# On a General Class of $A$-functions

Joseph Vade Burnett  The University of Texas at Dallas,
<EMAIL_ADDRESS>Alex Taylor  The University of Illinois at
Urbana-Champaign<EMAIL_ADDRESS>
###### Abstract

Expanding upon recent work, a new class of $A$-functions is introduced that
can be viewed as an appropriate generalization of the class of regular
$A$-functions, the class of structured $A$-functions, and the class of perfect
$A$-functions. Algebraic results are discussed and analysis is performed to
deduce new results concerning rational arithmetical functions.

##  1 Introduction

###  1.1 $A$-functions

Denote by $D(n)$ the set of positive divisors of $n$. Let
$A:\mathbb{N}\rightarrow\mathcal{P}(\mathbb{N})$ be any function such that
$1\in A(n)\subseteq D(n)$. Then $A$ is called an $A$-function.

Examples of $A$-functions have appeared in the literature for many years, but
usually in different forms. To each $A$-function $A$ a relation
$\operatorname{|}_{A}$ may be associated, where $d\operatorname{|}_{A}n$ if
and only if $d\in A(n)$. In this context we may call such $d$ ``$A$-divisors
of $n$'' and discuss the ``$A$-divisibility relation''.

Denote by $\mathcal{A}$ the set of complex-valued arithmetical functions. If
$A$ is an $A$-function, an arithmetical convolution-type operation $*_{A}$ may
be associated to it, where for $f,g\in\mathcal{A}$,

$(f*_{A}g)(n):=\sum_{d\in A(n)}f(d)g\left(\frac{n}{d}\right).$ (1)

In this case $*_{A}$ is called the ``$A$-convolution'', and we may discuss
algebraic properties of this operator. Since the term ``$A$-function'' was
only recently introduced, yet satisfies a need to be able to refer to
divisibility relations and arithmetical convolutions in the abstract, we shall
refer to such divisibility relations and arithmetical convolutions appearing
in the literature as $A$-functions, and speak of them in terms of
$A$-functions in order to have a uniform discussion.

###  1.2 Previous work

One of the first $A$-functions to be introduced was done so via the unitary
divisibility relation of Vaidyanathaswamy [33]. Subsequently, various works
were produced which studied and generalized the unitary divisibility relation
and associated convolution (see [8, 16, 19, 31]). In several cases, entire
classes of $A$-functions were introduced, viewing the usual divisibility
relation (which corresponds to the $A$-function $D$) and the unitary
divisibility relation as the first two elements of a sequence of $A$-functions
that preserved some property or theme (see [1, 3, 9, 10, 17, 28]).

Certain classes of $A$-functions have been the subject of many manuscripts
devoted to exploring their properties. The regular $A$-functions of Narkiewicz
[23] in particular have been heavily investigated and generalized (see [4, 14,
18, 22, 24, 29, 30, 32, 34]).

Other generalizations include the generalizations of the infinitary
divisibility relation of Cohen (see [5, 6, 9, 10, 21]), which will be actively
discussed in this work, and the generalization of convolutions to convolutions
involving weight functions (see [11, 12]).

###  1.3 Purpose

In this work, we describe the paradigm through which we view many of the
$A$-functions in the literature. In particular, the class of regular
$A$-functions, the class of structured $A$-functions, and the class of perfect
$A$-functions are portrayed as being the three sides of a diagrammatic
triangle (see figure below) whose vertices are the $A$-functions $D$, $D^{1}$,
and $\Theta$, corresponding to the Dirichlet, Unitary, and Infinitary
relations/convolutions, respectively.

One may then view the regular, structured, and perfect $A$-functions as being
those $A$-functions that preserve properties common to $D$ and $D^{1}$, to
$\Theta$ and $D$, and to $D^{1}$ and $\Theta$, respectively.

We introduce a new broad class of $A$-functions that contains as special
subclasses the regular, structured, and perfect $A$-functions. From this broad
class we extract a subclass, which we call the paradigmatic  $A$-functions,
whose members are $A$-functions possessing many properties common to the
regular, structured, and perfect $A$-functions. Algebraic properties
pertaining to the corresponding arithmetical convolution, numerical
factorization properties tied to the concept of multiplicativity, and analytic
properties resulting from a natural metric on $\mathcal{A}$ are explored.

###  1.4 Outline

In Section 2 we outline some preliminary definitions and results concerning
both $A$-functions in the general and in specific instances when $A$ is
regular or structured or perfect. The notion of the iterate is recalled for
later use, and we discuss various arithmetical functions which will appear
later, including rational arithmetical functions and a natural metric on
$\mathcal{A}$.

Special attention is given to the $A$-function $\Theta$, corresponding to the
infinitary divisibility relation/convolution, and in particular we introduce a
new equivalent definition of $\Theta$. Finally, we state the properties common
to the regular, structured, and perfect $A$-functions in order to guide the
next section.

In Section 3, we introduce our broad class of $A$-functions, termed complete
$A$-functions, in terms of certain generalized multiplicativity conditions. A
unique decomposition of each $n\in\mathbb{N}$ based off of $A$ follows,
enabling us to define the paradigmatic $A$-functions in a natural manner. It
is shown that paradigmatic $A$-functions correspond to commutative,
associative $A$ convolutions and that they may be used to characterize all
complete $A$-functions.

We introduce the concept of the ``maximal class'' of an arithmetical function
$f$ with $f(1)=1$, which is related to a concept of Ryden [25]. We prove
Ryden's theorem using our new notation and results, and deduce that all
paradigmatic $A$-functions may be realized as the maximal class of some
arithmetical function. We further show how every complete $A$-function may be
associated to a perfect $A$-function. Examples abound throughout this section,
including a special focus on the $A$-function $G$ from [4].

Finally, in Section 4 we bring together all of our tools in order to state a
result with an analytic flavor which is a generalization of the following
statement: Every multiplicative arithmetical function may be ``approximated''
using rational arithmetical functions. Here, ``approximation'' occurs with
respect to the metric of Schwab and Silberg [26].

##  2 Preliminary Definitions and Results

###  2.1 Properties $A$-functions can possess

Let $A\in\mathbb{A}$. $A$ is called simple if $1\in A(n)$ for all $n$ and
reflexive if $n\in A(n)$ for all $n$. $A$ is called symmetric if $d\in
A(n)\Rightarrow\frac{n}{d}\in A(n)$. $A$ is transitive if $A(d)\subseteq A(n)$
whenever $d\in A(n)$, which is to say that $\operatorname{|}_{A}$ is
transitive as a relation. If $A(n)=\\{1,n\\}$ ($n\neq 1$) then $n$ is called
$A$-primitive; 1 is not considered $A$-primitive for any $A$. We denote the
set of $A$-primitive elements by $\mathbb{P}_{A}$, with
$\mathbb{P}:=\mathbb{P}_{D}$ the set of prime numbers. Note that
$\mathbb{P}\subseteq\mathbb{P}_{A}$ as long as $A$ is reflexive.

$A$ is said to be multiplicative if
$A(mn)=A(m)\cdot(n):=\\{ab\operatorname{|}a\in A(m),b\in A(n)\\}$ whenever
$(m,n)=1$. Since $(m,n)=1\Leftrightarrow m\in D^{1}(mn)$, it is easy to see
that $A$ is multiplicative if and only if $m\in D^{1}(mn)\Rightarrow
A(mn)=A(m)\cdot A(n)$. This has been generalized by [6, 22, 34]. In ([6]), the
authors defined an $A$-function $A$ to be class-$B$, for $B\in\mathbb{A}$, if
$m\in B(mn)\Rightarrow A(mn)=A(m)\cdot A(n)$. We denote the set of all
class-$B$ $A$-functions by $C(B)$.

$A$ is called homogeneous if $A$ is multiplicative and for every
$p_{1},p_{2}\in\mathbb{P}$ and every $0\leqslant b\leqslant a$, $p_{1}^{b}\in
A(p_{1}^{a})\Leftrightarrow p_{2}^{b}\in A(p_{2}^{a})$. Examples of
homogeneous $A$-functions include the $k$-ary $A$-functions of Cohen [9], the
$k$-factorization $A$-functions of Litsyn and Shevelev [21], and the higher-
order $A$-functions of Alladi [1].

On the other hand, we call $A$ a cross-convolution if it is multiplicative but
not homogeneous. The concept of cross-convolution in the context of
$A$-functions is due to Tóth [30], and was formalized and generalized in [5].
If we let $\\{A_{\lambda}\\}_{\lambda\in\Lambda}$ be an indexed set of
homogeneous $A$-functions and let $f:\mathbb{P}\rightarrow\Lambda$ be a
function, then we may define $A_{f}$ to be the multiplicative $A$-function
such that $A_{f}(p^{a})=A_{f(p)}(p^{a})$ for all $p\in\mathbb{P}$ and all
$a\in\mathbb{N}$, with $A_{f}(1)=\\{1\\}$ by convention. We then call $A_{f}$
the $f$-cross convolution $A$-function with respect to
$\\{A_{\lambda}\\}_{\lambda\in\Lambda}$. If $f$ is a constant function, then
$A_{f}$ is homogeneous.

The original cross-convolution $A$-functions of Tóth [30], the structured
$A$-functions of [5], and the mixed-multiplicative $A$-functions of Litsyn and
Shevelev [21] are examples of families of cross-convolution $A$-functions
appearing in the literature.

###  2.2 The iterate

The divisibility relation corresponding to the unitary $A$-function $D^{1}$
was first introduced by Vaidyanathaswamy [33], who noted properties that it
possessed which were similar to the usual divisibility relation, and who
essentially initiated a search for other such divisibility relations. In
particular, this led to the introduction of entire families of $A$-functions
having $D$ and $D^{1}$ as elements:

  * •

The Suryanarayana $k$-ary $A$-functions $S_{k}$ of [28], where $d\in S_{k}(n)$
if and only if $d\operatorname{|}n$ and the largest $k^{th}$-power divisor of
$d$ and $\frac{n}{d}$ is 1.

  * •

The regular $A$-functions of Narkiewicz, discussed in Section 2.7.

  * •

The higher-order $A$-functions $A_{k}$ of Alladi [1], which are multiplicative
and are recursively defined such that $p^{b}\in A_{k}(p^{a})\Leftrightarrow
A_{k-1}(p^{b})\cap D(p^{a-b})=\\{1\\}$, where $A_{0}$ is $D$ and $A_{1}$
becomes $D^{1}$.

  * •

The $k$-ary $A$-functions of Cohen [9] (which were inspired by Suryanarayana's
introduction of the bi-unitary $A$-function $D^{2}$ and the recursive approach
taken by Alladi), which we denote by $D^{k}$, where $p^{b}\in D^{k}(p^{a})$ if
and only if $D^{k-1}(p^{b})\cap D^{k-1}(p^{a-b})=\\{1\\}$, where $D^{0}:=D$
and $D^{1}$ is the unitary divisibility relation.

Taking inspiration from the final family of $A$-functions, the $k$-ary
$A$-functions of Cohen, the authors of [5] introduced the following definition
for what they called the iterate of an $A$-function:

######  Definition 1.

Let $A\in\mathbb{A}$ be simple. Then the iterate of $A$, $A^{1}$, is the
$A$-function such that $d\in A^{1}(n)\Leftrightarrow A(d)\cap
A\left(\frac{n}{d}\right)=\\{1\\}$. For $k>1$, the $k$-fold iterate of $A$,
$A^{k}$, is the $A$-function such that $A^{k}=(A^{k-1})^{1}$.

The $k$-ary $A$-functions $D^{k}$ of Cohen are thus defined in terms of the
$k$-fold iterate. By applying the iterate recursively to an $A$-function $A$,
we may obtain a family of $A$-functions $\\{A^{k}\\}$ analogous to the $k$-ary
$A$-functions of Cohen. General results for arbitrary $A$ have not appeared in
the literature.

