\documentclass[12pt,reqno]{article} \usepackage[usenames]{color} \usepackage{amssymb} \usepackage{graphicx} \usepackage{amscd} %\usepackage{upref}%upright references in mathmode \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 A Note on the Relation Between Fixed Point \\ \vskip .1in and Orbit Count Sequences } \vskip 1cm \large Michael Baake and Natascha Neum\"arker \\ Fakult\"at f\"ur Mathematik \\ Universit\"at Bielefeld \\ Postfach 100131 \\ 33501 Bielefeld \\ Germany \\ \href{mailto:mbaake@math.uni-bielefeld.de}{\tt mbaake@math.uni-bielefeld.de} \\ \href{mailto:naneumae@math.uni-bielefeld.de}{\tt naneumae@math.uni-bielefeld.de} \\ \end{center} \vskip .2 in \begin{abstract} The relation between fixed point and orbit count sequences is investigated from the point of view of linear mappings on the space of arithmetic functions. Spectral and asymptotic properties are derived and several quantities are explicitly given in terms of Gaussian binomial coefficients. \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}} \newcommand{\CC}{\mathbb{C}} \newcommand{\NN}{\mathbb{N}} \newcommand{\RR}{\mathbb{R}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\id}{\mathrm{id}} \newcommand{\abs}[1]{\left|#1\right|} \newcommand{\norm}[1]{\left\|{#1}\right\|} \newcommand{\sga}[3]{\left[ \begin{smallmatrix} #1 \\ #2 \end{smallmatrix} \right]_{#3}} \newcommand{\gabi}[3]{\begin{bmatrix} #1\\ #2 \end{bmatrix}_{#3}} \newcommand{\fix}{\mathrm{fix}} \newcommand{\orb}{\mathrm{orb}} \newcommand{\diri}[2]{D^{}_{#1}(#2)} \newcommand{\zetfun}[1]{\zeta\left(#1\right)} \newcommand{\ts}{\hspace*{0.5pt}} %\newtheorem{theorem}{Theorem}[section] %\newtheorem{lemma}[theorem]{Lemma} \newtheorem{prop}[theorem]{Proposition} %\newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem*{rem}{Remark} \section{Introduction and general setting} Each map of an arbitrary set into itself gives rise to sequences $a=(a_n)_{n\in\NN}$ and $c=(c_n)_{n\in\NN}$ that count the number of periodic points with period $n$ and the orbits of length $n$, respectively. In many interesting cases, $a_n$ and $c_n$ are finite for all $n\in\NN$, turning $a$ and $c$ into sequences of non-negative integers. They are then related by M\"obius inversion as \begin{equation}\label{intTrans} a_n = \sum_{d|n} d\, c^{}_d \quad \text{and}\quad c_n = \frac{1}{n}\sum_{d|n}\mu\left(\frac{n}{d}\right) a^{}_d. \end{equation} Here, $\mu$ denotes the M\"obius function from elementary number theory \cite{apostol}. Puri and Ward \cite{PuriWard} study these relations as paired transformations between certain non-negative integer sequences, with special emphasis on arithmetic and asymptotic aspects. It is the purpose of this short note, which is based on \cite{NN}, to extend these transformations to the space of arithmetic functions, on which they act as linear operators, and to investigate their spectral properties and their asymptotic behaviour under iteration, thus generalising results from \cite{PuriWard}. Let $\mathcal{A} = \CC_{}^{\NN}$ denote the set of arithmetic functions. Its elements can be thought of either as functions $f\! :\,\NN\to\CC, n\mapsto f(n)$, or as sequences of complex numbers $(f_n)_{n\in\NN}$, hence the notations $f(n)$ and $f_n$ are used in parallel; see \cite{apostol} for details and general notation. Clearly, $\mathcal{A}$ is a complex vector space with respect to component-wise addition and the usual scalar multiplication. Additionally, it is equipped with the structure of a commutative algebra $(\mathcal{A},+,\ast)$ with unit element, where $\ast$ denotes \textit{Dirichlet convolution}, defined by \[ \bigl( f\ast g \bigr) (n) = \sum_{d|n} f(d) \, g\left(\frac{n}{d}\right) \] for all $n\in\NN$. The unit element with regard to this product is $\delta_1 = (1,0,0,\ldots)$. When $f\in\mathcal{A}$ is invertible, the notation $f^{-1}$ refers to its \textit{Dirichlet inverse}, defined by $f\ast f^{-1} = f^{-1}\ast f = \delta_1$. Furthermore, $f\cdot g$ denotes the pointwise product where $\bigl( f\!