<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[12pt,reqno]{article}

\usepackage[usenames]{color}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage{amscd}

\usepackage[colorlinks=true,
linkcolor=webgreen, filecolor=webbrown,
citecolor=webgreen]{hyperref}

\definecolor{webgreen}{rgb}{0,.5,0}
\definecolor{webbrown}{rgb}{.6,0,0}

\usepackage{color}
\usepackage{fullpage}
\usepackage{float}

\usepackage{psfig}
\usepackage{graphics,amsmath,amssymb}
\usepackage{amsthm}
\usepackage{amsfonts}
\usepackage{latexsym}
\usepackage{epsf}

\setlength{\textwidth}{6.5in} \setlength{\oddsidemargin}{.1in}
\setlength{\evensidemargin}{.1in} \setlength{\topmargin}{-.5in}
\setlength{\textheight}{8.9in}

\newcommand{\seqnum}[1]{\href{http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=#1}{\underline{#1}}}

\begin{document}

\begin{center}
\epsfxsize=4in \leavevmode\epsffile{logo129.eps}
\end{center}

\begin{center}
\vskip 1cm{\LARGE\bf On the Bi-Unitary Analogues 
of Euler's \\
Arithmetical Function 
and the Gcd-Sum \\
\vskip .17in
Function}
\vskip 1cm

\large L\'aszl\'o T\'oth \\
Institute of Mathematics and Informatics \\
University of P\'ecs \\
Ifj\'us\'ag u. 6 \\
7624 P\'ecs \\
Hungary \\
\href{mailto:ltoth@ttk.pte.hu}{\tt ltoth@gamma.ttk.pte.hu}\\
\end{center}

\vskip .2 in

\begin{abstract} We give combinatorial-type formulae for the bi-unitary
analogues of Euler's arithmetical function and the gcd-sum function
and prove asymptotic formulae for the latter one and for another
related function.
\end{abstract}

\newtheorem{theorem}{Theorem}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{defin}[theorem]{Definition}
\newenvironment{definition}{\begin{defin}\normalfont\quad}{\end{defin}}
\newtheorem{examp}[theorem]{Example}
\newenvironment{example}{\begin{examp}\normalfont\quad}{\end{examp}}
\newtheorem{rema}[theorem]{Remark}
\newenvironment{remark}{\begin{rema}\normalfont\quad}{\end{rema}}

\def\Z{{\Bbb Z}}
%\def\Reg{\operatorname{Reg}}

\def\RE{\operatorname{Re}}
\def\IM{\operatorname{Im}}

\newtheorem{thm}{Theorem}
\newtheorem{cor}{Corollary}
\newtheorem{lem}{Lemma}
\newtheorem{rem}{Remark}
\newtheorem{prop}{Proposition}
\newcommand{\ds}{\displaystyle}
\def\lcm{\operatorname{lcm}}
\def\N{{\Bbb N}}

%************************* section 1 *******************************************

\section{Introduction}

Euler's arithmetical function and the gcd-sum function (or Pillai's
function) are defined by
\begin{equation} \label{phi}
\phi(n):= \# \{k\in \N: 1\le k\le n, \gcd (k,n)=1\}
\end{equation}
and
\begin{equation} \label{P}
P(n):=\sum_{k=1}^n \gcd (k,n),
\end{equation}
respectively. The function $P$ was investigated in recent papers of
this journal; it is sequence \seqnum{A018804} in Sloane's On-Line
Encyclopedia of Integer Sequences. A direct connection between these two
functions is given by
\begin{equation}
P(n)=\sum_{d\mid n} d\phi(n/d).
\end{equation}

Let $n$ be a positive integer. Recall that a positive integer $d$ is
said to be a {\it unitary divisor} of $n$ if $d\mid n$ and $\gcd
(d,n/d)=1$, and we write $d \mid \mid n$. Let $(k,n)_*:=\max \{d\in \N:
d\mid k, d\mid \mid n\}$, and consider the function $\phi^*$
(\seqnum{A047994}) defined by
\begin{equation} \label{phi*}
\phi^*(n):= \# \{k\in \N: 1\le k\le n, (k,n)_*=1\},
\end{equation}
which is multiplicative and $\phi^*(p^\nu)=p^\nu-1$ for every prime
power $p^\nu$ ($\nu\ge 1$). Also,
\begin{equation} \label{phi*formula}
\phi^*(n)= \sum_{d\mid \mid n} d\mu^*(n/d), \quad \sum_{d\mid \mid
n} \phi^*(d)=n,
\end{equation}
where $\mu^*(n)=(-1)^{\omega(n)}$, where $\omega(n)$ is the number of
distinct prime factors of $n$. Various properties of the function
$\phi^*$ were investigated by several authors, cf.\ \cite{Coh1960,
Lal1974,McC1986,SitSur1973,SitSub2007,SivDix2006,Sko2008}.