######  Remark 1.

In using the notation $A(d)\cap A\left(\frac{n}{d}\right)$, the authors of [5]
have circumvented an awkward process present in much of the literature,
whereby in order to define a new family of $A$-functions it was necessary to
first define an analogue of the GCD-function and express which divisors $d$ of
$n$ were to be included in the set $A(n)$ on the basis of some condition on
this analogue of the GCD-function. (cf. [3, 9, 16])

Rather than define entire families of auxiliary GCD-functions whose sole
purpose is to serve as indicator functions for membership in $A(n)$ for each
$n$, we shall continue in the convention of [5] and use conditions on specific
values of $A$-functions themselves to define new $A$-functions, where needed,
as is done above.

###  2.3 Arithmetical Functions

Denote by $\mathcal{A}$ the set of complex-valued arithmetical functions, and
if $X$ is some set by $\mathcal{A}_{X}$ the set of $X$-valued arithmetical
functions. Typically, $X$ is taken to be a set with algebraic structure
allowing for the addition and multiplication of its elements; in our case, we
will be utilizing $X=\mathbb{R}$, the set of real numbers in Section 3. One
may think of $\mathcal{A}$ as either a set of arithmetical functions or as the
space $\mathbb{C}^{\infty}$ of sequences of complex numbers, the latter being
the natural context in which the Dirichlet convolution can be realized.

Throughout this paper we will be employing several arithmetical functions
commonly appearing in the literature, including the Euler totient function
$\varphi$, the divisor function $\tau$, and the Möbius function $\mu$. Let
$\gamma(n)$ be the multiplicative core of $n$, i.e.,

$\gamma(n)=\prod_{p\operatorname{|}n}p.$

We shall also make use of the function $u$, where $u(n)=1$ for all $n$, the
function $I$, where $I(n)=n$ for all $n$, and the function $\iota$, the
identity under the Dirichlet convolution, where

$\iota(n)=\begin{cases}1&\text{ if }n=1\\\ 0&\text{otherwise.}\end{cases}$

If $A$ is simple and reflexive, a result of Narkiewicz states that $\iota$ is
the unique identity element with respect to the $A$-convolution $*_{A}$ in the
sense that for all $f\in\mathcal{A}$,

$\iota*f=f*\iota=f.$ (2)

Narkiewicz [23] also remarked that for all simple and reflexive $A$-functions,
the corresponding $A$-convolution is such that every arithmetical function $f$
with $f(1)\neq 0$ possesses a unique left inverse and a unique right inverse
with respect to the $A$-convolution $*_{A}$; that is, for $f\in\mathcal{A}$
there is a unique $f^{r}$ and $f^{l}$ in $\mathcal{A}$ such that

$\displaystyle f*_{A}f^{r}=\iota$ (3) $\displaystyle f^{l}*_{A}f=\iota.$

In the case where $A$ is symmetric, $*_{A}$ is commutative and hence the
unique left and right inverses of a function coincide.

In keeping with the notation of [4], for a simple and reflexive $A$-function
$A$, we define $\mu_{A}$ to be the left inverse of $u$ under the
$A$-convolution $*_{A}$. Note that this generalizes the definition of the
usual Möbius function to all applicable $A$. In this paper, $\mu_{A}$ will
only appear in the context of symmetric $A$-functions, so the specification of
$\mu_{A}$ as the right inverse is redundant, as in this case there will be
only one Möbius function corresponding to $A$.

We let $\tau_{A}$ denote the $A$-convolution of $u$ with itself.
Combinatorically speaking, $\tau_{A}$ counts the number of ``$A$-divisors'' of
$n$, which is just the size of the set $A(n)$.

Just as $A$-functions can belong to a given class, generalizing the concept of
multiplicativity of $A$-functions, the conept of multiplicativity of
arithmetical functions has been generalized on several occasions. Recall that
an arithmetical function $f$ is multiplicative if $f$ is not identically zero
and $f(mn)=f(m)f(n)$ whenever $(m,n)=1$, which is to say whenever $m\in
D^{1}(mn)$. Yocom [34] called an arithmetical function $A$-multiplicative, for
a regular $A$-function $A$, if $f$ is not identically zero and $m\in
A(mn)\Rightarrow f(mn)=f(m)f(n)$. Burnett, Osterman, and Lewandowski [6]
defined an arithmetical function $f$ to be class-$A$ for an arbitrary
$A$-function $A$ if $f$ is not identically zero and $m\in A(mn)\Rightarrow
f(mn)=f(m)f(n)$. We then refer to $A$ as an arithmetical class of $f$. As this
is a more generalized version of the concept of multiplicativity, we shall
adopt the notation of [6] for our present work. Thus, we denote by $c(A)$ the
set of class-$A$ arithmetical functions. Then $c(D^{1})$ is the set of all
multiplicative functions and $c(D)$ is the set of all completely
multiplicative functions.

###  2.4 A metric on $\mathcal{A}$

We reformulate a definition of Schwab and Silberg [26] for our purposes:

######  Definition 2.

Let $d:\mathcal{A}\times\mathcal{A}\rightarrow\mathbb{R}$,

$d(f,g)=\begin{cases}0&\text{ if }f=g\\\
\frac{1}{\min(k\in\mathbb{N}\operatorname{|}f(k)\neq g(k))}&\text{ if }f\neq
g\end{cases}$ (4)

It was shown that (4) defines a metric on $\mathcal{A}$, and it is not
difficult to show that this metric is complete. In Section 4 we will perform
analysis on subsets of $\mathcal{A}$ with respect to this metric.

###  2.5 Rafts and $A$-rafts

Suppose $f\in\mathcal{A}$ is the Dirichlet convolution of some number of
completely multiplicative arithmetical functions and some number of inverses
(with respect to the Dirichlet convolution); that is,

$f=f_{1}*\cdots*f_{r}*(f_{r+1})^{-1}*\cdots*(f_{r+s})^{-1},$ (5)

where $(f_{i})^{-1}$ is the inverse of $f_{i}$ under the Dirichlet
convolution. Functions of this form have traditionally been called rational
arithmetical functions, and were introduced by Vaidyanathaswamy [33]. Because
of the unwieldiness of this expression, and since we intend to express
generalizations thereof, we shall abbreviate a rational arithmetical function
to be, simply, a raft.

Suppose instead of being completely multiplicative (i.e., class-$D$), we
restrict the $f_{i}$ in equation (5) to be class-$A$ for some simple,
symmetric, and reflexive $A\in\mathbb{A}$, and we similarly replace the usual
Dirichlet convolution with the $A$-convolution:

$f=f_{1}*_{A}\cdots*_{A}f_{r}*_{A}(f_{r+1})^{-1}*_{A}\cdots*_{A}(f_{r+s})^{-1},$
(6)

where $(f_{i})^{-1}$ is the inverse of $f_{i}$ under the $A$-convolution. Then
we refer to $f$ as an $A$-raft. Typically, if $s=0$ a raft (resp. $A$-raft) is
called $r$-ic [7] (resp. $A$-$r$-ic).

$A$-rafts have been studied in the literature before in the context of $A$
being a regular $A$-function; see [18] for a derivation of some relevant
analogues of rafts in the context of $A$-rafts. For the purpose of this paper,
$A$ will always be chosen in such a way that $*_{A}$ is associative,
justifying the lack of parentheses in (6).

Corresponding to each $A$-function we may define $\rm{Rat}(A)$ to be the set
of all $A$-rafts. We shall show in Section 4 that under certain conditions,
$\rm{Rat}(A)=c(B)$ for an appropriately chosen $B$.

For further properties and identities involving rafts, see [20].

###  2.6 The infinitary $A$-function $\Theta$

The infinitary $A$-function of Cohen, which we shall denote by $\Theta$
(keeping with the notation of [5, 6]), is of central importance to this work
and is more difficult to define than $D$ and $D^{1}$, the other two
$A$-functions that we will commonly reference.

There are many equivalent ways of defining $\Theta$ – see [5, 6, 9, 10, 21]
for a variety of ways $\Theta$ can be defined. We will discuss here a new
equivalent way to define $\Theta$.

######  Proposition 1.

Suppose $S\subseteq\mathbb{N}$ is a subset of $\mathbb{N}$ such that every
integer $n\in\mathbb{N}$ may be represented uniquely as a product of all the
elements of some finite subset $S_{n}\subseteq S$. Then if
$S^{\prime}:=\\{p^{2^{x}}\operatorname{|}p\in\mathbb{P},x\in\mathbb{N}_{0}\\}$,
then $S=S^{\prime}$.

###### Proof.

Clearly $1\notin S$ and $\mathbb{P}\subseteq S$, as every prime number must be
represented as itself. Furthermore, $p^{2}\in S$ for all primes $p$, since
there is no non-singleton subset of $\mathbb{N}$, the product of whose
elements is the representation of $p^{2}$ as $p\cdot p$. $p^{3}$ is
represented uniquely as $p\cdot p^{2}$, and $p^{4}$ must represent itself by
the same reasoning as for $p^{2}$. Continuing in this manner, we can see that
$S^{\prime}\subseteq S$.

To see that $S\subseteq S^{\prime}$, note that every integer $n\in\mathbb{N}$
may be represented uniquely as a product of elements of $S^{\prime}$ in the
following manner:

  1. 1.

First, decompose $n$ according to its prime decomposition:

$n=p_{1}^{a_{1}}\cdots p_{l}^{a_{l}}$

  2. 2.

Then, write each $a_{i}$ in terms of its binary representation as powers of 2:

$a_{i}=\sum_{j=0}^{\infty}b_{i,j}2^{j},$

where $b_{i,j}\in\\{0,1\\}$.

  3. 3.

Then

$n=p_{1}^{b_{1,0}2^{0}+b_{1,1}2^{1}+\dots}\cdots
p_{k}^{b_{l,0}2^{0}+b_{l,1}2^{1}+\dots}=q_{1}\cdots q_{r},$ (7)

where each $q_{k}$ is one of the $p_{i}^{b_{i,j}2^{j}}$ for which $b_{i,j}=1$.

Hence, $S=S^{\prime}$ and we are done. ∎

The set $S$ may then be identified with the $I$-components of Cohen [9, 10]
and hence with the $\Theta$-primitive elements $\mathbb{P}_{\Theta}$ as
mentioned in [5]. We may use $S$ to define $\Theta$ as follows:

######  Definition 3.

For all $n\in\mathbb{N}$, we say that $d\in\Theta(n)$ if $d$ may be
represented as a product of some subset of the elements in $S_{n}$.

Observe that if $d\in\Theta(n)$ then $\Theta(d)\subseteq\Theta(n)$ as
$S_{d}\subseteq S_{n}$. Since $S_{\frac{n}{d}}=S\setminus S_{d}$, it follows
that
$d\in\Theta(n)\Leftrightarrow\Theta(d)\cap\Theta\left(\frac{n}{d}\right)=\\{1\\}$.
This tells us that the iterate of $\Theta$ is $\Theta$, a fact observed in
[5].

###  2.7 The regular $A$-functions

Narkiewicz [23] formally introduced the concept of $A$-functions for the
purpose of analyzing a specific class of $A$-functions called regular
$A$-functions.

Narkiewicz defined an $A$-function to be regular if the following conditions
hold:

  1. 1.

The ring $(\mathcal{A},+,*_{A})$ is associative, commutative, and possesses a
unit element.

  2. 2.

The conovlution $*_{A}$ preserves multiplicativity of arithmetical functions.

  3. 3.

For all prime powers $p^{a}$, $\mu_{A}(p^{a})\in\\{0,-1\\}$.

Regular $A$-functions defined in this manner are difficult to work with.
Fortunately, Narkiewicz and others [4, 23, 30] have developed more workable
definitions, which we state as follows:

######  Proposition 2.

The $A$-function $A$ is regular if and only if

  1. 1.

$A$ is multiplicative.

  2. 2.

For every $p^{a}$ there is a unique integer $r_{A}(p^{a})$ such that
$A(p^{a})=\\{1,p^{r_{A}(p^{a})},p^{2\cdot r_{A}(p^{a})},\dots,p^{a}\\}$, and
if $p^{b}\in A(p^{a})$ then $r_{A}(p^{b})=r_{A}(p^{a})$.