\cdot g\bigr) (n) = f(n)\ts g(n)$ for all $n\in\NN$. Motivated by the transformations \eqref{intTrans}, we define `$\fix$' and `$\orb$' as mappings from $\mathcal{A}$ into itself by $a\mapsto a_{}^{\prime}$, with \begin{equation} a^{\prime}_n = \fix(a)_n = \sum_{d|n} d\, a^{}_d\quad \text{and}\quad a^{\prime}_n = \orb(a)_n = \frac{1}{n}\sum_{d|n} \mu{\left(\frac{n}{d}\right)} \ts a^{}_d, \end{equation} respectively, for all $n\in\NN$. Denoting by $N$ the arithmetic function $n\mapsto n$, $\fix$ and $\orb$ can be written as \begin{equation}\label{convRepres} \fix(f) = N\cdot \Bigl(\frac{1}{N}\ast f \Bigr) \quad \text{and}\quad \orb(f) = \frac{1}{N}\cdot \bigl( \mu\ast f \bigr). \end{equation} This is particularly useful when considering the Dirichlet series of $\fix(f)$ and $\orb(f)$. In general, the Dirichlet series $D_f$ of $f\in\mathcal{A}$ is defined by \[ D_f (s) = \sum_{m=1}^{\infty} \frac{f(m)}{m^s} \ts , \] which is viewed as a formal series at this stage. This generating function gives rise to the (formal) identities \cite{apostol} \[ D_{f+g} = D_f + D_g \quad \text{and}\quad D_{f\ast g} = D_f \cdot D_g \ts . \] Using these in connection with \eqref{convRepres}, a straightforward calculation shows \begin{lemma}\label{dirichFix} Let $D_a(s)$ denote the Dirichlet series of $a\in\mathcal{A}$. Then, \[ \diri{\fix(a)}{s} = \zetfun{s} \diri{a}{s-1} \quad \text{and} \quad \diri{\orb(a)}{s} = \frac{1}{\zetfun{s+1}} \, \diri{a}{s+1} \ts . \] where $\zeta$ is Riemann's zeta function. \hfill$\Box$ \end{lemma} For a discussion of the identities from Lemma~\ref{dirichFix} in a different context, and for interesting recent development on Dirichlet series as orbit counting generating functions, we refer the reader to \cite[Secs.~ 2 and 10]{finiteCombRank}. We now turn our attention to the structure of $\fix$ and $\orb$ as linear mappings on $\mathcal{A}$. \section{Spectral properties} When studying $\fix$ on $\mathcal{A}$, it is natural to ask for eigenvalues and eigenvectors. Let $x$ be an arithmetic function that satisfies the fixed point condition $\fix(x)_n = \sum_{d|n} d\, x_d = x_n$ for all $n\in\NN$. Solving for $x_n$ yields the recursive relation \begin{equation}\label{recursion} x_n = \frac{1}{1-n}\sum_{n>d | n} d\, x_d \end{equation} for $n> 1$, so that all $x_n$ are uniquely determined from $x^{}_{1}$. The $x_n$ depend linearly on $x_1$, which is the only degree of freedom in solving \eqref{recursion}, wherefore the eigenspace for the eigenvalue $1$ is one-dimensional. Setting $x_1 = a_1 = 1$, we denote the resulting solution of \eqref{recursion} by $ a = (a^{}_1,a^{}_2,a^{}_3,\ldots)$. Since $\fix$ maps multiplicative sequences to multiplicative sequences, it is an interesting question to what extent this is reflected by $a$. \begin{lemma}\label{multiplicativity} Let $a$ satisfy $\fix(a) = a$, with $a^{}_1 = 1$. The eigenvector $a$ is multiplicative, so that $a^{}_{nm} = a^{}_n\cdot a^{}_m$ for all $n,m \geq 1$ with $(m,n) = 1$. \end{lemma} \begin{proof} In order to show multiplicativity by induction on the index, we first note that the claim is true for trivial decompositions such as $n\cdot 1$, because $a^{}_1 = 1$. In particular, it is then true for all natural numbers $n \leq 5 = 2\cdot 3 - 1$. Let $(m,n) = 1$ with $mn > 1$ and suppose the $a_k$ satisfy the multiplicativity property for all $k \leq mn -1$ (so that $a^{}_k = a^{}_{d}\, a^{}_{d\ts '}$ for all $d,d'$ with $(d,d\ts ') = 1$ and $d\, d\ts ' = k$). This implies \begin{eqnarray*} a^{}_{mn} &=& \frac{1}{1-mn}\sum_{\substack{d|mn \\d < mn}} d\, a^{}_d = \frac{1}{1-mn}\sum_{\substack{d=d_md_n\\ d_m|m,\ d_n|n \\ d_md_n < mn}}(d_m a^{}_{d_m})\, (d_n a^{}_{d_n})\\ &=&\frac{1}{1-mn}\Biggl(\Bigl(\underbrace{\sum_{d|m}d\ts a^{}_d}_{=a^{}_m} \underbrace{\sum_{d\ts ' |n}d\ts\ts ' a^{}_{d\ts '}}_{=a^{}_n}\Bigr)- mn\,a^{}_m a^{}_n \Biggr) = a^{}_m\, a^{}_n \ts . \end{eqnarray*} This argument shows the induction step from $mn - 1$ to $mn$. In general, the step from $N$ to $N+1$ is trivial when $N+1$ is prime, and of the above form otherwise, which completes the induction. \end{proof} For the remainder of this section, $a$ always denotes the unique solution of $\fix(a) = a$ with $a_1 = 1$. The recursion in \eqref{recursion} runs over all proper divisors of $n$ and yields an explicit expression for $a^{}_{p^r}$ with $p$ prime and $r\in\NN$. By multiplicativity, this extends to a closed formula for $a_n$. \begin{prop}\label{productRepres1} Let $n > 1$ with prime decomposition $n = p_1^{r_1}\cdot\ldots\cdot p_s^{r_s}$. Then, the $n$-th entry of the eigenvector $a^{}$ of Lemma~\ref{multiplicativity} reads \[ a_n = \prod_{k=1}^{s} \ \prod_{\ell=1}^{r_k} \frac{1}{1-p_k^\ell} \ts . \] \end{prop} \begin{proof} Observing that $a^{}_p = \frac{a^{}_1}{1-p}$, induction on the exponent $r$ leads to $a^{}_{p^r} = \prod_{\ell = 1}^{r} (1 - p^{\ell})^{-1}$, from which the statement follows by Lemma~\ref{multiplicativity}. \end{proof} The sequence of denominators is now entry \seqnum{A153038} of the Online Encyclopedia of Integer Sequences (OEIS) \cite{OEIS}. In particular, $\abs{a_n} \leq 1$ for all $n$, and Proposition~\ref{productRepres1} suggests that the $a^{}_n$ become `asymptotically small' in some sense, as $n\to\infty$. Indeed, with \[ \ell^{\ts q} = \bigl\{f\in\CC_{}^{\NN} \, \big| \sum_{n=1}^{\infty}|f_n|^q < \infty \bigr\} , \] one obtains the following result. \begin{prop}\label{ell1pluseps} The eigenvector $a$ is an element of $\ell^{1+\varepsilon}$ for all $\varepsilon > 0$. \end{prop} \begin{proof} Substituting $p^k -1 = (p-1)\sum_{j=0}^{k-1}p^j$ in the denominator of $\abs{a_{p^r}}$ leads to \[ \prod_{k=1}^{r} (p^k -1 )= (p-1)^r\prod_{k=1}^{r}{\sum_{j=0}^{k-1}p^j}\geq (p-1)^r p^{\sum_{k=1}^{r}{(k-1)}} = (p-1)^r p^{r(r-1)/2}. \] Since $(p-1)^r p^{r(r-1)/2}\geq p^r$ for $r>2$ or $p>2$, the inequality $\abs{a_{p^r}}\leq \frac{1}{p^r}$ holds in these cases. Besides, one has $\abs{a_p}=\frac{1}{p-1}=\frac{p}{p-1}\cdot\frac{1}{p}$ for every prime $p$. The remaining term $a_{2^2} = \frac{4}{3}\frac{1}{2^2}$ introduces an additional factor of $c = \frac{4}{3}$ in the estimate of $\abs{a_n}$, to account for the case when $2^2$ is the highest power of $2$ that divides $n$. For $n=p_{1}^{r^{}_{1}}\cdot\ldots\cdot p_{s}^{r^{}_{s}}$, this gives \[ \abs{a_n} \le \frac{c}{n} \prod_{\{i\mid r_i = 1 \}}\frac{p_i}{p_i-1}. \] This product can be related to Euler's $\phi$-function, \[ \phi(n) = n\prod_{p|n}\left(1-\frac{1}{p}\right) = n\prod_{p|n}\left(\frac{p-1}{p}\right) \leq n \ \prod_{\substack{p|n\\p^2 \nmid\ts n}}^{r}\left(\frac{p-1}{p}\right), \] giving $\prod_{\{i\mid r_i = 1 \}}\frac{p_i}{p_i-1} \leq \frac{n}{\phi(n)}$. Using \[ \limsup_{n\to\infty} \frac{n}{\phi(n)\log(\log n)} = e^{\gamma}, \] where $\gamma = \lim_{n\to\infty}\left(1+ \frac{1}{2} + \cdots + \frac{1}{n} - \log{n}\right)$ denotes Euler's constant, compare \cite[Thm.~328]{HardyWright}, we obtain \[ \abs{a_n}\leq \frac{C\cdot \log(\log(n))}{n}\ \] for some constant $C$, which implies $\sum_{n=1}^{\infty}\abs{a_n}^{1+\varepsilon} <\infty$. \end{proof} In particular, $a\in\ell^2$, and (numerically) one finds $\sum_{n=1}^{\infty} \lvert a_n \rvert^2 \approx 2.99635 < 3$. Since $\sum_{p} \frac{1}{p}$ diverges and $\abs{a^{}_p} = \frac{1}{p-1} > \frac{1}{p}$, it is clear that the statement of Proposition~\ref{ell1pluseps} is not true for $\varepsilon = 0$. \begin{lemma}\label{dirichConvergence} Let $D_a(s)$ be the Dirichlet series of the eigenvector $a$, with $s = \sigma + i t$. \begin{itemize} \item[\rm (i)] $\diri{a}{s}$ \text{converges absolutely in} $\,\mathcal{S} = \{s\in\CC\mid\sigma > 1\}$.