Furthermore, consider the gcd-sum type function given by
\begin{equation} \label{P*}
P^*(n):=\sum_{k=1}^n (k,n)_*,
\end{equation}
which was introduced in our paper \cite{Tot1989}; also see
\cite{Tot1996,Tot1998}. The function $P^*$ (\seqnum{A145388}) is
also multiplicative; we have $P^*(p^\nu)=2p^\nu-1$ for every prime power
$p^\nu$ ($\nu\ge 1$) and
\begin{equation}
P^*(n)=\sum_{d\mid \mid n} d \phi^*(n/d).
\end{equation}

The functions $\phi^*$ and $P^*$ defined by \eqref{phi*} and
\eqref{P*} are called the unitary Euler function and unitary gcd-sum
function (or unitary Pillai function), respectively. Many of their
properties are analogous to the functions $\phi$ and $P$. Note that
$\mu^*$ is the unitary analogue of the M\"obius function $\mu$.

A natural question is the following: Why do we not consider the
greatest common unitary divisor of $k$ and $n$ when defining these
functions? The answer is, because if we do so, the resulting functions are
not multiplicative, and their properties are not so close to those
of Euler's function \eqref{phi} and of the gcd-function \eqref{P},
as it turns out from the results of the present paper.

Let $(k,n)_{**}= \max \{d\in \N: d\mid \mid k, d\mid \mid n\}$ stand
for the greatest common unitary divisor of $k$ and $n$. The function
\begin{equation} \label{P**}
\phi^{**}(n):= \# \{k\in \N: 1\le k\le n, (k,n)_{**}=1\},
\end{equation}
called the bi-unitary Euler function, was introduced by Subbarao and
Suryanarayana in 1971, cf.\ \cite{SurSub1980} and
\cite{Hau1998}. We have
\begin{equation} \label{phi**}
\phi^{**}(n)=\sum_{d\mid \mid n} \mu^*(d)\phi(n/d,d),
\end{equation}
where $\phi(x,n):=\# \{k\in \N: 1\le k\le x, \gcd (k,n)=1\}$ is the
Legendre function. Note that $\phi(n,n)=\phi(n)$.

Here $(k,n)_{**}\le (k,n)_*\le (k,n)$ for every $k,n\ge 1$, and hence
\begin{equation}
\phi(n)\le \phi^*(n)\le \phi^{**}(n).
\end{equation}

The average order of $\phi^{**}$ is given by
\begin{equation}
\sum_{n\le x} \phi^{**}(n)=\frac{Ax^2}{2}+{\cal O}(x\log^2 x),
\end{equation}
where
\begin{equation}
A:=\prod_p \left( 1-\frac{p-1}{p^2(p+1)}\right)=\zeta(2) \prod_p
\left(1-\frac{2}{p^2}+\frac{2}{p^3}-\frac1{p^4}\right),
\end{equation}
the products being over the primes and $\zeta$ denoting the Riemann
zeta function; see \cite[Corollary 3.6.2]{SurSub1980}. Here
$A\approx 0.8073$, this value is the asymptotic density of ordered
pairs $(m,n)\in \N^2$ such that $(m,n)_{**}=1$, i.e., there is no
prime power $p^\nu$ appearing in the prime factorizations of both
$m$ and $n$; see \cite[Corollary 3.6.3]{SurSub1980}, \cite[Theorems
2.2, 2.3]{Tot2001}.

Now let
\begin{equation}
P^{**} (n):=\sum_{k=1}^n (k,n)_{**}
\end{equation}
be the bi-unitary gcd-sum function, introduced recently by Haukkanen
\cite{Hau2008}. It is known (see \cite[Corollary 3.1]{Hau2008}) that
\begin{equation} \label{P**_formula}
P^{**} (n)= \sum_{d\mid \mid n} \phi^{*}(d)\phi(n/d,d).
\end{equation}

Note that for every $n\ge 1$,
\begin{equation}
P^{**}(n)\le P^*(n)\le P(n).
\end{equation}

Haukkanen \cite{Hau2008} considered also the function
\begin{equation} \label{P(f)**}
P_f^{**}(n): = \sum_{k=1}^n f((k,n)_{**}),
\end{equation}
and showed that for any arithmetical function $f$,
\begin{equation} \label{Pf**}
P_f^{**}(n)= \sum_{d\mid \mid n} (f\times \mu^{*})(d)\phi(n/d,d),
\end{equation}
where $\times$ stands for the unitary convolution. The unitary
convolution of the functions $f$ and $g$ is given by
\begin{equation} \label{unitary_convo}
(f\times g)(n)= \sum_{d\mid \mid n} f(d)g(n/d).
\end{equation}

For the properties of convolution \eqref{unitary_convo} and for
background material on arithmetical functions involving unitary
divisors we refer to the book \cite{McC1986}.