For any reflexive $A$-function $A$ and any prime power $p^{a}$, we may define
$r_{A}(p^{a})$ to be the smallest element of $A(p^{a})$ larger than 1. This
extends the original definition of $r_{A}$ from [23] to all $A$-functions
relevant to this paper.

Examples of regular $A$-functions include $D$ and $D^{1}$, whose properties
are well-known, the cross-convolutions of Tóth [30], and the so-called ternary
$A$-function (not to be confused with the tri-unitary $A$-function, $D^{3}$)
of [13], defined to be the regular $A$-function such that

$r_{A}(p^{a})=\begin{cases}1&\text{ if }a\text{ is odd or a multiple of }4\\\
2&\text{ if }a\text{ is twice an odd number}.\end{cases}$ (8)

Notably, $\Theta$ is not regular. This fact can be seen any number of ways,
the simplest of which is by observing that $\mu_{\Theta}(p^{3})=1$, which
disagrees with the condition that $\mu_{A}(p^{a})\in\\{0,-1\\}$.

The regular $A$-functions possess special combinatorial properties with
respect to the corresponding $A$-convolution. Their main property is best
exhibited in the following result, which can be seen, for example, as a
consequence of Theorem 3 of [4]:

######  Theorem 1.

Let $\varphi_{A}:=\mu_{A}*_{A}I$. Then $\varphi_{A}(n)$ counts the number of
positive integers $k\leqslant n$ such that $D(k)\cap A(n)=\\{1\\}$.

The regular $A$-functions are not the only $A$-functions with this property.
In [4], the authors introduced a more general class of $A$-function which they
called semi-regular. A semi-regular $A$-function satisfies the following
conditions:

  1. 1.

$A$ is simple, reflexive, and multiplicative.

  2. 2.

$A(p^{a})$ contains exactly one $A$-primitive element for all prime powers
$p^{a}$.

  3. 3.

$p^{r_{A}(p^{a})}$ is $A$-primitive for all prime powers $p^{a}$.

Theorem 5 of [4] guarantees that the semi-regular $A$-functions are the only
multiplicative $A$-functions for which Theorem 1 holds. Although our present
work does not deal with semi-regular $A$-functions in particular, we will
point to one example in particular of a semi-regular $A$-function with
interesting properties in Section 3.

We introduce the following new characterization of regular $A$-functions:

######  Theorem 2.

For $A\in\mathbb{A}$, $A$ is regular if and only if $A$ is semi-regular and
$*_{A}$ is associative and commutative.

To show this, we borrow a very useful lemma from Narkiewicz:

######  Lemma 1 (Associativity condition).

Suppose $A\in\mathbb{A}$. Then $*_{A}$ is associative if and only if the
following two conditions are equivalent:

  1. 1.

$d\in A(m)$ and $m\in A(n)$

  2. 2.

$d\in A(n)$ and $\frac{m}{d}\in A\left(\frac{n}{d}\right)$.

Note that $*_{A}$ is commutative if and only if $A$ is symmetric (cf. [23],
2.ii). We now prove the theorem.

###### Proof.

The forward implication is immediate and follows from the definition of
regularity in Narkiewicz [23] and Proposition 1 of [4]. To see the reverse
implication, observe that if we can show condition 2 of Proposition 2 then our
result follows.

To this end, let $p$ be an arbitrary prime. For $a=1$ and $a=2$, there is
nothing to show. Assume that we have demonstrated condition 2 of Proposition 2
for all $a<a_{0}$. Then for $a_{0}$, let $r_{0}:=r_{A}(p^{a_{0}})$. If
$r_{0}=a_{0}$ we are done. Otherwise, $r_{0}<a_{0}$ and so $p^{a_{0}-r_{0}}\in
A(p^{a_{0}})$ is a non-trivial statement. Observe that $A$ is transitive
(since associativity of $*_{A}$ implies transitivity), so
$A(p^{a_{0}-r_{0}})\subseteq A(p^{a_{0}})$. Hence,
$r_{A}(p^{a_{0}-r_{0}})=r_{0}$, and so
$\\{1,p^{r_{0}},p^{2r_{0}},\cdots,p^{a_{0}}\\}\subseteq A(p^{a_{0}})$.

To see that no other elements are present in $A(p^{a_{0}})$, note that any
$p^{b}\in A(p^{a_{0}})$ such that $b$ is not a multiple of $r_{0}$ will carry
with it (by transitivity) a different $A$-primitive element $p^{r_{1}}\in
A(p^{a_{0}})$, contradicting semi-regularity. Thus, by induction, we are done.
∎

Thus, we may think of regular $A$-functions as those multiplicative
$A$-functions for which Theorem 1 holds true, whose $A$-convolutions are
associative and commutative.

###  2.8 The structured $A$-functions

The structured $A$-functions were first introduced by Burnett and Osterman [5]
as a generalization of the infinitary $A$-function. Originally, two distinct
but equivalent constructions were provided.

First, we must recall several definitions from [5], foremost of which is the
concept of the Cohen Triangle of $A$ at $p$.

For a multiplicative $A$-function $A$ and a prime number $p$, the Cohen
Triangle of $A$ at $p$ is an infinite lower triangular binary array
$C_{A,p}=\\{c_{ab}\\}$, indexed from zero, such that

$c_{ab}=\begin{cases}1&p^{b}\in A(p^{a})\\\ 0&\text{ otherwise.}\end{cases}$

In general, if $A$ is homogeneous then $C_{A,p_{1}}=C_{A,p_{2}}$ for all
primes $p_{1},p_{2}\in\mathbb{P}$ (providing an alternative definition for
homogeneity), so we refer to the Cohen Triangle of $A$ in this case. Just as a
cross convolution may be viewed as being formed out of homogeneous
$A$-functions, we may view the Cohen Triangle of $A$ for various homogeneous
$A$ to be the ``building blocks'' of the Cohen Triangles of cross-
convolutions.

The concept of the Cohen Triangle of $A$ was first introduced by Cohen [9],
who displayed progressively larger upper left submatrices that provided a
visualization of the manner in which the $k$-ary divisibility relations
``tended towards'' the infinitary divisibility relation. Any finite lower
triangular matrix $\gamma$ consisting entirely of 1's will be called a
structure.

######  Definition 4.

Let $A\in\mathbb{A}$ be homogeneous. We say that $A$ is (homogeneously)
$N$-finitely structured for $N\in\mathbb{N}$ if there exist structures
$\gamma_{1},\gamma_{2},\dots,\gamma_{N}$ such that

$C_{A}=C_{D}\otimes\gamma_{N}\otimes\cdots\otimes\gamma_{2}\otimes\gamma_{1},$

where $\otimes$ denotes the Kronecker product of matrices. We allow $N$ to be
0, making $D$ a structured $A$-function.

If, on the other hand, there exists an infinite sequence of structures
$\gamma_{1},\gamma_{2},\cdots$ and an infinite sequence of $A$-functions
$A_{1},A_{2},\dots$ such that

$\displaystyle C_{A}$ $\displaystyle=C_{A_{1}}\otimes\gamma_{1}$
$\displaystyle C_{A_{1}}$ $\displaystyle=C_{A_{2}}\otimes\gamma_{2}$
$\displaystyle\vdots$

then we call $A$ (homogeneously) infinitely structured. If $A$ is any cross-
convolution of a set of homogeneously finite or infinite structures, we simply
call $A$ structured.

Examples of structured $A$-functions include $D$, $\Theta$, and the
$k$-factorization $A$-functions of Litsyn and Shevelev $\Theta_{k}$. In
general, every structured $A$-function may be viewed as a cross-convolution of
homogeneous structured $A$-functions, each of which is uniquely determined by
a so-called structure sequence $\\{s_{1},s_{2},\cdots\\}$, which may be finite
or infinite, where $s_{k}$ is the size of the structure $\gamma_{k}$.

In the remarks following Definition 2.3 of [5], the concept of the character
of an $B$-primitive element $q$ was introduced for structured $A$-functions
$B$. We extend this to general $A$-functions:

######  Definition 5.

If $A\in\mathbb{A}$ is reflexive with $\mathbb{P}_{A}$ the set of
$A$-primitive elements, then for each $q\in\mathbb{P}_{A}$ let $\chi_{A}(q)$
denote the least integer $a$ larger than 1 such that $q^{a}\in\mathbb{P}_{A}$,
or $\infty$ if no such power exists.

In [5], Burnett and Osterman demonstrated

######  Theorem 3 (Proposition 2.5 of [5]).

If $B$ is a structured $A$-function and $n\in\mathbb{N}$, then there exists a
unique decomposition of $n$ into the $B$-primitive elements,

$n=q_{1}^{c_{1}}\cdots q_{l}^{c_{l}},$ (9)

such that $c_{i}<\chi_{A}(q_{i})$ for all $i$.

The following algebraic properties of structured $A$-functions were also
demonstrated (cf., [5]):

######  Theorem 4.

Suppose $B$ is a structured $A$-function. Then

  1. 1.

(Theorem 4.3, [5]) both $(c(B^{1}),*_{B})$ and $(c(B^{1}),*_{B^{1}})$ form
Abelian groups.

  2. 2.

(Corollary 4.6, [5]) $*_{B}$ preserves $c(B)$ if and only if $B=\Theta$

In Section 4 we will prove an analogue of Theorem 4.3 of [5] for a much more
general class of $A$-functions.

###  2.9 The perfect $A$-functions

In Theorem 10 of [10], Cohen observed that the $A$-function $\Theta$ has the
special property of the infintary convolution $*_{\Theta}$ preserving
$c(\Theta)$ – that is, it is an instance of an $A$-function $A$ preserving its
own arithmetical class. Corollary 4.6 of [5] makes it clear that this property
is not shared among any other structured $A$-function. Yet, it appears to hold
for $D^{1}$ as well, since the unitary convolution preserves multiplicativity
of arithmetical functions (i.e., $*_{D^{1}}$ preserves $c(D^{1})$).
Furthermore, it can be shown that $(c(\Theta),*_{\Theta},\cdot)$ and
$(c(D^{1}),*_{D^{1}},\cdot)$ are both rings (here $\cdot$ is usual
multiplication).

In Theorem 3 of [6], the authors completely characterized all $A$-functions
with this property: $A$ is such that $(c(A),*_{A},\cdot)$ forms a ring if and
only if $A$ is perfect, which is to say that $A\in C(A)$ and $A\subseteq
A^{1}$.

It is furthermore demonstrated that all the rings created in this way are
isomorphic to $(\mathcal{A},+,\cdot)$.

Examples of perfect $A$-functions include $D^{1}$ and $\Theta$. Corollary 6 of
[6] demonstrated that if $B$ is structured, then $B^{1}$ is perfect.
Interestingly, $D^{3}$ is perfect, and it is currently unknown if $D^{2k-1}$
is perfect for all $k\in\mathbb{N}$. For an example of a perfect $A$-function
which is not multiplicative, Example 7 of [6] gives us the $A$-function $Z$,
where $Z(n)=\\{1,n\\}$ for all $n$. Note that $c(Z)$ consists of all
arithmetical functions $f$ for which $f(1)=1$.

In [6], the authors deduced further results for general $A$-functions. Call a
symmetric $A\in\mathbb{A}$ split if $m\in A(mn)$, $a\in A(m)$, and $b\in A(n)$
means $a\in A(ab)$. Then

######  Theorem 5 (Lemma 5 of [6]).

If $A\in C(A)$ and $A$ is split, then $*_{A}$ is associative.

###  2.10 Connections between regular, structured, and perfect $A$-functions

First, we shall demonstrate a result that reveals to us a paradigm by which we
may connect regular, structured, and perfect $A$-functions.

######  Theorem 6.

The only regular and structured $A$-function is $D$. The only structured and
perfect $A$-function is $\Theta$. The only perfect and regular $A$-function is
$D^{1}$.