\label{absConvSet} \vspace{1.5mm} \item[\rm (ii)] $\lim_{\sigma\to\infty}\diri{a}{\sigma + it} = 1$, \text{uniformly in} $t\in\RR$. \end{itemize} \end{lemma} \begin{proof} We employ the methods of \cite[Chapter 11]{apostol}. The first claim follows from the absolute convergence of $\zeta(\sigma)$ for $\sigma > 1$, because $\sum_{m}\abs{\frac{a_m}{m^s}}\leq \sum_{m}\frac{\abs{a_m}}{m^{\sigma}}\leq \zeta(\sigma)$. For (ii), observe that $a_1=1$, wherefore one has \[ \abs{\diri{a}{s}-1}\leq \sum_{m=2}^{\infty}\frac{1}{m^{\sigma}} = \zeta(\sigma)- 1, \] which tends to $0$ (independently of $t$) as $\sigma\to\infty$. \qedhere \end{proof} Using the fixed point relations $\fix(a) = a$ and $\orb(a) = a$, the first equation of Lemma~\ref{dirichFix} becomes a `recursion' that can be solved for an explicit expression in terms of the Riemann zeta function. \begin{theorem}\label{dirich_a} The Dirichlet series of the fixed point $a$ satisfies \[ \diri{a}{s} = \prod_{\ell=1}^{\infty} \frac{1}{\zeta(s+\ell)} = \prod_{p} \ \prod_{\ell\geq 1}\left(1-\frac{1}{p^{s+\ell}}\right), \] where the infinite product converges on the set\/ $\mathcal{S}$ from Lemma~\ref{dirichConvergence}. \end{theorem} \begin{proof} Using Lemma \ref{dirichFix}, one inductively obtains $ \diri{a}{s} = \frac{\diri{a}{s+m}}{\prod_{\ell=1}^{m}\zeta(s+\ell)} $ for arbitrary $m\in\NN$. For $m\to\infty$, the numerator converges to $1$, since \[ \diri{a}{s+m} = a_1 + \sum_{n\geq 2}\frac{a_n}{n^s}\frac{1}{n^m}\leq a_1 + \frac{1}{2^m}\diri{a}{s}. \] Due to the asymptotic behaviour $\zeta(\sigma + i t) = 1+ \mathcal{O}(1/2^{\sigma})$ for $\sigma\to\infty$, the product in the denominator of the resulting expression $D_a(s) = \prod_{\ell=1}^{\infty} \frac{1}{\zeta(s+\ell)}$ converges as well, giving the first part of the claim. The second equality follows from the Euler product representation $ \zeta(s+\ell)^{-1} = \prod_{p} (1 - p^{-(s+\ell)}) $ for $\sigma > 1$, in which the order of taking products over $p$ and $\ell$ may be exchanged due to absolute convergence. \end{proof} \begin{rem} Since $a^{}_1\not= 0$, $a$ admits a Dirichlet inverse. Its terms can be calculated via to the general recursion formula, compare \cite[Thm.~2.8]{apostol}. The Dirichlet inverse satisfies $D^{}_{f^{-1}} = 1/D^{}_{f}$, wherefore $b := a^{-1}$ has the Dirichlet series $D^{}_{b} = \prod_{\ell \geq 1} \zeta(s + \ell).$ Determining the coefficients of this series leads to an analogue of Proposition~\ref{productRepres1}; the $n$-th entry of $b$ is \[ b^{}_n = \prod_{j=1}^{s} \ \prod_{k=1}^{r_j} \frac{p_j^{k-1}}{p_j^{k}-1} =\prod_{j=1}^{s} \frac{p_j^{\frac{1}{2} r_j(r_j-1)}}{\prod_{\ell = 1}^{r_j} (p_j^{\ell}-1)}, \] with $n = p_1^{r^{}_1}\cdot\ldots\cdot p_s^{r^{}_s}$ as before; see \cite{NN} for details. The first few terms of $a$ and $b$ read \[ a = \Bigl( 1,-1,-\frac{1}{2},\frac{1}{3},-\frac{1}{4}, \frac{1}{2},-\frac{1}{6},-\frac{1}{21}, \ldots \Bigr) \quad \text{and} \quad b = \Bigl( 1,1,\frac{1}{2},\frac{2}{3},\frac{1}{4},\frac{1}{2}, \frac{1}{6},\frac{8}{21}, \ldots \Bigr). \] Although it appears that $a$ and $b$ share the same denominator sequence, this is not true when the terms are represented as reduced fractions, because cancellation can occur for the $b_n$. The first instance of this phenomenon is $b_{12} = \frac{1}{3}$, whereas $a_{12}=-\frac{1}{6}$. \end{rem} Returning to the task of identifying all eigenvalues and their corresponding eigenspaces, we consider the equation $x = \lambda \cdot\fix(x)$ for $\lambda \not= 1$ or, equivalently, \[ (\lambda - n) \, x^{}_n = \lambda \sum_{n > d|n}d\, x^{}_d \] for all $n\in\NN$. Since $\fix(x)^{}_1 = x^{}_1$ by definition, the first term of such an eigenvector has to be $0$. In general, if $x^{}_d = 0$ for all $d|n$ with $dd|n}}d\, x^{}_d\ \qquad \text{(for}\quad n\not= m \text{)}, \end{equation} which shows that $x^{}_n \not= 0$ if and only if $n = km$ for some $k\geq 1$. The recursion for the non-vanishing terms simplifies to \[ x_{km}^{} = \frac{1}{m(1-k)}\sum_{k>d|k}^{} d\ts m\, x^{}_{dm} = \frac{1}{1-k}\sum_{k>d|k}^{} d\, x^{}_{dm} \] for $k>1$,where the only free parameter is now the first non-zero