Furthermore, let $S^{**}$ denote the function given by
\eqref{P(f)**} in case $f(n)=\tau(n)$, the number of divisors of
$n$, i.e.,
\begin{equation}
S^{**}(n)= \sum_{k=1}^n \tau((k,n)_{**}).
\end{equation}

It is the aim of this paper to give combinatorial-type formulae for
$\phi^*(n)$, $\phi^{**}(n)$, $P^{**}(n)$, $S^{**}(n)$ and to
investigate the average orders of the functions $P^{**}$ and
$S^{**}$.

%********************  section 2  *****************************************

\section{Combinatorial-type formulae}

Let $\phi^*(x,n)=\# \{k\in \N: 1\le k\le x, (k,n)_*=1\}$ be the
unitary Legendre function, considered by Haukkanen \cite{Hau2002},
and let $\phi^{**}(x,n)=\# \{k\in \N: 1\le k\le x, (k,n)_{**}=1\}$
stand for the bi-unitary Legendre function, investigated in
\cite{SurSub1980}. Observe that $\phi^*(n,n)=\phi^*(n)$ and
$\phi^{**}(n,n)=\phi^{**}(n)$.

Let $n=p_1^{\nu_1}\cdots p_r^{\nu_r}&gt;1$ be a fixed integer and let
$A=A_n=\{1,\ldots,r\}$, $r=\omega(n)$.

We give explicit formulae for the values $\phi(x,n)$, $\phi^*(x,n)$
and $\phi^{**}(x,n)$ which are of combinatorial type and do not
involve the M\"obius function $\mu$, its unitary analogue $\mu^*$ or
other similar arithmetical functions.

In the following formulae $\lfloor y \rfloor$ stands for the largest
integer less than or equal to $y$ (floor function) and we use the
convention that the empty products are equal to $1$.
\begin{thm} \label{Th_1}
\begin{equation} \label{phi(x,n)}
\phi(x,n)= \sum_{B\subseteq A} (-1)^{\# B} \left\lfloor x
\prod_{j\in B} p_j^{-1} \right\rfloor,
\end{equation}
\begin{equation} \label{phi*(x,n)}
\phi^*(x,n)= \sum_{B\subseteq A} (-1)^{\# B} \left\lfloor x
\prod_{j\in B} p_j^{-\nu_j} \right\rfloor,
\end{equation}
\begin{equation} \label{phi**(x,n)}
\phi^{**}(x,n)= \sum_{B\subseteq A} (-1)^{\# B} \sum_{C\subseteq B}
(-1)^{\# C} \left\lfloor x \prod_{j\in B} p_j^{-\nu_j} \prod_{j\in
C} p_j^{-1} \right\rfloor.
\end{equation}
\end{thm}

\begin{proof} It is well known that $\phi(x,n)= \sum_{d\mid n}
\mu(d) \lfloor x/d \rfloor$. We give its proof for the sake of
completeness and in order to compare it with the following proofs. Using
that $d\mid \gcd(k,n)$ iff $d\mid k$, $d\mid n$,
\[
\phi(x,n)=\sum_{k\le x} \sum_{d\mid \gcd (k,n)} \mu(d)= \sum_{k\le
x} \sum_{\substack{d\mid k\\ d\mid n}} \mu(d)= \sum_{d\mid n} \mu(d)
\sum_{\substack{k\le x\\ d\mid k}} 1 =\sum_{d\mid n} \mu(d) \lfloor
x/d \rfloor.
\]

Now let $d=p_1^{\beta_1}\cdots p_r^{\beta_r}$ stand for a
square-free divisor of $n$, i.e., $\beta_j=1$ or $0$ for every $j$,
and let $B$ be the set of those $j$ for which $\beta_j=1$ and we
obtain \eqref{phi(x,n)} by the definition of the M\"obius function.

For the proof of \eqref{phi*(x,n)} we use that $d\mid \mid (k,n)_*$
iff $d\mid k$, $d\mid \mid n$ and have
\[
\phi^*(x,n)=\sum_{k\le x} \sum_{d\mid \mid (k,n)_*} \mu^*(d)=
\sum_{k\le x} \sum_{\substack{d\mid k\\ d\mid \mid n}} \mu^*(d)=
\sum_{d\mid \mid  n} \mu^*(d) \sum_{\substack{k\le x\\ d\mid k}} 1=
\sum_{d\mid \mid n} \mu^*(d) \lfloor x/d \rfloor.
\]

Now let $d=p_1^{\gamma_1}\cdots p_r^{\gamma_r}$ be a unitary divisor
of $n$, i.e., $\gamma_j=\nu_j$ or $0$ for every $j$, and let $B$ be
the set of those $j$ for which $\gamma_j=\nu_j$ and we obtain the
formula \eqref{phi*(x,n)}.