###### Proof.

By the comments following Remark 4.7 of [5], no regular $A$-function except
$D$ can be structured.

By Corollary 4.6 in [5], $\Theta$ is the only structured $A$-function $A$
whose $A$-convolution preserves its arithmetical class.

Finally, to see that the only perfect and regular $A$-function is $D^{1}$,
observe that if $A$ is regular and not $D^{1}$, then there is an
$n\in\mathbb{N}$ such that $\mu_{A}(n)=0$, but by Corollary 5 of [6] if $A$ is
perfect then $\mu_{A}(n)\in\\{-1,1\\}$, so since $D^{1}$ is the only regular
$A$-function with this property, ∎

We present the following paradigm for understanding the three families of
$A$-functions appearing in the above theorem:

  1. 1.

The regular $A$-functions are those which generalize the properties of $D$ and
$D^{1}$.

  2. 2.

The structured $A$-functions are those which generalize the properties of
$\Theta$ and $D$.

  3. 3.

The perfect $A$-functions are those which generalize the properties of $D^{1}$
and $\Theta$.

In this paper, we will identify a class of $A$-functions to which the regular
$A$-functions, the structured $A$-functions, and the perfect $A$-functions
belong, but which keeps many relevant properties common to all three of these
families. We state these properties below:

######  Proposition 3.

If $A$ is regular, structured, or perfect, then $A$ has the following
properties:

  1. (P1)

$A\in C(A)$

  2. (P2)

When constructing an arbitrary arithmetical function $f\in c(A)$, one may
freely choose the value of $f$ at $A$-primitive elements.

  3. (P3)

Each $n\in\mathbb{N}$ has a unique representation as a product of
$A$-primitive elements $n=q_{1}^{a_{1}}\cdots q_{r}^{a_{r}}$ such that for
every $f\in c(A)$, $f(n)=f(q_{1})^{a_{1}}\cdots f(q_{r})^{a_{r}}$.

In the next section, we will introduce a very general class of $A$-functions
satisfying (P2) and (P3), and then refine that class to exclude $A$-functions
not satisfying (P1) as well.

##  3 Complete $A$-functions

###  3.1 Complete $A$-functions and the $A$-decomposition

For the remainder of this paper, we will be considering $A$-functions that are
reflexive. Observe that in this case $\mathbb{P}_{A}$ is a countably-infinite
set since $\mathbb{P}\subseteq\mathbb{P}_{A}$. We introduce the following
concept to aid us in understanding class-$A$ arithmetical functions:

######  Definition 6.

Consider the enumeration of $A$-primitive elements in increasing order (i.e.
$q_{1}=2$, $q_{2}=3$, etc.). Let $V(A)$ be the set of all real-valued
arithmetical functions $f$ such that there exists $g\in c(A)$ with the
property that for all $k\in\mathbb{N}$, $log(g(q_{k}))=f(k)$. We call $V(A)$
the arithmetical span of $A$.

In general we will make use of the above enumeration of a set of $A$-primitive
elements unless otherwise mentioned.

Since $\mathcal{A}_{\mathbb{R}}$ may be viewed as $\mathbb{R}^{\infty}$, we
may discuss $V(A)$ in terms of its properties relating to this vector space:

######  Proposition 4.

$V(A)$ is a vector subspace of $\mathbb{C}^{\infty}$.

###### Proof.

Let $A$ be an $A$-function. Since $f(n)\equiv 0$ is in $V(A)$, $V(A)$ is non-
empty. Now suppose $f_{1}$ and $f_{2}$ are in $V(A)$. Then there is $g_{1}$
and $g_{2}$ in $c(A)$ such that $\log(g_{i}(q_{k}))=f_{i}(k)$. But then
$g_{1}g_{2}\in c(A)$ and so
$\log(g_{1}(q_{k})g_{2}(q_{k}))=\log(g_{1}(q_{k}))+\log(g_{2}(q_{k}))=f_{1}(k)+f_{2}(k)$.
Hence, $f_{1}+f_{2}\in V(A)$.

Now suppose $f\in V(A)$ and $c\in\mathbb{R}$. Then if $g$ is the associated
arithmetical function in $c(A)$, then $g^{c}\in c(A)$ as well, so that
$\log(g^{c}(q_{k}))=c\log(g(q_{k}))=cf(k)$. Hence, $S_{A}$ is a vector
subspace of $\mathcal{A}_{\mathbb{R}}$. ∎

The following defines what will become our most general class of
$A$-functions, from which we will refine the aforementioned paradigmatic
$A$-functions.

######  Definition 7.

We say that $A$ is complete if $V(A)=\mathcal{A}_{\mathbb{R}}$.

The following weak version of (P2), can be immediately realized:

######  Proposition 5.

If $A$ is complete, then when constructing a generic $g\in c(A)$ which is
positive and real-valued, we may freely choose the values of $g(q_{k})$
independent of all other choices of $g(q_{k})$.

######  Theorem 7.

The following are equivalent:

  1. 1.

$A$ is complete.

  2. 2.

For every $n\in\mathbb{N}$, $n$ may be uniquely represented as a product of
$A$-primitive elements $n=q_{1}^{a_{1}}\cdots q_{l}^{a_{l}}$ in such a way
that for every $f\in c(A)$, $f(n)=f(q_{1})^{a_{1}}\cdots f(q_{l})^{a_{l}}$.

  3. 3.

For every $f\in\mathcal{A}$ (complex-valued arithmetical functions) there is a
$g\in c(A)$ such that for all $k\in\mathbb{N}$, $g(q_{k})=f(k)$.

Observe that the second property in Theorem 7 is (P3). In order to prove this,
we need the following Lemma, which essentially guarantees that a weak version
of the second property of Theorem 7 holds for every reflexive $A$-function.

######  Lemma 2.

Suppose $A\in\mathbb{A}$ is simple and reflexive. Then for every
$n\in\mathbb{N}$ there exists a representation of $n$ as a (not necessarily
unique) product of $A$-primitive elements $n=q_{1}^{a_{1}}\cdots
q_{l}^{a_{l}}$ such that for every $f\in c(A)$, $f(n)=f(q_{1})^{a_{1}}\cdots
f(q_{l})^{a_{l}}$.

###### Proof.

We may ``represent'' 1 as an empty product by convention. For
$q\in\mathbb{P}_{A}$ we have $f(q)=f(q)$, establishing our base case. Now
assuming we have shown this to be true for all $k<n$, we examine $A(n)$. If
$n\in\mathbb{P}_{A}$ then we are done. Otherwise, if $d\in A(n)$ with $1<d<n$
then for every $f\in c(A)$, $f(d)f(\frac{n}{d})=f(n)$. But then if
$f(d)=f(q_{1})^{b_{1}}\cdots f(q_{l})^{b_{l}}$ and
$f(\frac{n}{d})=f(q_{1})^{c_{1}}\cdots f(q_{l})^{c_{l}}$ then
$f(n)=f(q_{1})^{a_{1}}\cdots f(q_{l})^{a_{l}}$ for $a_{i}=b_{i}+c_{i}$, so by
induction we are done. ∎

Now we may prove our first main result.

###### Proof.

Suppose $A$ is complete. We will show that the representation of each
$n\in\mathbb{N}$ guaranteed by Lemma 2 is unique. Let $n\in\mathbb{N}$ be
arbitrary and suppose $n=q_{1}^{a_{1}}\cdots q_{l}^{a_{l}}=q_{1}^{b_{1}}\cdots
q_{l}^{b_{l}}$ are two representations of $n$, where we allow some $a_{i}$ and
$b_{j}$ to be zero if needed in order to maintain identical enumerations on
the left and right hand side. Then by Lemma 2, for every $g\in c(A)$,
$g(q_{1})^{a_{1}}\cdots g(q_{l})^{a_{l}}=g(q_{1})^{b_{1}}\cdots
g(q_{l})^{b_{l}}$. Consider the $g\in c(A)$ which are positive real-valued.
Then taking logarithms of the above equation gives us

$a_{1}\log(g(q_{1}))+\cdots+a_{l}\log(g(q_{l}))=b_{1}\log(g(q_{1}))+\cdots+b_{l}\log(g(q_{l})),$
(10)

which holds for all positive real-valued $g\in c(A)$.

We may write

$(a_{1}-b_{1})log(g(q_{1}))+\dots+(a_{l}-b_{l})\log(g(q_{l}))=0.$ (11)

As we range over all positive real-valued $g$, the vectors whose $i^{th}$
components are the values $(a_{i}-b_{i})log(g(q_{i}))$ span a subspace
$\mathbb{R}^{l}$. If this subspace is non-trivial then there is a $k$ such
that $(a_{k}-b_{k})\neq 0$, allowing us to express $log(g(q_{k}))$ in terms of
the other $log(g(q_{i}))$, $i\neq k$. However, this contradicts Proposition 5,
so $a_{i}=b_{i}$ for all $i$ and the representation is unique.

Now assume $A$ is such that we have a unique representation of every
$n\in\mathbb{N}$ in the manner described above. Let
$f\in\mathcal{A}=\mathbb{C}^{\infty}$ be given. Define $g\in\mathcal{A}$ on
$A$-primitive elements $q_{k}$ to be $g(q_{k})=f(k)$ for all $k\in\mathbb{N}$.
Then for $n=q_{1}^{a_{1}}\cdots q_{l}^{a_{l}}$ the unique representation of
$n$, let $g(n)=\prod_{k=1}^{l}g(q_{k})^{a_{k}}$. By definition, this function
is in $c(A)$, implying that we may find such a $g$ for every
$f\in\mathcal{A}$.

Finally, suppose that $A$ is such that for every $f\in\mathcal{A}$ there is
$g\in c(A)$ such that $g(q_{k})=f(k)$ for all $k\in\mathbb{N}$. Then in
particular, this holds for all positive real-valued $g$, and hence
$V(A)=\mathcal{A}_{\mathbb{R}}$ so $A$ is complete. ∎

The unique representation of $n$ demonstrated to exist for complete $A$ will
be called the $A$-decomposition of $n$. The following are immediate
consequences of Theorem 7:

######  Corollary 1.

If $A$ is complete, then $A$ satisfies (P2) and (P3).

######  Corollary 2.

If $A$ is regular, structured, or perfect, then $A$ is complete.

Let us now give some examples of complete $A$-functions, as well as examples
of $A$-functions which are not complete.

######  Example 1.

The $A$-functions $D$, $D^{1}$, and $\Theta$ are all complete. Every integer
$n$ is naturally decomposed as

$\begin{aligned}
n&=(p_{1})^{a_{1}}\cdots(p_{l})^{a_{l}}&\quad(p_{i}\in\mathbb{P}_{D}=\mathbb{P})\\\
&=(p_{1}^{a_{1}})\cdots(p_{l}^{a_{l}})&\quad(p_{i}^{a_{i}}\in\mathbb{P}_{D^{1}})\\\
&=(p_{1}^{2^{0}})^{b_{1,0}}(p_{1}^{2^{1}})^{b_{1,1}}\cdots(p_{l}^{2^{0}})^{b_{l,0}}(p_{1}^{2^{1}})^{b_{l,1}})\cdots&\quad(p_{i}^{2^{j}}\in\mathbb{P}_{\Theta})\end{aligned},$

and it is well-known that

$f\in c(D)\Rightarrow f(n)=f(p_{1})^{a_{1}}\cdots f(p_{l})^{a_{l}}$

and

$f\in c(D^{1})\Rightarrow f(n)=f(p_{1}^{a_{1}})\cdots f(p_{l})^{a_{l}}.$

By Corollary 4 of [6],

$f\in c(\Theta)\Rightarrow
f(n)=f(p_{1}^{2^{0}})^{b_{1,0}}f(p_{1}^{2^{1}})^{b_{1,1}}\cdots
f(p_{l}^{2^{0}})^{b_{l,0}}f(p_{1}^{2^{1}})^{b_{l,1}}),$

where we are able to find $f\in c(A)$ in each case to show that this is unique
(for example, $f=I$ for $D$ and $f=\varphi$ for $D^{1}$). In Section 3.3 we
will deduce for any complete $A$-function $A$ that a single function can be
found whose ``largest'' arithmetical class is $c(A)$.