term $x_m$, indicating a one-dimensional eigenspace. Choosing $x_m = a^{}_1 = 1$, the resulting eigenvector $a^{(m)}$ with eigenvalue $m$ can be expressed in terms of $a = a^{(1)}$: \begin{equation} a^{(m)} (n) = \begin{cases} a\left(\frac{n}{m}\right), & \text{if } m|n ; \\ 0, & \text{otherwise}. \end{cases} \end{equation} In particular, $a^{(m)}\in\ell^{1+\varepsilon}$ for all $\varepsilon > 0$ and all $m\geq 1$, where $a^{(m)}$ and $a$ clearly have the same norm. Being eigenvectors for different eigenvalues, the $a^{(m)}$ are linearly independent. The next proposition states that they even generate the space of arithmetic functions in the sense of formal linear combinations. \begin{prop} The eigenvectors of\/ $\fix$ form a basis of $\mathcal{A}$, so that $\left_{\CC} = \mathcal{A}$, and the representation of $f\in\mathcal{A}$ as a formal linear combination of the $a^{(m)}$ is unique. \end{prop} \begin{proof} Let $f\in\mathcal{A}$, $a = a^{(1)}$ and $b := (a^{})^{-1}$, and set $\alpha = f \ast b$. Then, $\alpha * a^{} = (f*b)*a = f$. On the other hand, one has \[ \bigl(\alpha* a \bigr) (k) = \sum_{d|k} \alpha^{}_{d} \, a^{(d)} (k) = \sum_{m\geq 1} \alpha^{}_{m}\, a^{(m)}(k) \] for all $k\in\NN$, so $f\in \left_{\CC} $. Uniqueness of the linear combination follows from the uniqueness of the Dirichlet inverse. \end{proof} The spectrum of an operator $T\!: \, \mathcal{A}\to\mathcal{A}$ is defined by \[ \mathrm{spec}(T) := \{\lambda\in\CC\mid T-\lambda\,\id \text{ is not invertible}\, \} \ts . \] Note that $\fix$ and $\orb$ could be investigated as operators on $\ell^2$, restricting their domains to appropriate invariant subspaces. But since we need a more general setting later when considering their behaviour under iteration, we work with the spectrum as a purely algebraic notion. \begin{theorem} The spectra of\/ $\fix$ and $\orb$ are \[ \mathrm{spec}(\fix) = \NN \quad and \quad \mathrm{spec}(\orb) = \left\{\frac{1}{N} \mid N\in\NN \right\}\ts . \] \end{theorem} \begin{proof} Obviously, $\fix(f) = \lambda f$ implies $f=0$ if and only if $\lambda\in\CC\setminus\NN$, so $\fix - \lambda\,\id$ is injective for these $\lambda$. Surjectivity can be seen by constructing a preimage $f\in\mathcal{A}$ of an arbitrary $g\in\mathcal{A}$ under $\fix-\lambda\, \id$. To this end, let $f_1 = \frac{g^{}_1}{1-\lambda}$ and, having already defined $f_1,\ldots,f_{n-1}$ for some $n>1$, set \[ f^{}_n = \frac{1}{n-\lambda} \Bigl(g^{}_n - \sum_{n>d|n} d\, f^{}_d\Bigr). \] When $\lambda\notin\NN$, the resulting arithmetic function $f$ is a term-wise well-defined element of $\mathcal{A}$ with $\fix(f) - \lambda f = g$. In summary, $\fix-\lambda\,\id$ is invertible if and only if $\lambda\in\CC\setminus\NN$. The spectrum of $\orb$ can be derived from that of $\fix$ by the equivalence \[ \fix(a^{(N)}) = N a^{(N)} \, \Longleftrightarrow \; \orb(a^{(N)}) = \frac{1}{N} \, a^{(N)}, \] which follows from $\fix\circ\orb = \orb\circ\fix = \id$. \end{proof} \section{Iterations of the operator $\fix$} Puri and Ward \cite{PuriWard} raised the question of the nature of the orbits under $\fix$. In particular, the asymptotic properties of such orbits are sought. In our terminology, this is to ask how elements of $\mathcal{A}$ behave under iteration of the operator $\fix$. Starting from $\delta_{1} = (1,0,0,\ldots)$, the first iterates are the sequences \seqnum{A000007}, \seqnum{A000012}, \seqnum{A000203}, \seqnum{A001001} and \seqnum{A038991} -- \seqnum{A038999} of the OEIS, see \cite{PuriWard, OEIS}. Here, we start from the sequences $\delta_m$, $m\geq 1$, defined via $\delta_m (n) = \delta_{m,n}$ for all $n$, and give term-wise exact representations of $\fix^n(\delta_m)$, from which the corresponding quantities of general starting sequences can later be constructed. Let $y^{(0)} = \delta_1$ and $y^{(n)} := \fix(y^{(n-1)})$ for $n\geq 1$, so that \[ y^{(n)}(m) = \bigl(\fix(y^{(n-1)})\bigr)_m = \bigl(\fix^n(y^{(0)})\bigr)_m \] holds for all $m\ge 1$. \begin{prop}\label{y(primePower)} Let $p$ be a prime and $r\in\NN$. Then \[ y^{(n)}(p^r) = \gabi{n+r-1}{r}{p} = \prod_{i=0}^{r-1}\frac{1-p^{n+r-1-i}}{1-p^{i+1}}, \] where $\sga{n}{r}{q}$ denotes the Gaussian or $q$-binomial coefficient. \end{prop} \begin{proof} The claim can be verified by induction in $n$. By definition of $\fix$, we have \begin{align}\label{E:sumBin} y^{(n+1)}(p^r) &= \fix(y^{(n)})(p^r) = \sum_{d|p^r}d\cdot y^{(n)}(d)= \sum_{k=0}^{r}p^k \cdot y^{(n)}(p^k) \notag \\ &= 1+ p\gabi{n}{1}{p}+\sum_{k=2}^{r}p^k \cdot y^{(n)}(p^k) \ts . \end{align} Applying \textit{Pascal's identity} for Gaussian binomials, \[ \gabi{m}{\ell}{q} = q^r \gabi{m-1}{\ell}{q} + \gabi{m-1}{\ell-1}{q}, \] to the $k$-th summand in \eqref{E:sumBin} yields \[ p^k\gabi{n+k-1}{k}{p}=\gabi{n+k}{k}{p}- \gabi{n+k-1}{k-1}{p}. \] Plugging this and the relation $\sga{n}{1}{p}=\sum_{k=0}^{n-1} p^k$ into \eqref{E:sumBin}, we obtain \[ y^{(n+1)}(p^r) = \gabi{n+1}{1}{p} + \sum_{k=2}^{r}\gabi{n+k}{k}{p}-\gabi{n+k-1}{k-1}{p} =\gabi{n+r}{r}{p} \] which completes the proof. \end{proof} \begin{corollary}\label{y(m)} For $1\not= m\in\NN$ with prime decomposition $m = p_1^{r^{}_1}p_2^{r^{}_2}\cdot\ldots\cdot p_k^{r^{}_k}$, one has \[ y^{(n)}(m)=y^{(n)}(p_1^{r^{}_1})\, y^{(n)}{(p_2^{r^{}_2})}\cdot\ldots\cdot y^{(n)}(p_k^{r^{}_k}) =\prod_{j=1}^{k} \ \prod_{i=0}^{r_j -1} \frac{1-p_j^{n+r_j-1-i}}{1-p_j^{i+1}}. \] \end{corollary} \begin{proof} It is easy to check that $\fix$, and hence all iterates $\fix^n$ with $n\geq 1,$ preserve multiplicativity. Thus, in view of Proposition~\ref{y(primePower)}, the claim follows from $\delta_1$ being multiplicative. \end{proof} \begin{rem} A different approach to calculating $y^{(n)}$ exploits the fact that its Dirichlet series $D^{}_{y^{(n)}}(s)$ is known from the context of counting sublattices of $\ZZ^n$. Let \[ h^{}_n(m) = \big\lvert\{\varLambda \mid \varLambda \text{ is a sublattice of $\ZZ^n$ of index $m$}\}\big\rvert \] and $D_{h_n}(s) = \sum_{m\geq 1} \frac{h_n(m)}{m^s}$. Proposition A.1 of \cite{coincApp} states that \[ h_{n}(m) = \sum_{d_1\cdot\ldots\cdot d_n = m} d_1^{\ts 0}\cdot d_2^{\ts 1}\cdot\ldots\cdot d_n^{\ts n-1} \quad \text{and}\quad D^{}_{h_n}(s) = \zeta(s)\cdot D^{}_{h_{n-1}}(s-1), \] where the first sum runs over all $n$-tuples $(d_1,\ldots, d_n)$ of positive integers with $d_1\cdot\ldots\cdot d_n = m$. In particular, $D^{}_{h_1}(s) = \zeta(s) = D^{}_{y^{(1)}}(s)$. On the other hand, Lemma~\ref{dirichFix} gives $D^{}_{y^{(n+1)}}(s) = \zeta(s) D^{}_{y^{(n)}}(s-1)$, so $D^{}_{y^{(n)}}(s) = D^{}_{h_n}(s)$ for all $n\geq 1$ and thus $h_n(m) = y^{(n)} (m)$ for all $n,m\geq 1$. For $m = p_1^{r^{}_1}\cdot\ldots\cdot p_{k}^{r^{}_k}$, the coefficient $h_n(m)$ can be written as \[ h_n(m) = \prod_{j=1}^{k}\ \prod_{i=1}^{r_j} \frac{p_j^{n+i-1}-1}{p_j^i-1} = \prod_{j=1}^{k}\gabi{n+r_j-1}{r_j}{p_j} \ts , \] as shown in \cite{Gruber} and \cite{Zou}, leading to the result of Corollary~\ref{y(m)}. This also provides a concrete interpretation of the iteration sequences mentioned earlier. \end{rem} Similarly to calculating $\fix$-eigenvectors $a^{(m)}$ for $m > 1$, the sequences arising from $\fix$-iterations on starting sequences $\delta_m$ with $m > 1$ can be related to the case $m=1$. From $\bigl(\fix(\delta_m)\bigr)_k = m\cdot \delta_{k,jm}$ for some $j\in\NN$, one can conclude inductively \[ \bigl(\fix^{n}(\delta_m)\bigr)_k = \begin{cases} m^{n}\! \cdot y^{(n)}\bigl(\tfrac{k}{m} \bigr), &\text{if $m | k$}; \\ 0, &\text{otherwise.