For the proof of \eqref{phi**(x,n)} we apply that $d\mid \mid
(k,n)_{**}$ iff $d\mid \mid k$, $d\mid \mid n$ and obtain
\[
\phi^{**}(x,n)=\sum_{k\le x} \sum_{d\mid \mid (k,n)_{**}} \mu^*(d)=
\sum_{k\le x} \sum_{\substack{d\mid \mid k\\ d\mid \mid n}}
\mu^*(d)= \sum_{d\mid \mid  n} \mu^*(d) \sum_{\substack{k\le x\\
d\mid k\\ \gcd (d,k/d)=1}} 1
\] \[=  \sum_{d\mid \mid n} \mu^*(d) \phi(x/d,d)= \sum_{d\mid \mid n}
\mu^*(d) \sum_{e\mid d} \mu(e) \lfloor x/(de) \rfloor.
\]

Let $d=p_1^{\gamma_1}\cdots p_r^{\gamma_r}$ be a unitary divisor of
$n$ i.e., $\gamma_j=\nu_j$ or $0$ for every $j$, and let $B$ be the
set of those $j$ for which $\gamma_j=\nu_j$. Furthermore, let
$e=p_1^{\eta_1}\cdots p_r^{\eta_r}$ be a square-free divisor of $d$,
where $\eta_k=1$ or $0$ for every $k$ and let $C$ be the set of
those $k$ for which $\eta_k=1$, where $C\subseteq B$.
\end{proof}

\begin{cor}
\begin{equation}
\phi^{**}(n)= \sum_{B\subseteq A} (-1)^{\# B} \sum_{C\subseteq B}
(-1)^{\# C} \left\lfloor \prod_{j\in A\setminus B} p_j^{\nu_j}
\prod_{j\in C} p_j^{-1} \right\rfloor.
\end{equation}
\end{cor}

This is obtained from \eqref{phi**(x,n)} for $x=n$. If $r=1$, then
we deduce that $\phi^{**}(p_1^{\nu_1})= p_1^{\nu_1}-1=
\phi^*(p_1^{\nu_1})$, which can be seen directly from the definition
\eqref{P**}. For $r=2$ we obtain
\begin{equation}
\phi^{**}(p_1^{\nu_1}p_2^{\nu_2}) = (p_1^{\nu_1}-1) (p_2^{\nu_2}-1)
+ \left\lfloor p_1^{\nu_1}p_2^{-1}\right\rfloor + \left\lfloor
p_2^{\nu_2}p_1^{-1}\right\rfloor = \phi^*(p_1^{\nu_1} p_2^{\nu_2}) +
\left\lfloor p_1^{\nu_1}p_2^{-1}\right\rfloor + \left\lfloor
p_2^{\nu_2}p_1^{-1}\right\rfloor,
\end{equation}
therefore the function $\phi^{**}$ is not multiplicative.

For $P^{**}(n)$ and $S^{**}(n)$ we have the following formulae. First
note that $\tau \times \mu^*=\beta$, where $\beta(n)$ is the number
of square-full divisors of $n$ (\seqnum{A005361}), $\beta$ is
multiplicative and $\beta(p^\nu)=\nu$ for any prime power $p^\nu$
($\nu \ge 1$). Therefore we have from \eqref{Pf**},
\begin{equation} \label{S**}
S^{**}(n)= \sum_{d\mid \mid n} \beta(d) \phi(n/d,d).
\end{equation}

\begin{thm}
\begin{equation} \label{P**combo_formula}
P^{**}(n)= \sum_{B\subseteq A} \left( \sum_{D\subseteq B} (-1)^{\#
D} \prod_{j\in B\setminus D} p_j^{\nu_j}\right) \left(
\sum_{C\subseteq B} (-1)^{\# C} \left\lfloor \prod_{j\in A\setminus
B} p_j^{\nu_j} \prod_{j\in C} p_j^{-1} \right\rfloor \right),
\end{equation}
\begin{equation} \label{S**combo_formula}
S^{**}(n)= \sum_{B\subseteq A} \left( \prod_{j\in B} \nu_j \right)
\sum_{C\subseteq B} (-1)^{\# C} \left\lfloor \prod_{j\in A\setminus
B} p_j^{\nu_j} \prod_{j\in C} p_j^{-1} \right\rfloor.
\end{equation}
\end{thm}

\begin{proof} From \eqref{P**_formula} and \eqref{phi*formula}
we obtain
\[
P^{**} (n)= \sum_{d\mid \mid n} \phi^{*}(d)\phi(n/d,d) =\sum_{d\mid
\mid n} \left( \sum_{e\mid \mid d} \mu^*(e) \frac{d}{e} \right)
\left( \sum_{\delta \mid d} \mu(\delta) \lfloor n/(d\delta) \rfloor
\right),
\]
and use similar arguments as in Theorem 1. To obtain
\eqref{S**combo_formula} use \eqref{S**}.