######  Example 2.

Any cross-convolution of complete $A$-functions is also complete.

######  Example 3.

For an $A$-function which is not complete, consider the homogeneous
$A$-function $A$ defined on prime powers as

$A(p^{a})=\begin{cases}\\{1,p,p^{2},p^{3},p^{4}\\}&a=4\\\
D^{1}(p^{a})&\text{\rm otherwise}\end{cases}$

Notice that every $f\in c(A)$ must satisfy
$f(p^{4})=f(p^{2})^{2}=f(p)f(p^{3})$ (since $p^{2}\in A(p^{4})$ and $p\in
A(p^{4})$) for every prime $p$, so in particular, even though
$p,p^{2},p^{3}\in\mathbb{P}_{A}$, choosing $f(p)$ and $f(p^{3})$ restricts
choice of $f(p^{2})$. As we will see in Section 3.3, this excludes $A$ from
being regular, structured, or perfect.

However, note that $A$ is associative, indicating that completeness is not a
sufficiently general property to capture all associative $A$-functions.

######  Example 4.

To see a non-symmetric complete $A$-function, consider the homoegeneous
$A$-function $G$ introduced in [4], which is the $A$-function satisfying

$d\in G(n)\Leftrightarrow D(d)\cap\Theta\left(\frac{n}{d}\right)=\\{1\\}.$

By examining the Cohen Triangle of $G$, one can deduce that

  1. 1.

The $G$-convolution $*_{G}$ is associative.

  2. 2.

The $G$-decomposition of any integer $n$ is equal the $\Theta$-decomposition.

Neither of these assertions are immediate. Nevertheless, we leave it to the
reader to check these facts.

###  3.2 Paradigmatic $A$-functions

Note that any complete $A$-function $A$ induces an $A$-decomposition of every
$n\in\mathbb{N}$, but the $A$-decomposition does not necessarily uniquely
determine $A$ (cf. Examaple 4).

######  Definition 8.

Suppose $A$ is complete. Then for all $n\in\mathbb{N}$, expressed via the
$A$-decomposition $n=q_{1}^{a_{1}}\cdots q_{l}^{a_{l}}$,
$q_{k}\in\mathbb{P}_{A}$, suppose $A(n)=A(q_{1})^{a_{1}}\cdots
A(q_{l})^{a_{l}}$. In such a case we call $A$ paradigmatic .

It follows via an inductive argument that paradigmatic $A$-functions are
complete, which relies on the use of a complete $A$-function in their
construction.

We immediately obtain the following:

######  Proposition 6.

If $A$ is regular, structured, or perfect, then $A$ is paradigmatic .

###### Proof.

In each case, since $A$ is complete we may decompose each $n\in\mathbb{N}$ via
the $A$-decomposition as $n=q_{1}^{a_{1}}\cdots q_{l}^{a_{l}}$. Note that
$A(q_{i})=\\{1,q_{i}\\}$ since $q_{i}$ is $A$-primitive. Then regardless of
whether $A$ is regular, stuctured, or perfect, $q_{1}^{b_{1}}\cdots
q_{l}^{b_{l}}\in A(n)$ for all $0\leqslant b_{i}\leqslant a_{i}$ for all $i$,
and $d\in A(n)$ if and only if it is of this form. But the set of all such $d$
may also be written as $A(q_{1})^{a_{1}}\cdots A(q_{l})^{a_{l}}$, so $A$ is
paradigmatic . ∎

We then have the following main result:

######  Theorem 8.

$A$ is paradigmatic if and only if $A\in C(A)$.

###### Proof.

First, notice that by definition, if $A$ is paradigmatic and $d\in A(n)$, then
we may write $n=q_{1}^{a_{1}}\cdots q_{l}^{a_{l}}$, $d=q_{1}^{d_{1}}\cdots
q_{l}^{d_{l}}$, and $\frac{n}{d}=q_{1}^{c_{1}}\cdots q_{l}^{c_{l}}$, where
$a_{k}=d_{k}+c_{k}$. But then, $A(d)=A(q_{1})^{d_{1}}\cdots A(q_{l})^{d_{l}}$
and $A(\frac{n}{d})=A(q_{1})^{c_{1}}\cdots A(q_{l})^{c_{l}}$, so we indeed
have $A(d)A(\frac{n}{d})=A(n)$, so $A\in C(A)$.

On the other hand, if $A\in C(A)$ then we proceed by induction on $n$. For any
$A$-primitive element $q$, the unique $A$-decomposition is trivially $q=q$,
giving us $A(q)=A(q)$. If for all $k<n$, $k=q_{1}^{k_{1}}\cdots
q_{l}^{k_{l}}$, we have deduced a unique $A$-decomposition giving us
$A(k)=A(q_{1})^{k_{1}}\cdots A(q_{l})^{k_{l}}$ for all $k<n$, then examine
$A(n)$. If $n\in\mathbb{P}_{A}$ then we are done, so assume there is $d\in
A(n)$ such that $1<d<n$. Then $A(d)A(\frac{n}{d})=A(n)$. If we let
$d=q_{1}^{d_{1}}\cdots q_{l}^{d_{l}}$, $\frac{n}{d}=q_{1}^{c_{1}}\cdots
q_{l}^{c_{l}}$, and define $a_{i}=d_{i}+c_{i}$, then
$A(n)=A(d)A(\frac{n}{d})=A(q_{1})^{a_{1}}\cdots A(q_{l})^{a_{l}}$.

To complete the induction step we claim that $n=q_{1}^{a_{1}}\cdots
q_{l}^{a_{l}}$ is the $A$-decomposition of $n$. Suppose $n=q_{1}^{a_{1}}\cdots
q_{l}^{a_{l}}=q_{1}^{b_{1}}\cdots q_{l}^{b_{l}}$. Let $d$ be the least element
of $A(n)$ larger than one, so that $d\in\mathbb{P}_{A}$. Hence, the first
$a_{i}$ and the first $b_{i}$ that are not zero must occur at the same $i$,
say $i_{0}$. Then observe that $q_{i_{0}}\cdot q_{i_{0}}^{a_{i_{0}}-1}\cdot
q_{i_{0}+1}^{a_{i_{0}+1}}\cdots q_{l}^{a_{l}}=q_{i_{0}}\cdot
q_{i_{0}}^{b_{i_{0}}-1}\cdot q_{i_{0}+1}^{b_{i_{0}+1}}\cdots q_{l}^{b_{l}}$.
But then since the $A$-decomposition is unique for all $k<n$, $a_{i}=b_{i}$
for all $i$. Hence, we are finished by induction. ∎

By Proposition 6, the following is immediate.

######  Corollary 3.

If $A$ is regular, structured, or perfect, then $A\in C(A)$.

######  Corollary 4.

(P1) implies (P2) and (P3).

###### Proof.

$A\in C(A)$ implies $A$ is paradigmatic , so $A$ is complete, which implies
(P2) and (P3). ∎

######  Corollary 5.

If $A$ is paradigmatic then $A$ is split.

###### Proof.

Let $A$ be paradigmatic and let $m\in A(mn)$, $a\in A(m)$, and $b\in A(n)$.
Then $ab\in A(mn)$, and since $A\in C(A)$ we have that
$A(m)A(n)=A(a)A(b)A(\frac{m}{a})A(\frac{n}{b})=A(mn)$. Hence, $ab$ has as its
$A$-decomposition the product of the $A$-decomposition of $a$ and $b$, and so
since $A$ is fractured, $a\in A(ab)$. ∎

Of course, this means that we may refine Lemma 5 of [6]:

######  Theorem 9.

Suppose $A\in C(A)$. Then $*_{A}$ is associative.

The only example appearing in the literature of a non-paradigmatic
$A$-function for which $*_{A}$ is associative is the $A$-function $G$ from [4]
(cf. Example 4 above), which demonstrates also that $A$ need not be symmetric
in order for $*_{A}$ to be associative, a fact previously unstated.

We summarize the preliminary results on paradigmatic $A$-functions in the
following:

######  Theorem 10.

If $A$ is paradigmatic , then:

  1. 1.

$A\in C(A)$

  2. 2.

$A$ is complete.

  3. 3.

$*_{A}$ is associative

  4. 4.

$A$ is symmetric

Since all regular, structured, or perfect $A$-functions are paradigmatic , we
present examples of $A$-functions that belong to none of these classes, yet
are paradigmatic .

######  Example 5.

Any cross-convolution of paradigmatic $A$-functions is paradigmatic . In
particular, suppose $A$ is the cross-convolution of $D$, $D^{1}$, and
$\Theta$, where, say, $A(2^{a})=D(2^{a})$, $A(p^{a})=D^{1}(p^{a})$ if $p\equiv
1$ (mod 4), and $A(p^{a})=\Theta(p^{a})$ if $p\equiv 3$ (mod 4). Then $A$
belongs to none of the three families.

######  Example 6.

Let $A$ be the $A$-function which for all $n\in\mathbb{N}$ not a power of 2 is
$A(n)=Z(n)=\\{1,n\\}$, but if $n=2^{a}$ is $A(2^{a})=D(2^{a})$ for all
$a\in\mathbb{N}$. Note that $A$ is obviously not multiplicative in this case.

######  Example 7.

To see an example of a homogeneous $A$-function that is not regular,
structured, or perfect and yet is paradigmatic , consider the $A$-function $A$
defined on prime powers via

$A(p^{a})=\begin{cases}\\{1,p^{2},p^{4}\\}&a=4\\\
\\{1,p^{2},p^{3},p^{5}\\}&a=5\\\ \\{1,p^{2},p^{3},p^{4},p^{5},p^{7}\\}&a=7\\\
D^{1}(p^{a})&\text{\rm otherwise}\end{cases}$

Then this $A$-function is paradigmatic .

###  3.3 The maximal class of an arithmetical function

One can notice that, when speaking of arithmetical classes for a given
arithmetical function $f$, a natural ordering emerges in which those functions
$f\in c(Z)$ (i.e., $f(1)=1$) constitute a very general class of arithmetical
function, whereas the functions $f\in c(D)$ are in a sense the least general
class. Indeed, $f\in c(D)$ implies $f\in c(A)$ for all $A$, and in general if
$A\subseteq B$ then $c(B)\subseteq c(A)$. We wish to switch the focus from
finding an $A$ for which $f\in c(A)$ to finding the maximal $A$ for which this
is true.

######  Definition 9.

Let $f\in c(Z)$. Denote by $m(f):=A$ the $A$-function such that $m\in A(mn)$
if and only if $f(mn)=f(m)f(n)$. We call this the maximal class of $f$.

It is clear from the definition that for any $f\in\mathcal{A}$ with $f(1)=1$,
$m(f)$ is well-defined and if $A$ is any $A$-function with $f\in c(A)$, then
$A\subseteq m(f)$.

######  Example 8.

Consider the Euler $\varphi$ function. A well-known identity (cf. [2])
involving $\varphi$ is

$\varphi(mn)=\varphi(m)\varphi(n)\frac{(m,n)}{\varphi(m,n)}.$

We can immediately see that unless $(m,n)=1$, then
$\varphi(mn)\neq\varphi(m)\varphi(n)$, but of course $(m,n)=1\Leftrightarrow
m\in D^{1}(mn)$, so $m(\varphi)=D^{1}$. Functions of this type were examined
in a very general setting by [15]

######  Example 9.

Consider the divisor function $\tau$. Then for $m=p_{1}^{a_{1}}\cdots
p_{l}^{a_{l}}$ and $n=p_{1}^{b_{1}}\cdots p_{l}^{b_{l}}$,
$\tau(m)\tau(n)=(a_{1}+1)\cdots(a_{l}+1)(b_{1}+1)\cdots(b_{l}+1)$, which must
be strictly larger than $(a_{1}+b_{1}+1)\cdots(a_{l}+b_{l}+1)=\tau(mn)$ unless
$(m,n)=1$ (i.e., $a_{i}>0\Rightarrow b_{i}=0$), which means $m(\tau)=D^{1}$ as
well.