} \end{cases} \] Since arbitrary elements from $\mathcal{A}$ can be written as complex linear combinations of the $\delta_m$, we find the following behaviour of general arithmetic sequences under $\fix$-iterations. \begin{lemma}\label{nthIterationGeneral} Let $f = (f_1,f_2,\ldots)\in\mathcal{A}$. Then, for all $M\geq 1$, \[ \bigl(\fix^n(f)\bigr)_M = \sum_{k=1}^{M} f_k\, \bigl(\fix^n(\delta_k)\bigr)_M = \sum_{k|M} f_k\, k^{n} y^{(n)}\bigl(\tfrac{M}{k}\bigr). \] As a convolution product, this is \[ \fix^n (f) = \bigl(f\cdot N^n \bigr) * y^{(n)}, \quad \text{for all $n\in\NN$}, \] where $N^n$ denotes the arithmetic function defined by $m\mapsto m^n$. \end{lemma} \begin{proof} Note first that, for all $M\geq 1$, the value $\fix(f)_M$ depends on the $f_n$ with $n|M$ only. Each single term of $\fix(f)_M$ can thus be obtained by applying $\fix$ to the finite linear combination $\sum_{k=1}^M f_k \delta_k$, resulting in a component-wise well-defined arithmetic function. The last identity follows from the definition of the convolution product. \end{proof} Given the unbounded spectrum of $\fix$, it is not surprising that the image sequences $\fix^{n}(f)$ of a non-negative $f$ component-wise tend to infinity when $n\to\infty$. Nevertheless, the `quotient sequences' $\left(\frac{\fix^{n+1}(f)_M}{\fix^{n}(f)_M}\right)_n$ have a well-defined convergence behaviour for each $M$ as follows. \begin{theorem} Let $f$ be a non-negative arithmetic function and let $M\in\NN$ be such that $\bigl(\fix^r(f)\bigr)_M \not= 0$ for some $r\in\NN$. Then, \[ \lim_{n\to\infty} \frac{\bigl(\fix^{n+1}(f)\bigr)_{M}} {\bigl(\fix^{n}(f)\bigr)_{M}} = M \ts . \] \end{theorem} \begin{proof} Expanding the quotient in question yields \[ \frac{\bigl(\fix^{n+1}(f)\bigr)_{M}}{\bigl(\fix^{n}(f)\bigr)_{M}} \, = \, \sum_{d|M} d\ \frac{\bigl(\fix^{n}(f)\bigr)_{d}}{\bigl(\fix^{n}(f)\bigr)_{M}} \, = \, M + \!\! \sum_{M> d|M} d\ \frac{\bigl(\fix^{n}(f)\bigr)_{d}}{\bigl(\fix^{n}(f)\bigr)_{M}} . \] The number of terms in the last sum being independent of $n$, it suffices to show that, for each divisor $d$ of $M$, the corresponding summand approaches $0$ as \text{$n\to\infty$.} Let $d^{}_1=1, \ldots, d^{}_R = d$ denote the divisors of $d$. Then, using Lemma~\ref{nthIterationGeneral}, one gets \[ \frac{\bigl(\fix^{n}(f)\bigr)_{d}}{\bigl(\fix^{n}(f)\bigr)_{M}} \, = \, \sum_{i=1}^R \frac{f_{d_i}\ d_i^n\ y^{(n)}\bigl(\frac{d}{d_i}\bigr)}{\sum_{k|M}f_k\, {k}_{}^{n} y^{(n)}\bigl(\frac{M}{k}\bigr)} \,\leq\, \sum_{i=1}^R \frac{d_i^n\ f_{d_i} y^{(n)}(d/d_i)}{d_i^n\ f_{d_i} y^{(n)}(M/d_i)} \, = \, \sum_{i=1}^{R}\frac{y^{(n)}(d/d_i)}{y^{(n)}(M/d_i)} \ts . \] This upper bound can be seen to converge to $0$ by splitting numerator and denominator of each summand in the last sum into the factors that arise from the prime decomposition of $d/d_i$ and $M/d_i$. For $r > s$, the quotient \begin{equation}\label{singleFactor} \frac{y^{(n)}(p^s)}{y^{(n)}(p^r)}= \prod_{i=0}^{s -1}\frac{1-p^{n+s-1-i}}{1-p^{n+r-1-i}} \prod_{i=s}^{r-1}\frac{1-p^{i+1}}{1-p^{n+r-1-i}} \end{equation} converges to $0$ as $n\to\infty$, since the first product tends to $p^{s-r} > 0$, while the second tends to $0$. Being composed of factors such as \eqref{singleFactor}, the quotient $\bigl(\fix^n(f)\bigr)_{d} \, /\bigl(\fix^n(f)\bigr)_M $ tends to $0$ for all $d\ts |M$ as ${n\to\infty}$, which proves the claim. \end{proof} \begin{rem} Unlike $\fix$, the operator $\orb$'s spectrum is bounded by $0$ from below and by $1$ from above. One would expect $\orb^n(f)$ to converge to an element of the eigenspace for the largest eigenvalue ($\lambda = 1$), hence to a scalar multiple of $a^{(1)}=a$. Indeed, employing the eigenvector basis representation $f = \sum_{m\geq 1} \alpha^{}_m a^{(m)}$, one obtains \[ \lim_{n\to\infty} (\orb^n f)^{}_M = \lim_{n\to\infty} \sum_{m=1}^{M} \alpha^{}_m \orb^n(a^{(m)})_M = \lim_{n\to\infty}\sum_{m=1}^{M}\alpha^{}_m \frac{1}{m^n} a^{(m)}_{M}%\\ = \alpha^{}_1 a^{(1)}_M %$ \] for all $M\in\NN$. In other words, $\orb^n(f) \xrightarrow{\; n\to\infty \; } f^{}_1\, a$, for all $f\in\mathcal{A}$. \end{rem} \section{Concluding remarks} The eigenvector $a$ has an interesting consequence in the setting of dynamical (or Artin-Mazur) zeta functions, where one considers the generating function \[ \exp\left(\sum_{m=1}^{\infty} \frac{f_m}{m}\ z^m\right) = \prod_{m=1}^{\infty} (1-z^m)^{-\orb(f)^{}_m}, \] usually