For $r=1$ we deduce from \eqref{P**combo_formula} that
$P^{**}(p_1^{\nu_1})=2p_1^{\nu_1}-1=P^*(p_1^{\nu_1})$, which can be
seen also directly from the definitions. For $r=2$ we obtain
\[
P^{**}(p_1^{\nu_1}p_2^{\nu_2})= (2p_1^{\nu_1}-1)
\left(2p_2^{\nu_2}-1\right) -(p_1^{\nu_1}-1) \left\lfloor
p_2^{\nu_2}p_1^{-1}\right\rfloor - (p_2^{\nu_2}-1) \left\lfloor
p_1^{\nu_1}p_2^{-1}\right\rfloor \] \[ = P^*(p_1^{\nu_1}
p_2^{\nu_2}) -(p_1^{\nu_1}-1) \left\lfloor
p_2^{\nu_2}p_1^{-1}\right\rfloor - (p_2^{\nu_2}-1) \left\lfloor
p_1^{\nu_1}p_2^{-1}\right\rfloor.
\]

Also, for $r=1$ we have from \eqref{S**combo_formula},
$S^{**}(p_1^{\nu_1})=p_1^{\nu_1}+\nu_1$. For $r=2$ we obtain
\[
S^{**}(p_1^{\nu_1}p_2^{\nu_2})= (p_1^{\nu_1}+\nu_1)
(p_2^{\nu_2}+\nu_2) - \nu_1 \left\lfloor
p_2^{\nu_2}p_1^{-1}\right\rfloor - \nu_2 \left\lfloor
p_1^{\nu_1}p_2^{-1}\right\rfloor \] \[ = S^{**}(p_1^{\nu_1}
p_2^{\nu_2}) - \nu_1 \left\lfloor p_2^{\nu_2}p_1^{-1}\right\rfloor -
\nu_2 \left\lfloor p_1^{\nu_1}p_2^{-1}\right\rfloor.
\]
\end{proof}

Therefore the functions $P^{**}$ and $S^{**}$ are not
multiplicative.


%********************  section 3  *****************************************

\section{The average order of $P^{**}$}

We show that
\begin{thm} \label{Th_3}
\begin{equation}
\sum_{n\le x} P^{**} (n)= \frac1{2} B x^2 \log x+{\cal O}(x^2),
\end{equation}
where
\begin{equation}
B: = \prod_p \left(1-\frac{3p-1}{p^2(p+1)}\right) = \zeta(2) \prod_p
\left(1-\frac{(2p-1)^2}{p^4}\right) \approx 0.35823.
\end{equation}
\end{thm}

\begin{proof} We will use the fact that for any real numbers $s\ge 0$ and $0\le
\varepsilon &lt;1$,
\begin{equation} \label{estimate_eps}
\sum_{\substack{n\le x\\\gcd (n,k)=1}} n^s =
\frac{x^{s+1}\phi(k)}{(s+1)k}+{\cal O}(x^s
\min(x^{\varepsilon}\sigma_{-\varepsilon}(k), 2^{\omega(k)})),
\end{equation}
this estimate being uniform for $x$ and $k$, where
$\sigma_{-\varepsilon}(k)$ is the sum of $\varepsilon$-th powers of
the reciprocals of divisors of $k$. A similar treatment was applied
also in \cite{Tot1989}. We have by \eqref{P**_formula}, and by
\eqref{estimate_eps} applied for $s=0$ and $s=1$, respectively,
\begin{equation*}
\sum_{n\le x} P^{**} (n)= \sum_{\substack{de\le x\\ \gcd (d,e)=1}}
\phi^*(d) \phi(e,d)= \sum_{\substack{de\le x\\ \gcd (d,e)=1}}
\phi^*(d) \left(\frac{e\phi(d)}{d}+ {\cal
O}\left(e^{\varepsilon}\sigma_{-\varepsilon}(d)\right)\right)
\end{equation*}
\begin{equation*}
=\sum_{d\le x} \frac{\phi(d)\phi^*(d)}{d}\sum_{\substack{e\le x/d
\\ \gcd (e,d)=1}} e+ {\cal O}\left(\sum_{d\le x} \phi^*(d)
\sigma_{-\varepsilon}(d) \sum_{e\le x/d} e^{\varepsilon}\right)
\end{equation*}
\begin{equation*}
=\sum_{d\le x} \frac{\phi(d)\phi^*(d)}{d}
\left(\frac{x^2\phi(d)}{2d^3}+{\cal
O}\left((x/d)^{\varepsilon+1}\sigma_{-\varepsilon}(d)\right)\right)+
{\cal O}\left(\sum_{d\le x} d \sigma_{-\varepsilon}(d)
(x/d)^{\varepsilon+1} \right)
\end{equation*}
\begin{equation*}
=\frac{x^2}{2} \sum_{d\le x} \frac{\phi^2(d)\phi^*(d)}{d^4} + {\cal
O}\left(x^{\varepsilon+1} \sum_{d\le x}
\frac{\sigma_{-\varepsilon}(d)}{d^{\varepsilon}}\right),
\end{equation*}
where the error term is ${\cal O}(x^{\varepsilon+1}\cdot
x^{1-\varepsilon})={\cal O}(x^2)$, cf.\ \cite[Lemma 2.2]{Tot1989}.