######  Example 10.

Consider the Möbius function $\mu$. Then the equation $\mu(mn)=\mu(m)\mu(n)$
can be split into two cases, according to whether or not the left and right
hand sides are equal to zero. If they are not equal to zero, then we need
$(m,n)=1$ in addition to $m$ and $n$ being individually square-free.

If $\mu(mn)=0$, then since $\mu(m)\mu(n)=0$, one of $m$ or $n$ must not be
square-free, and for all such $m$ and $n$, $\mu(mn)=\mu(m)\mu(n)$. So $m(\mu)$
is the $A$-function $A$, where $m\in A(mn)$ if either $m\in D^{1}(mn)$ for $m$
and $n$ square-free or $m$ or $n$ is not square-free.

Note that $m(\mu)$ is not multiplicative even though $\mu$ is, and in general
it can be true that $m(f)$ appears to ``lose'' properties analogous to those
that $f$ possesses.

######  Example 11.

Initially, one might think that $m(f)$ ought to be complete for any $f$.
Unfortunately, this is not the case. Define

$f(n):=\begin{cases}\exp(\pi^{n})&n\neq 2,4,8,16\\\ 2i&n=2\\\ 4&n=4\\\
-8i&n=8\\\ 16&n=16.\end{cases}$

It is then clear that

$m(f)(n)=\begin{cases}Z(n)&n\neq 16\\\ \\{1,2,4,8,16\\}&n=16,\end{cases}$

and yet $m(f)$ is not complete.

######  Example 12.

On the other hand, even if $A$ is complete, it does not mean that $f\in c(A)$
can be found such that $m(f)=A$. Indeed, any complete $A$-function which is
not symmetric will lack this property, as $m(f)$ is symmetric by definition.

We do, however, have the following:

######  Theorem 11.

If $A$ is paradigmatic , then there exists a function $f\in c(A)$ with
$m(f)=A$.

###### Proof.

Let $A$ be paradigmatic and define for any real number $x$ the function
$f_{x}\in c(A)$ on $A$-primitive elements to be $f_{x}(q_{k})=\exp(x^{k})$.
For a given expressed via its $A$-decomposition as $n=q_{1}^{c_{1}}\cdots
q_{l}^{c_{l}}$, it is clear that $f_{x}(n)$ is the exponential of a polynomial
in $x$ for all $x$, dependent only on $n$. Call this polynomial $P_{n}$. Then
$P_{n}(x)=c_{1}x^{1}+c_{2}x^{2}+\cdots c_{l}x^{l}$.

Let $m=q_{1}^{b_{1}}\cdots q_{l}^{b_{l}}$ and let $mn=q_{1}^{a_{1}}\cdots
q_{l}^{a_{l}}$ be the $A$-decomposition of $m$ and $mn$, respectively. Suppose
that $f_{x}(mn)=f_{x}(m)f_{x}(n)$ for all $x\in\mathbb{R}$. Then
$log(f_{x}(mn))=log(f_{x}(m))+log(f_{x}(n))$, and hence $P_{mn}=P_{m}+P_{n}$
(as non-zero polynomials may only possess a finite number of solutions).
However, this implies that $a_{i}=b_{i}+c_{i}$ for all $i$, which is to say
that the $A$-decomposition of $mn$ is the product of the $A$-decompositions of
$m$ and $n$.

Finally, since $A$ is paradigmatic it follows that $m\in A(mn)$, as each
$b_{i}$ satisfies $0\leqslant b_{i}\leqslant a_{i}$ for all $i$. Choose $x$ to
be a non-algebraic real number. Then for all $m,n\in\mathbb{N}$,
$P_{mn}=P_{m}+P_{n}$ if and only if $P_{mn}(x)=P_{m}(x)+P_{n}(x)$, as all such
polynomials have coefficients in $\mathbb{N}_{0}$. Thus, $m(f_{x})=A$. ∎

An analogous version of our $m(f)$ has appeared in the literature before, in
the work of Ryden [25]. In that work, for $f\in\mathcal{A}$ the set $T(f)$ is
defined to be the set of all ordered pairs $(a,b)$ such that $f(ab)=f(a)f(b)$.
Clearly, $(a,b)\in T(f)$ is equivalent to $a\in m(f)(ab)$ for $f(1)=1$.

The scope of Ryden's work was confined to investigating groups of the form
$(c(A),*)$, that is, groups formed from arithmetical classes under Dirichlet
convolution. We may reformulate and prove his main result using results from
[6], but first need the following definition from that work.

######  Definition 10 (Definition 10 from [6]).

Let $f\in\mathcal{A}$ be an arithmetical function with $f(1)=1$. Let $A$ and
$B$ be symmetric $A$-functions such that $m\in B(mn)$, $a\in A(m)$, and $b\in
A(n)$ together imply that $f(ab)=f(a)f(b)$. We then call $f$ $(A,B)$-split. If
under the same premises, $a\in H(ab)$ for an $A$-function $H$, then we say
that $H$ is $(A,B)$-split.

######  Theorem 12 (Restatement of Theorem 3.8 of Ryden).

Suppose $A\subseteq D^{1}$ is a symmetric, reflexive $A$-function. Then
$(c(A),*)$ is a group if and only if for all $f\in c(A)$, $f$ is
$(D,A)$-split.

###### Proof.

By Theorem 1 of [6], $*_{D}=*$ preserves $c(A)$ if and only if

  1. 1.

$D\in C(A)$

  2. 2.

$\tau_{D}=\tau\in c(A)$

  3. 3.

$\forall f\in c(A)$, $f$ is $(D,A)$-split.

The third condition is explicitly satisfied by assumption. Since $D\in C(D)$,
$D\in C(A)$ for all $A\in\mathbb{A}$. Finally, $\tau\in c(D^{1})$ and since
$A\subseteq D^{1}$, $\tau\in c(A)$. Then by Corollary 3 of [6], since $*$ is
associative, $(c(A),*)$ forms a group.

Going the other way, we see that if $(c(A),*)$ forms a group then $*$ must
preserve $c(A)$, so in particular since $m(\tau)=D^{1}$, we need $A\subseteq
D^{1}$ and we need all $f\in c(A)$ to be $(D,A)$-split, so we are done. ∎

###  3.4 Arithmetical functions associated with complete $A$-functions

######  Definition 11.

Let $A$ be complete and let $n=q_{1}^{a_{1}}\cdots q_{l}^{a_{l}}$ be the
$A$-decomposition of $n$. Define $\omega_{A}(n):=l$ and define, for
$q\in\mathbb{P}_{A}$, $\nu_{A,q}(n)$ to be $a_{i}$ if $q=q_{i}$ for some $i$,
and 0 otherwise.

######  Definition 12.

Let $A$ be complete. Let $P_{A}(n)$ be the set of $A$-primitive elements in
the $A$-decomposition of $n$.

It is easy to see $\omega_{A}(n)=|P_{A}(n)|$. We also have the following:

######  Proposition 7.

Suppose $A$ is complete and let $B:=P(A)$ be the $A$-function defined at each
$n\in\mathbb{N}$ to be, for $n=q_{1}^{a_{1}}\cdots q_{l}^{a_{l}}$ the
$A$-decomposition of $n$,

$B(n)=\begin{cases}\\{1,n\\}&\text{ if }\omega_{A}(n)=1\\\
B(q_{1}^{a_{1}})\cdots B(q_{l}^{a_{l}})&\text{otherwise.}\end{cases}$

Then $B$ is perfect.

###### Proof.

The fact that $B=P(A)$ is such that $B\in C(B)$ follows from the definition.
We show that $B\subseteq B^{1}$. If $n\in\mathbb{P}_{B}$ then $B(n)=B^{1}(n)$,
so assume $n\notin\mathbb{P}_{B}$. We need to show that $d\in B(n)\Rightarrow
B(d)\cap B(\frac{n}{d})=\\{1\\}$. If $d=q_{1}^{b_{1}}\cdots q_{l}^{b_{l}}$ is
the $A$-decomposition of $d$ and $\frac{n}{d}=q_{1}^{c_{1}}\cdots
q_{l}^{c_{l}}$ is the $A$-decomposition of $\frac{n}{d}$, then for each $i$,
$b_{i}$ or $c_{i}$ is 0, for the elements of $B(n)$ are of the form
$q_{i_{1}}^{a_{i_{1}}}\cdots q_{i_{k}}^{a_{i_{k}}}$. Hence, $B(d)\cap
B\left(\frac{n}{d}\right)=\\{1\\}$ and we are done. ∎

Hence, for example, $P(D)=D^{1}$ and $P(G)=\Theta$. For any structured
$A$-function $B$, $P(B)=B^{1}$. We also have the following:

######  Proposition 8.

Suppose $A$ is complete and let $F:=F(A)$ be the $A$-function defined at each
$n\in\mathbb{N}$ to be, for $n=q_{1}^{a_{1}}\cdots q_{l}^{a_{l}}$,

$F(n)=A(q_{1})^{a_{1}}\cdots A(q_{l})^{a_{l}}$

Then $F$ is paradigmatic .

###### Proof.

This $A$-function is well-defined due to the unique $A$-decomposition of each
$n\in\mathbb{N}$, and so by construction is paradigmatic . ∎

Propositions 7 and 8 tell us that each complete $A$-function may be associated
with a perfect $A$-function and a paradigmatic $A$-function.

######  Lemma 3.

Suppose $A$ is complete and suppose $f\in\mathcal{A}$ is such that $f(1)\neq
0$ and $f(mn)=f(m)f(n)$ whenever $P_{A}(m)\cap P_{A}(n)=\emptyset$. Then $f\in
c(P(A))$.

###### Proof.

Let $B=P(A)$ and suppose $m\in B(mn)$. Then since $B$ is perfect, $m\in
B(mn)\Rightarrow m\in B^{1}(mn)\Rightarrow B(m)\cap B(n)=\\{1\\}\Rightarrow
P_{B}(m)\cap P_{B}(n)=\emptyset$ since $P_{B}(m)$ and $P_{B}(n)$ are subsets
of $B(m)$ and $B(n)$, respectively, which do not contain 1. But then our
result follows. ∎

For the purposes of the next result, let $A$ be complete. We say that
$n\in\mathbb{N}$ is $A$-square-free if in the $A$-decomposition of
$n=q_{1}^{a_{1}}\cdots q_{l}^{a_{l}}$, each $a_{i}$ is either 0 or 1.

######  Theorem 13.

Let $A$ be complete and let $B:=P(A)$ and $F:=F(A)$. Then $\mu_{F},\mu_{B}\in
c(B)$, $\omega_{A}=\omega_{F}=\omega_{B}$, $\mu_{B}(n)=(-1)^{\omega_{A}(n)}$,
and furthermore

$\mu_{F}(n)=\begin{cases}\mu_{B}(n)&n\text{is A-square-free}\\\
0&\text{otherwise.}\end{cases}$ (12)

###### Proof.

The fact that $\omega_{A}=\omega_{F}=\omega_{B}$ is immediate from the
definition of $\omega_{A}$ and $P(A)$.

Suppose $m,n\in\mathbb{N}$ are such that $P_{A}(m)\cap P_{A}(n)=\emptyset$. If
$m\in B(mn)$, then $P_{A}(mn)=P_{A}(m)\cup P_{A}(n)$ and so
$\omega_{A}(mn)=\omega_{A}(m)+\omega_{A}(n)$. Then
$(-1)^{\omega_{A}(mn)}=(-1)^{\omega_{A}(m)}(-1)^{\omega_{A}(n)}$, so
$g(n):=(-1)^{\omega_{A}(n)}$ is class-$B$.

