is an arithmetic function additive, then
by using the theorem $(\ref{the-2-1})$ we have :

$\Phi_{Id.f}(n)=n\sum\limits_{p^{\alpha}||n}\frac{Id\big{(}p^{\alpha}\big{)}f\big{(}p^{\alpha}\big{)}}{p^{\alpha}}=n\sum\limits_{p^{\alpha}||n}\frac{p^{\alpha}f\big{(}p^{\alpha}\big{)}}{p^{\alpha}}=n\sum\limits_{p^{\alpha}||n}f\big{(}p^{\alpha}\big{)}=nf(n)$

∎

######  Theorem 2.2.

Let $n$ a positive integer , for every arithmetic function additive $f$ we
have :

$\big{(}\mu*f.Id\big{)}(n)=\varphi(n)f(n)+\varphi(n)\sum\limits_{p^{\alpha}||n}\frac{f(p^{\alpha})-f(p^{\alpha-1})}{p-1}$
(19)

###### Proof.

Let $n$ a positive integer , and $f$ an arithmetic function additive, by using
proposition $(\ref{prop-2-6})$ we have :

$\displaystyle\big{(}\mu\ast\Phi_{Id.f}\big{)}(n)$
$\displaystyle=\varphi(n)\sum\limits_{p^{\alpha}||n}\frac{p^{\alpha}f(p^{\alpha})-p^{\alpha-1}f(p^{\alpha-1})}{p^{\alpha}-p^{\alpha-1}}$
$\displaystyle=\varphi(n)\sum\limits_{p^{\alpha}||n}\frac{p^{\alpha}f(p^{\alpha})-p^{\alpha-1}f(p^{\alpha})+p^{\alpha-1}f(p^{\alpha})-p^{\alpha-1}f(p^{\alpha-1})}{p^{\alpha}-p^{\alpha-1}}$
$\displaystyle=\varphi(n)\sum\limits_{p^{\alpha}||n}\frac{f(p^{\alpha})(p^{\alpha}-p^{\alpha-1})}{p^{\alpha}-p^{\alpha-1}}+\frac{p^{\alpha-1}(f(p^{\alpha})-f(p^{\alpha-1}))}{p^{\alpha-1}(p-1)}$
$\displaystyle=\varphi(n)\sum\limits_{p^{\alpha}||n}f(p^{\alpha})+\varphi(n)\sum\limits_{p^{\alpha}||n}\frac{f(p^{\alpha})-f(p^{\alpha-1})}{p-1}$
$\displaystyle=\varphi(n)f(n)+\varphi(n)\sum\limits_{p^{\alpha}||n}\frac{f(p^{\alpha})-f(p^{\alpha-1})}{p-1}$

Since $\big{(}\mu\ast\Phi_{Id.f}\big{)}(n)=nf(n)$ by the proposition
$(\ref{prop-2-7})$ then we have :

$\big{(}\mu*f.Id\big{)}(n)=\varphi(n)f(n)+\varphi(n)\sum\limits_{p^{\alpha}||n}\frac{f(p^{\alpha})-f(p^{\alpha-1})}{p-1}$

∎

If f is completely additive then $f(p^{\alpha})-f(p^{\alpha-1})=\alpha
f(p)-(\alpha-1)f(p)=f(p)$ , so we have this corollary :

######  Corollary 2.2.

Let $n$ a positive integer not null , If $f$ is completely additive then we
have :

$\big{(}\mu*Id.f\big{)}(n)=\varphi(n)f(n)+\varphi(n)\sum\limits_{p^{\alpha}||n}\frac{f(p)}{p-1}$
(20)

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