Now let $E_k(n)=n^k$ and $\phi^2 \phi^* =f* E_3$ in terms of the
Dirichlet convolution. Thus $f= \phi^2 \phi^* * \mu E_3$. Here $f$
is multiplicative and direct computations show that
$f(p)=-3p^2+3p-1$ and $f(p^\nu)=p^{2\nu+1}(1-1/p)^3$ for every prime
$p$ and every $\nu \ge 2$. We obtain that the Dirichlet series
\begin{equation*}
\sum_{n=1}^{\infty} \frac{f(n)}{n^s}
\end{equation*}
is absolutely convergent for $\RE s&gt;3$ and by $\phi^2 \phi^*/E_4=
f/E_4 * E_{-1}$,
\begin{equation*}
\sum_{n\le x} \frac{\phi^2(n)\phi^*(n)}{n^4} =\sum_{d\le x}
\frac{f(d)}{d^4} \sum_{e\le x/d} \frac1{e} = \sum_{d\le x}
\frac{f(d)}{d^4} \left(\log(x/d) +{\cal O}(1) \right)
\] \[ = \log x \sum_{d=1}^{\infty} \frac{f(d)}{d^4} + {\cal
O}\left(\log x \sum_{d&gt;x} \frac{|f(d)|}{d^4}\right) + {\cal
O}\left(\sum_{d\le x} \frac{|f(d)|\log d}{d^4} \right) = B \log x +
{\cal O}(1),
\end{equation*}
therefore
\begin{equation} \label{phiphi}
\sum_{n\le x} \frac{\phi^2(n)\phi^*(n)}{n^4}=B \log x + {\cal O}(1),
\end{equation}
by the Euler product formula. This completes the proof.
\end{proof}

\begin{rem}\upshape \label{rem_1}
The asymptotic formula \eqref{phiphi} can be obtained by using known
theorems on mean values of certain multiplicative functions. The
following useful theorem, which is a particular case of a more
general result, has been proved in \cite{Mar2002} using Iwaniec's
ideas.

{\it Theorem {\rm \cite[Proposition A.3]{Mar2002}}. Let f be any
nonnegative multiplicative function such that $f(n)\ll n^{\alpha}$
for some $\alpha &lt; 1/2$ and satisfying
\begin{equation*}
\sum_{p\le x} \frac{f(p)\log p}{p} = \kappa \log x + {\cal O}_f(1)
\quad (x\ge 2),
\end{equation*}
where $\kappa=\kappa_f&gt;0$. Then we have uniformly for all $x \ge 2$,
\begin{equation} \label{propA.3}
\sum_{n\le x} \frac{f(n)}{n} ={\cal M}_{f,\kappa}(\log x)^{\kappa}
+{\cal O}_f((\log x)^{\kappa-1}),
\end{equation}
where
\begin{equation*}
{\cal M}_{f,\kappa}:= \frac1{\Gamma(\kappa+1)}\prod_p \left(1
-\frac1{p}\right)^{\kappa} \left(1+\sum_{\nu=1}^{\infty}
\frac{f(p^\nu)}{p^\nu} \right).
\end{equation*}
}

Now \eqref{phiphi} follows at once by applying \eqref{propA.3} to
$f(n) =\frac{\phi^2(n)\phi^*(n)}{n^3}$ with $\kappa = 1$ and
noticing that $f(p) = (1 - 1/p)^3 = 1 + {\cal O}(1/p)$.
\end{rem}


%********************  section 4  *****************************************

\section{The average order of $S^{**}$}

For the function $S^{**}$ we have
\begin{thm} \label{Th_4}
\begin{equation}
\sum_{n\le x} S^{**} (n)= \frac1{2} C x^2 +{\cal O}(x\log^2 x),
\end{equation}
where
\begin{equation}
C: = \prod_p \left(1+\frac1{(p+1)^2}\right) = \zeta^2(2) \prod_p
\left(1-\frac1{p^2}-\frac{2}{p^3}+\frac{2}{p^4}\right)\approx
1.266558.
\end{equation}
\end{thm}