To see that $g=\mu_{B}$, we may check that for $q\in\mathbb{P}_{B}$,
$1+g(q)=1-1=0$, since $g\in c(B)$, so $g=\mu_{B}$ and we are done. To see that
(12) gives the correct expression for $\mu_{F}$, first apply the same
reasoning as for $\mu_{B}$ to see that $\mu_{F}\in c(B)$. Then we may check,
for $q\in\mathbb{P}_{A}=\mathbb{P}_{F(A)}$,
$(u*_{F}\mu_{F})(q^{a})=1+\mu_{F}(q)=1-1=0$ for all $a$ such that $q^{a}$ is
the $A$-decompositon of $n=q^{a}$. ∎

###  3.5 Classification of complete $A$-functions

The next few results enable us to understand the relationship between complete
$A$-functions, the subclass of paradigmatic $A$-functions and the further
subclass of perfect $A$-functions.

######  Lemma 4.

Suppose $A$ is complete and $B$ is such that

  1. 1.

$\mathbb{P}_{B}=\mathbb{P}_{A}$

  2. 2.

For each $n\in\mathbb{N}$, $n\notin\mathbb{P}_{B}$, $n>1$, there is some $d\in
A(n)$, $1<d<n$, such that $B(n)=\\{1,d,n\\}$

Then $B$ is complete and $c(B)=c(A)$.

In general, if $B$ is a complete $A$-function such that $\tau_{B}(n)\leqslant
3$ for all $n$, we call $B$ minimally complete.

###### Proof.

Observe that by the proof of Lemma 2, we may decompose any $n\in\mathbb{N}$
using $B$ in the same manner as $A$, since $B\subseteq A$. However, $B$ is
such that each $n\in\mathbb{N}$ has only a single ``path'' along which it can
be decomposed. Namely, if $n>1$ is not $B$-primitive, then there is only one
$d\in B(n)$ such that $1<d<n$, enabling one to inductively show, starting from
the $B$-primitive elements (which are equal to the $A$-primitive elements by
assumption), that $B$-decomposition of $n$ is unique and equal to the
$A$-decomposition of $n$ for all $n$. ∎

######  Theorem 14.

If $A$ is complete, then $c(A)=c(F(A))$.

###### Proof.

It is obvious that $\mathbb{P}_{A}=\mathbb{P}_{F(A)}$ by the definition of
$F(A)$. For each $n\notin\mathbb{P}_{A}$, choose $d\in A(n)$ with $1<d<n$ and
construct a minimally complete $A$-function $B$ via $B(n)=\\{1,d,n\\}$ for
these $n$, letting the $B$-primitive elements be equal to the $A$-primitive
elements. Then the conditions of Lemma 4 hold for $A$ and $B$, as well as for
$F(A)$ and $B$, giving us $c(A)=c(B)=c(F(A))$. ∎

######  Theorem 15.

Suppose $A$ is complete and $B$ is such that $c(A)=c(B)$. Then $B$ is
complete.

###### Proof.

Let $A$ be complete and let $B$ be such that $c(A)=c(B)$. We argue that
$B\subseteq F(A)$. Indeed, were this not the case there would be some minimal
$n\in\mathbb{N}$ such that $d\in B(n)$ but $d\notin F(A)(n)$. Hence $f\in
c(B)$ would mean $f(d)f(\frac{n}{d})=f(n)$, a relation not satisfied by every
$f\in c(F(A))$ since there is such an $f$ with $m(f)=F(A)$ by Theorem 11. Thus
there can be no such value and $B\subseteq F(A)$.

We can easily show that $\mathbb{P}_{B}=\mathbb{P}_{F(A)}=\mathbb{P}_{A}$;
else, there would be relations satisfied by all $f\in c(B)$ that are not
satisfied by all $f\in c(A)$, contradicting the assumption. This being said,
if $n\notin\mathbb{P}_{B}$ then there is some proper non-trivial $m\in
F(A)(n)$ such that $m\in B(n)$, and this would be true for each such $n$.
Hence there is some minimally complete $\tilde{B}$ contained in $B$ such that
$\mathbb{P}_{\tilde{B}}=\mathbb{P}_{B}$. Indeed, $F(B)=F(A)$. ∎

######  Corollary 6.

Suppose $A$ is complete, $B$ is minimally complete with $c(A)=c(B)$, and
$B^{\prime}$ is such that $B\subseteq B^{\prime}\subseteq F(A)$. Then
$B^{\prime}$ is complete and $c(B^{\prime})=c(A)$.

###### Proof.

Since $B\subseteq B^{\prime}\subseteq F(A)$, $c(F(A))\subseteq
c(B^{\prime})\subseteq c(B)$. However, $c(B)=c(F(A))$ by Theorem 14, so
$c(B^{\prime})=c(F(A))=c(A)$, and so by Theorem 15 $B^{\prime}$ is complete. ∎

We may now easily see that $F(G)=P(G)=\Theta$ since $c(G)=c(\Theta)$.

######  Example 13.

Let $A$ be complete and let $\Gamma(A)$ be the reflexive $A$-function such
that $\Gamma(A)(n)=\\{1,n\\}\cup P_{A}(n)$. Then $\Gamma$ is complete by
Corollary 6 and $c(\Gamma(A))=c(A)$.

The following will be needed in the next section:

######  Lemma 5.

If $A$ is complete, $F:=F(A)$, and $B:=P(A)$, then $F$ is $(F,P)$-split.

###### Proof.

Suppose $m\in B(mn)$, $a\in F(m)$, and $b\in F(n)$. Then since $B\subseteq F$,
$m\in F(mn)$, and since $F\in C(F)$, $ab\in F(mn)$, so $F(ab)\subseteq F(mn)$.
An argument involving the $F$-decomposition of $ab$ reveals that we must have
$a\in F(ab)$, so we are done. ∎

##  4 Results for $A$-rafts and examples

###  4.1 Main result for $A$-rafts

We have the following:

######  Theorem 16.

Suppose $A$ is complete. Then the closure of $\rm{Rat}(F(A))$ under the above
metric is $c(P(A))$.

###### Proof.

Assume $A$ is complete and let $F=F(A)$ and let $B=P(A)$. First we show that
$*_{F}$ preserves $c(B)$ since this will show that $\rm{Rat}(F)\subseteq
c(B)$. By Theorem 1 of [6], $*_{F}$ preserves $c(B)$ if and only if

  1. 1.

$F\in C(B)$

  2. 2.

$\tau_{F}\in c(B)$

  3. 3.

For every $f\in c(B)$, $f$ is $(F,B)$-split.

By Theorem 8, $F\in C(F)$ and hence $F\in C(B)$ since $B\subseteq F$.
Furthermore, Lemma 3 combined with the definition of $F$ implies that
$\tau_{F}\in c(B)$. Finally, let $f\in c(B)$. If $m\in B(mn)$ $a\in F(m)$ and
$b\in F(n)$ then the following is true: Since $m\in B(mn)$, we may relabel the
$A$-primitive elements $q_{i}$ and write this generally as
$q_{1}^{a_{1}}\cdots q_{l}^{a_{l}}\in B(q_{1}^{a_{1}}\cdots
q_{l}^{a_{l}}q_{l+1}^{a_{l+1}}\cdots q_{k}^{a_{k}})$, and we may write $a$ as
$q_{1}^{c_{1}}\cdots q_{l}^{c_{l}}$ and $b$ as $q_{l+1}^{c_{l+1}}\cdots
q_{k}^{c_{k}}$.

But since $F$ is $(F,B)$-split by Lemma 5, $a\in F(ab)$ and we have
$q_{1}^{c_{1}}\cdots q_{l}^{c_{l}}\in B(q_{1}^{c_{1}}\cdots
q_{l}^{c_{l}}\cdots q_{l+1}^{a_{l+1}}\cdots q_{k}^{c_{k}})$ which itself
implies $B$ is $(F,B)$-split but in particular implies $f(ab)=f(a)f(b)$. So
indeed, $*_{A}$ preserves $c(B)$. This implies that the closure of
$\rm{Rat}(A)$ is a subset of $c(B)$, as $c(B)$ is closed under the
$A$-convolution and hence all $A$-convolutions of arithmetical functions are
in $c(B)$.

We demonstrate that the closure of $\rm{Rat}(A)$ is $c(B)$. First, note that
$c(B)$ is closed, since every sequence $\\{f_{n}\\}$ of functions in $c(B)$
will consist entirely of functions which are class-$B$, and hence cannot
converge to a function lacking this property.

Now take $f\in c(B)$. We demonstrate that this $f$ may be approximated by a
sequence of rational functions from $\rm{Rat}(A)$. Since we may freely choose
$A$-primitive elements for some class-$A$ arithmetical function $g_{1}$, we
may choose these in such a way that $g_{1}(n)=f(n)$ if $n\in\mathbb{P}_{A}$.
Since $f\in c(B)$, it is sufficient to construct a sequence $g_{k}$ in
$\rm{Rat}(F)$ such that in the limit $g_{k}(n)\rightarrow f(n)$ for all
$n\in\mathbb{P}_{B}$.

Let $h_{1}=g_{1}$. Then suppose $n$ has $A$-decomposition $q^{2}$. We will
show that we can find class-$A$ functions $g_{2,1}$ and $g_{2,2}$ such that
$(g_{2,1}*_{A}g_{2,2})(q)=f(q)$ and $(g_{2,1}*_{A}g_{2,2})(q^{2})=f(q^{2})$.
Notice that this turns into the following system of equations:

$\displaystyle f(q)$ $\displaystyle=g_{2,1}(q)+g_{2,2}(q)$ $\displaystyle
f(q^{2})$ $\displaystyle=g_{2,1}(q^{2})+g_{2,1}(q)g_{2,2}(q)+g_{2,2}(q^{2}).$

We may express $g_{2,1}(q)$ in terms of $g_{2,2}(q)$ as
$g_{2,1}(q)=-g_{2,2}(q)+f(q)$, which turns our second equation into

$\displaystyle f(q^{2})$
$\displaystyle=-g_{2,2}(q)+f(q))^{2}+(-g_{2,2}(q)+f(q))g_{2,2}(q)+g_{2,2}(q^{2})$
$\displaystyle=f(q)^{2}-2f(q)g_{2,2}(q)+g_{2,2}(q)^{2}+-g_{2,2}(q)^{2}+f(q)g_{2,2}(q)+g_{2,2}(q)^{2}.$

Since we may choose $g_{2,2}(q)$ freely in $\mathbb{C}$ and since the above
can be viewed as a quadratic equation in $g_{2,2}(q)$, we may choose such a
value that satisfies it. Letting $g_{2}=g_{2,1}*_{A}g_{2,2}$ gives us
$g_{2}(q)=f(q)$ and $g_{2}(q^{2})=f(q^{2})$.

Proceeding in a similar fashion, we can construct $g_{k}$ using $k$ class-$A$
functions $g_{k,1},\cdots,g_{k,k}$ such that $g_{k}(q^{a})=f(q^{a})$ for all
$a\leqslant k$ and for all $q\in\mathbb{P}_{A}$. In the limit $g_{k}$
approaches $f$ and hence the closure of $\rm{Rat}(A)$ is $c(B)$. ∎

######  Corollary 7.

For all paradigmatic $A$-functions $A$ and for all $f\in c(P(A))$, $f$ may be
approximated by a sequence $\\{f_{k}\\}\subseteq\rm{Rat}(A)$ such that for
each $k$, $f_{k}$ is $A$-$k$-ic.

###### Proof.

The proof of Theorem 16 was undertaken by constructing such a sequence, so we
are done. ∎

######  Corollary 8.

Every multiplicative function $f$ may be approximated by a sequence of
functions $f_{n}$, where each $f_{n}$ is $n$-ic. In particular, even though
$\varphi=\mu*I$ is rational, $\varphi$ may be approximated by such a sequence
of $n$-ic functions.

######  Corollary 9.

For all complete $A$-functions $A$, $(c(P(A)),*_{F(A)})$ and
$(\rm{Rat}(F(A)),*_{F(A)})$ form groups.