for arithmetic functions $f$ that count (isolated) fixed points of a discrete dynamical system, see \cite{Fel} for background material. Using $a$ together with $a=\orb(a)$, one finds the product identity \[ \exp\left(\sum_{m=1}^{\infty} \frac{a_m}{m}\ z^m\right) = \prod_{m=1}^{\infty} (1-z^m)^{-a_m} \] or, equivalently, the series identity \[ \sum_{m=1}^{\infty}\frac{a_m}{m}\, z^m = -\sum_{m=1}^{\infty}a_m\,\log(1-z^m), \] with absolute convergence for $\abs{z} < 1$. It would be interesting to know whether this gives rise to another interpretation of the meaning of the eigenvector $a$\ in this setting. \section{Acknowledgements} It is our pleasure to thank Tom Ward for his interest in this work and to John A.\ G.\ Roberts and Johan Nilsson for useful hints on the manuscript. This work was supported by the German Research Council (DFG), within the CRC 701. \bigskip \begin{thebibliography}{xx} \bibitem{apostol} T.~M.~Apostol, \textit{Introduction to Analytic Number Theory}, Springer, New York, 1976. \bibitem{coincApp} M.~Baake, Solution of the coincidence problem in dimensions $d\le 4$, in R.\ts V.\ Moody, ed., \textit{The Mathematics of Long-Range Aperiodic Order}, NATO-ASI Series C\ts 489, Kluwer, Dordrecht (1997), pp.\ 9--44; revised version available at \href{http://arxiv.org/abs/math/0605222}{\tt http://arxiv.org/abs/math/0605222}. \bibitem{finiteCombRank} G.~Everest, R.~Miles, S.~Stevens and T.~Ward, Dirichlet series for finite combinatorial rank dynamics, to appear, {\it Trans. Amer. Math. Soc.}. Preprint available at \href{http://arxiv.org/abs/0705.1067}{\tt http://arxiv.org/abs/0705.1067}. \bibitem{Fel} A.~Fel'shtyn, \textit{Dynamical Zeta Functions, Nielsen Theory and Reidemeister Torsion}, {\it Memoirs of the American Mathematical Society}, Vol.\ 147, no.\ 699, American Mathematical Society, Providence, RI, 2000. \bibitem{Gruber} B.~Gruber, Alternative formulae for the number of sublattices, {\it Acta Cryst.} \textbf{A53} (1997), 807--808. \bibitem{HardyWright} G.~H.~Hardy and E.~M.~Wright, \textit{An Introduction to the Theory of Numbers}, 5th ed., Clarendon Press, Oxford, 1979. \bibitem{mossDiss} P.~Moss, \textit{Algebraic realizability problems}, PhD thesis, University of East Anglia, Norwich, 2003. \bibitem{NN} N.~Neum\"arker, \textit{Orbitstatistik und relative Realisierbarkeit}, Diplomarbeit, Universit\"at Bielefeld, 2007. \bibitem{purisDiss} Y.~Puri, \textit{Arithmetic and growth of periodic orbits}, PhD thesis, University of East Anglia, Norwich, 2000. \bibitem{PuriWard} Y.~Puri and T.~Ward, \href{http://www.cs.uwaterloo.ca/journals/JIS/VOL4/WARD/short.html}{Arithmetic and growth of periodic orbits}, {\it J.\ Integer Sequences} \textbf{4} (2001), Article 01.2.1. \bibitem{OEIS} N.~J.~A.~Sloane, \textit{The Online Encyclopedia of Integer Sequences}, \newline \href{http://www.research.att.com/~njas/sequences/}{\tt http://www.research.att.com/$\sim$njas/sequences/}. \bibitem{Zou} Y.~M.~Zou, Gaussian binomials and the number of sublattices, {\it Acta Cryst.} \textbf{A62} (2006), 409--410. Preprint available at \href{http://arxiv.org/abs/math/0610684}{\tt http://arxiv.org/abs/math/0610684}. \end{thebibliography} \bigskip \hrule \bigskip \noindent 2000 {\it Mathematics Subject Classification}: Primary 37A45; Secondary 11A25. \noindent \emph{Keywords: } dynamical systems, M\"obius inversion, arithmetic functions, spectra, asymptotics. \bigskip \hrule \bigskip \noindent (Concerned with sequences \seqnum{A000007}, \seqnum{A000012}, \seqnum{A000203}, \seqnum{A001001}, \seqnum{A038991}, \seqnum{A038992}, \seqnum{A038993}, \seqnum{A038994}, \seqnum{A038995}, \seqnum{A038996}, \seqnum{A038997}, \seqnum{A038998}, \seqnum{A038999}, and \seqnum{A153038}.) \bigskip \hrule \bigskip \vspace*{+.1in} \noindent Received December 23 2008; revised version received May 6 2009. Published in {\it Journal of Integer Sequences}, May 12 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}