\begin{proof} Similar to the proof of Theorem 3, here it is
enough to apply \eqref{estimate_eps} for $\varepsilon =0$. From
\eqref{S**} we obtain
\begin{equation*}
\sum_{n\le x} S^{**} (n)= \sum_{\substack{de\le x\\ (d,e)=1}}
\beta(d) \phi(e,d)= \sum_{\substack{de\le x\\(d,e)=1}} \beta(d)
\left(\frac{e\phi(d)}{d}+ {\cal O}\left(2^{\omega(d)}\right)\right)
\end{equation*}
\begin{equation*}
=\sum_{d\le x} \frac{\phi(d)\beta(d)}{d}\sum_{\substack{e\le x/d
\\ (e,d)=1}} e+ {\cal O}\left( \sum_{de\le x} \beta(d)2^{\omega(d)} \right)
\end{equation*}
\begin{equation*}
=\sum_{d\le x} \frac{\phi(d)\beta(d)}{d}
\left(\frac{x^2\phi(d)}{2d^3}+{\cal O}\left((x/d)2^{\omega(d)}
\right)\right) + {\cal O}\left(x \sum_{d\le x} \frac{\beta(d)
2^{\omega(d)}}{d} \right)
\end{equation*}
\begin{equation} \label{errors}
=\frac{x^2}{2} \sum_{d=1}^{\infty} \frac{\beta(d)\phi^2(d)}{d^4}+
{\cal O}\left(x^2\sum_{d&gt;x} \frac{\beta(d)}{d^2}\right) + {\cal
O}\left(x \sum_{d\le x} \frac{\beta(d)2^{\omega(d)}}{d} \right).
\end{equation}

We need here the estimate $\sum_{n\le x} \beta(n)\ll x$. The
function $\beta$ is multiplicative and increasing on prime powers,
i.e., $\beta(p^\nu)=\nu &gt;\nu -1 = \beta(p^{\nu-1})$ for every prime
$p$ and every $\nu \ge 1$. It follows by well known results on sums
of multiplicative functions, that
\begin{equation*}
\sum_{n\le x} \beta(n) \le x\prod_{p\le x} \left(1-\frac1{p}\right)
\sum_{\nu=0}^{\infty} \frac{\beta(p^\nu)}{p^{\nu}},
\end{equation*}
cf. for ex. \cite[p.\ 23]{IK2004}, and obtain that
\begin{equation*}
\sum_{n\le x} \beta(n) \le x\prod_{p\le x} \left(1-\frac1{p}\right)
\left(1+ \sum_{\nu=1}^{\infty} \frac{\nu}{p^{\nu}}\right) =
x\prod_{p\le x} \left(1+\frac1{p(p-1)}\right)\ll x.
\end{equation*}

By partial summation we obtain
\begin{equation*}
\sum_{d&gt;x} \frac{\beta(d)}{d^2}= - \frac1{x^2} \sum_{d\le x}
\beta(d) +2 \int_x^{\infty} \left(\sum_{d\le t} \beta(d)
\right)\frac{dt}{t^3} \ll \frac1{x},
\end{equation*}
therefore the first ${\cal O}$-term in \eqref{errors} is ${\cal
O}(x)$. The second ${\cal O}$-term in \eqref{errors} is ${\cal
O}(x\log^2 x)$. This follows from the estimate
\begin{equation*}
\sum_{d\le x} \frac{\beta(d)2^{\omega(d)}}{d} \le \prod_{p\le x}
\left(1+ \sum_{\nu=1}^{\infty}
\frac{\beta(p^{\nu})2^{\omega(p^{\nu})}}{p^\nu} \right) =\prod_{p\le
x} \left(1+\frac{2}{p-1}+ \frac{2}{(p-1)^2} \right)\ll \log^2 x,
\end{equation*}
by using Mertens' theorem.

The given representations of $C$ follow by the Euler product
formula. For the numerical value of $C$ consult \cite[item
56]{NM2002}.
\end{proof}

\begin{rem}\upshape
It is known that $\sum_{n\le x} \beta(n) \sim cx$, where
$c=\zeta(2)\zeta(3)/\zeta(6)$, which can be obtained using the
representation $\beta(n)=\sum_{de=n} \sum_{a^2b^3=d} \mu^2(b)$. For
an asymptotic formula with remainder term for $\beta(n)$, see
\cite{SurSit1972}.
\end{rem}


%********************  section 5  *****************************************

\section{\bf Acknowledgement}
The author thanks the referee for improving the error term of
Theorem \ref{Th_4} and for Remark \ref{rem_1}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{thebibliography}{99}

\bibitem{Coh1960} E. Cohen, Arithmetical functions
associated with the unitary divisors of an integer, {\it Math. Z.}
{\bf 74} (1960), 66--80.

\bibitem{Hau1998} P. Haukkanen, Basic properties of the bi-unitary 
convolution and
the semi-unitary convolution, {\it Indian J. Math.} {\bf 40}
(1998), 305--315.