The latter result is in fact true for any reflexive $A$-function which is
associative – hence, for example, $(\rm{Rat}(G),*_{G})$ forms a (non-Abelian)
group.

######  Remark 2.

It is unknown to the authors if results like this are known in the literature.
Technically, one need not invoke the analytical language used here in order to
realize Theorem 16, but the authors found no references to any such results in
the literature.

Indeed, naming functions which satisfy (5) ``rational'' seems to be even more
fitting if one realizes that, under a natural discrete metric, these functions
``approximate'' the class of multiplicative arithmetical functions in a manner
directly analogous to the rational approximation of a real number.

###  4.2 Examples

We first extend the concept of the characteristic function $\chi_{A}$ to
complete $A$-functions:

######  Definition 13.

Suppose $A$ is complete. Then for each $q\in\mathbb{P}_{A}$, let $\chi_{A}(q)$
denote the least $a$ such that the $A$-decomposition of $q^{a}$ is not
$(q)^{a}$, or $\infty$ if no such $a$ exists.

Note that this definition agrees with the original definition of $\chi_{A}$
from [5].

######  Definition 14.

Let $A$ be complete. We say $A$ is $N$-finitely primitive if
$\chi_{A}(q)\leqslant N$ for all $q\in\mathbb{P}_{A}$. Otherwise, if
$\chi_{A}(q)<\infty$ for all $q\in\mathbb{P}_{A}$ but $A$ is not $N$-finitely
primitive for any $N$, we call $A$ quasi-infinitely primitive. If
$\chi_{A}(q)=\infty$ for some $q\in\mathbb{P}$, we call $A$ infinitely
primitive.

If $\chi_{A}(q)=N$ for all $q\in\mathbb{P}_{A}$, we call $A$ strictly
$N$-finitely primitive, and if $\chi_{A}(q)=\infty$ for all
$q\in\mathbb{P}_{A}$, we call $A$ strictly infinitely primitive.

######  Proposition 9.

If $A$ is $N$-finitely primitive, then $\rm{Rat}(F(A))=c(P(A))$, and
furthermore for all $f\in c(P(A))$, $f$ may be represented as the
$F(A)$-convolution of $N$ or fewer functons $f_{k}\in c(F(A))$. If $A$ is
quasi-infinitely primitive or infinitely primitive, then $\rm{Rat}(F(A))$ is
not equal to $c(P(A))$.

###### Proof.

In the proof for Theorem 16, if $A$ is $N$-finitely primitive then our
approximations $f_{k}$ may stop at $k=N$ due to the fact that at each
$q\in\mathbb{P}_{A}$, a system of equations with at most $N$ unknowns is to be
solved. Hence, $Rat(F(A))=c(P(A))$.

By the same token if $A$ is quasi-infinitely primitive or infinitely
primitive, then we may apply diagonal-type arguments to choose $f\in c(P(A))$
so as to avoid any finite sequence of $f_{k}$ that end in $f_{N}=f$, giving us
our result. ∎

The remainder of this work will be dedicated to a few examples of applying
Theorem 16 and Proposition 9 to various $A$-functions.

######  Example 14.

In the most basic application of Theorem 16, we have that the closure of
$\rm{Rat}(D)$ is $c(D^{1})$, but since $D$ is strictly infinitely primitive,
$\rm{Rat}(D)\neq c(D^{1})$, which is to say that not every multiplicative
arithmetical function is rational.

######  Example 15.

Consider $\Theta_{k}$, the $k$-factorization $A$-functions of Litsyn and
Shevelev. For $k=2$, $\Theta_{k}=\Theta$ is perfect, but this is not the case
for $k>2$. Nevertheless, for each $k\in\mathbb{N}$, $\Theta_{k}$ is
$k$-finitely primitive, so
$\rm{Rat}(\Theta_{k})=P(\Theta_{k})=(\Theta_{k})^{1}$.

######  Example 16.

In general, if $A$ is finitely structured, $A$ is infinitely primitive,
whereas if $A$ is infinitely structured, then $A$ may be quasi-infinitely
primitive or $N$-finitely primitive. According to the class into which the
structured $A$-function $A$ falls, we may have $\rm{Rat}(A)=c(A^{1})$ or
$\rm{Rat}(A)\neq c(A^{1})$.

######  Example 17.

We shall construct a paradigmatic $A$-function which is strictly infinitely
primitive but not multiplicative. We observe that, prior to this introduction,
no such $A$-function with these properties exists – $D$ is strictly infinitely
primitive, but is multiplicative.

Let $\Sigma$ be the paradigmatic $A$-function such that $\mathbb{P}_{\Sigma}$
is the set of all square-free integers, with the $\Sigma$-decomposition of a
non-square-free integer $n$ proceeding according to the following algorithm:

  1. 1.

Let $a_{1}$ denote the largest power of $q_{1}:=\gamma(n)$ which divides $n$.

  2. 2.

Let $n_{1}=\frac{n}{(\gamma(n))^{a_{1}}}$.

  3. 3.

Repeat steps 1 and 2 for $n:=n_{1}$ until $n=1$, incrementing the index each
iteration.

  4. 4.

Write $n=q_{1}^{a_{1}}\cdots q_{l}^{a_{l}}$.

Then $\Sigma(n):=\Sigma(q_{1})^{a_{1}}\cdots\Sigma(q_{l})^{a_{l}}$. That this
$A$-function is well-defined is clear, and by construction it is paradigmatic
.

Furthermore, for all $q\in\mathbb{P}_{\Sigma}$ and any $a\in\mathbb{N}$,
$(q)^{a}$ is the $A$-decomposition of $q^{a}$, which implies that $\Sigma$ is
strictly infinitely primitive, as desired. Since
$\Sigma(12)=\\{1,2,6,12\\}\neq\\{1,2,4\\}\cdot\\{1,3\\}$, it is clear that
$\Sigma$ is not multiplicative.

##  5 Acknowledgements

We would like to thank the Directed Research Program at The University of
Texas at Dallas for facilitating the collaboration that led to this
manuscript.

## References

  * [1] Alladi, Krishnaswami. ``On arithmetic functions and divisors of higher order." Journal of the Australian Mathematical Society 23.1 (1977): 9-27.
  * [2] Apostol, Tom M. Introduction to analytic number theory. Springer Science & Business Media, 2013.
  * [3] Burnett, Joseph Vade, Sam Grayson, Zachary Sullivan, Richard Van Natta, and Luke Bang. ``Arithmetical Functions Associated with the $k$-ary Divisors of an Integer.'' International Journal of Mathematics and Mathematical Sciences (2018), doi:10.1155/2018/9349245.
  * [4] Burnett, Joseph Vade, and Otto Vaughn Osterman. ``A generalization of regular convolutions and Ramanujan sums." The Ramanujan Journal (2020): 1-9.
  * [5] Burnett, Joseph Vade, and Otto Vaughn Osterman. ``A generalization of the infinitary divisibility relation: Algebraic and analytic properties.'' International Journal of Number Theory 15.9 (2019): 1771-1792.
  * [6] Burnett, Joseph Vade, Otto Vaughn Osterman, and Reece Lewandowski. ``On a special class of A-functions." New York J. Math 25 (2019): 1405-1420.
  * [7] Carroll, T. B., and A. A. Gioia. ``On a subgroup of the group of multiplicative arithmetic functions." Journal of the Australian Mathematical Society 20.3 (1975): 348-358.
  * [8] Cohen, Eckford. ``Arithmetical functions associated with the unitary divisors of an integer." Mathematische Zeitschrift 74.1 (1960): 66-80.
  * [9] Cohen, Graeme L. ``On an integer’s infinitary divisors.'' mathematics of computation 54.189 (1990): 395-411.
  * [10] Cohen, Graeme L., and Peter Hagis, Jr. ``Arithmetic Functions Associated with the Infinitary Divisors of an Integer.'' International Journal of Mathematics and Mathematical Sciences 16.2 (1993): 373-374, doi:10.1155/S0161171293000456.
  * [11] Davison, Thomas MK. ``On arithmetic convolutions." Canadian Mathematical Bulletin 9.3 (1966): 287-296.
  * [12] Fotino, Ingrid Popa. ``Generalized convolution ring of arithmetic functions." Pacific Journal of Mathematics 61.1 (1975): 103-116.
  * [13] Gawell, Elin. Rings of Arithmetic Functions with Regular Convolutions. Matematiska institutionen, Stockholms universitet, 2005.
  * [14] Hanumanthachari, J. ``On an arithmetic convolution." Canadian Mathematical Bulletin 20.3 (1977): 301-305.
  * [15] Haukkanen, Pentti. ``An arithmetical equation with respect to regular convolutions." Aequationes mathematicae 91.3 (2017): 527-536.
  * [16] Haukkanen, Pentti. ``Basic properties of the bi-unitary convolution and the semi-unitary convolution." Indian J. Math 40 (1998): 305-315.
  * [17] Haukkanen, Pentti. ``On the k-ary convolution of arithmetical functions.'' Fibonacci Quarterly 38.5 (2000): 440-445.
  * [18] Haukkanen, Pentti. ``Rational arithmetical functions of order (2, 1) with respect to regular convolutions." Portugaliae Mathematica 56.3 (1999): 329-344.
  * [19] Johnson, Kenneth. ``Unitary analogs of generalized Ramanujan sums." Pacific Journal of Mathematics 103.2 (1982): 429-432.
  * [20] Laohakosol, Vichian, and Nittiya Pabhapote. ``Properties of rational arithmetic functions.'' International Journal of Mathematics and Mathematical Sciences 2005 (2005).
  * [21] Litsyn, Simon, and Vladimir Shevalev. ``On Factorization of Integers with Restrictions on the Exponents.'' Integers: Electronic Journal of Combinatorial Number Theory 7 (2007).
  * [22] McCarthy, Paul J. ``Regular arithmetical convolutions." Portugaliae mathematica 27.1 (1968): 1-13.
  * [23] Narkiewicz, Wroclaw. ``On a Class of Arithmetical Convolutions.'' Colloquium Mathematicum X (1963), doi:10.4064/cm-10-1-81-94.
  * [24] Ramaiah, V. Sita. ``Arithmetical sums in regular convolutions." Journal für die reine und angewandte Mathematik 1978.303-304 (1978): 265-283.
  * [25] Ryden, Roy. ``Groups of arithmetic functions under Dirichlet convolution." Pacific Journal of Mathematics 44.1 (1973): 355-360.
  * [26] Schwab, Emil D., and Gheorghe Silberberg. ``The valuated ring of the arithmetical functions as a power series ring." Archivum Mathematicum 37.1 (2001): 77-80.
  * [27] Subbarao, M. V. ``On some arithmetic convolutions." The theory of arithmetic functions. Springer, Berlin, Heidelberg, 1972. 247-271.
  * [28] Suryanarayana, D. ``The number of k-ary divisors of an integer." Monatshefte für Mathematik 72.5 (1968): 445-450.
  * [29] Tóth, László. ``A generalization of Pillai's arithmetical function involving regular convolutions." Acta Mathematica et Informatica Universitatis Ostraviensis 6.1 (1998): 203-217.
  * [30] Tóth, László. ``Asymptotic formulae concerning arithmetical functions defined by crossconvolutions, I. Divisor-sum functions and Euler-type functions." Publ. Math. Debrecen 50 (1997): 159-176.
  * [31] Tóth, László, and Pentti Haukkanen. ``On the binomial convolution of arithmetical functions." arXiv preprint arXiv:0806.0508 (2008).
  * [32] Tóth, L. ``Sum functions of certain generalized divisors." Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math 41 (1998): 165-180.
  * [33] Vaidyanathaswamy, R. ``The theory of multiplicative arithmetic functions." Transactions of the American Mathematical Society 33.2 (1931): 579-662.
  * [34] Yocom, K. L. ``Totally multiplicative functions in regular convolution rings." Canadian Mathematical Bulletin 16.1 (1973): 119-128.