\bibitem{Hau2002} P. Haukkanen, On an inequality related to the Legendre totient function,
{\it J. Inequal. Pure Appl. Math.} {\bf 3} (2002), Article 37.

\bibitem{Hau2008} P. Haukkanen, On a gcd-sum function, {\it Aequationes Math.}
{\bf 76} (2008), 168--178.

\bibitem{IK2004} H. Iwaniec and  E. Kowalski, {\it Analytic Number Theory},
American Mathematical Society, 2004.

\bibitem{Lal1974} M. Lal, Iterates of the unitary totient
function, {\it Math. Comp.} {\bf 28} (1974), 301--302.

\bibitem{Mar2002} G. Martin, An asymptotic formula for the number of smooth values
of a polynomial, {\it J. Number Theory} {\bf 93} (2002), 108--182.

\bibitem{NM2002} G. Niklasch and P. Moree, Some number-theoretical
constants, 2002, webpage: \\
\url{http://www.gn-50uma.de/alula/essays/Moree/Moree.en.shtml} .

\bibitem{McC1986} P. J. McCarthy, {\it Introduction to Arithmetical
Functions}, Springer, 1986.

\bibitem{Pil1933} S. S. Pillai, On an arithmetic function,
{\it J. Annamalai Univ.} {\bf 2} (1933), 243--248.

\bibitem{SitSur1973} R. Sitaramachandrarao and
D. Suryanarayana, On $\sum_{n\le x} \sigma^*(n)$ and
$\sum_{n\le x} \phi^*(n)$, {\it Proc. Amer. Math. Soc.} {\bf 41}
(1973), 61--66.

\bibitem{SitSub2007} V. Sitaramaiah and M. V. Subbarao, Unitary analogues of some formulae of
Ingham, {\it Ars Combin.} {\bf 84} (2007), 33--49.

\bibitem{SivDix2006} V. Siva Rama Prasad and
U. Dixit, Inequalities related to the
unitary analogue of Lehmer problem, {\it J. Inequal. Pure Appl.
Math.} {\bf 7} (2006), no. 4, Article 142.

\bibitem{Sko2008} M. Skonieczna, Some results on the unitary
analogue of the Lehmer problem, {\it J. Inequal. Pure Appl. Math.}
{\bf 9} (2008), no. 2, Article 55.

\bibitem{SurSit1972} D. Suryanarayana and
R. Sitaramachandra Rao, The number of
square-full divisors of an integer, {\it Proc. Amer. Math. Soc.}
{\bf 34} (1972), 79--80.

\bibitem{SurSub1980} D. Suryanarayana and
M. V. Subbarao, Arithmetical functions associated with
the bi-unitary $k$-ary divisors of an integer, {\it Indian J. Math.}
{\bf 22} (1980), 281--298.

\bibitem{Tot1989} L. T\'oth, The unitary analogue of Pillai's
arithmetical function, {\it Collect. Math.} {\bf 40} (1989), 19--30.

\bibitem{Tot1996} L. T\'oth, The unitary analogue of Pillai's arithmetical function II., {\it
Notes Number Theory Discrete Math.} {\bf 2} (1996), no 2, 40--46,
available at \\
\url{http://www.ttk.pte.hu/matek/ltoth/Toth_Pillai2_1996.pdf} .

\bibitem{Tot1998} L. T\'oth, A generalization of Pillai's arithmetical function
involving regular convolutions, {\it Acta Math. Inform. Univ.
Ostraviensis} {\bf 6} (1998), 203--217, available at \\
\url{http://www.ttk.pte.hu/matek/ltoth/TothGeneralizationPillai(1998).pdf} .

\bibitem{Tot2001} L. T\'oth,
On the asymptotic densities of certain subsets of $\N^k$,
{\it Riv. Mat. Univ. Parma} (6) {\bf 4} (2001), 121--131, available
at \url{http://arxiv.org/abs/math.NT/0610582} .
\end{thebibliography}

\bigskip
\hrule
\bigskip

\noindent 2000 {\it Mathematics Subject Classification}: Primary
11A25; Secondary 11N37, 05A15.

\noindent \emph{Keywords:} Euler's arithmetical function, M\"obius
function, divisor function, gcd-sum function, unitary divisor,
average order.

\bigskip
\hrule
\bigskip

\noindent (Concerned with sequences 
\seqnum{A000010},
\seqnum{A001221},
\seqnum{A005361},
\seqnum{A008683},
\seqnum{A018804},
\seqnum{A047994}, and
\seqnum{A145388}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in} 
\noindent Received April 24 2009; revised version received June
22 2009.  Published in {\it Journal of
Integer Sequences}, June 23 2009.

\bigskip
\hrule
\bigskip

\noindent Return to \htmladdnormallink{Journal of Integer Sequences
home page}{http://www.cs.uwaterloo.ca/journals/JIS/}. \vskip .1in

\end{document}
</pre></body></html>