\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}{-.1in} \setlength{\textheight}{8.4in} \newcommand{\seqnum}[1]{\href{http://oeis.org/#1}{\underline{#1}}} \usepackage{tkz-euclide} \usepackage{tikz-cd} \usepackage{mathtools} \begin{document} \begin{center} \epsfxsize=4in \leavevmode\epsffile{logo129.eps} \end{center} \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{conjecture}[theorem]{Conjecture} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \begin{center} \vskip 1cm{\LARGE\bf Divisors on Overlapped Intervals and\\ \vskip .1in Multiplicative Functions} \vskip 1cm \large Jos\'e Manuel Rodr\'iguez Caballero\\ D\'epartement de Math\'ematiques\\ UQAM\\ Case Postale 8888, Succ.\ Centre-ville \\ Montr\'eal, Qu\'ebec H3C 3P8 \\ Canada\\ \href{mailto:rodriguez\_caballero.jose\_manuel@uqam.ca}{\tt rodriguez\_caballero.jose\_manuel@uqam.ca}\\ \end{center} \def\DD{\mathcal{D}} \def\SS{\mathcal{S}} \vskip .2 in \begin{abstract} Reutenauer and Kassel introduced a family $P_n(q)$ of polynomials defined in terms of divisors of $n$ on overlapped intervals. The evaluation of $P_n(q)$ at roots of unity of order 2, 3, 4, 6 form well-known integer sequences related to the number of integer solutions of the equations $x^2 + y^2 = n$, $x^2 + 2y^2 = n$, and $x^2 + xy + y^2 = n$. Also, $P_n(1)$ is the sum of divisors of $n$. In this paper we define a new family $L_n(q)$ of polynomials defined in terms of divisors of $n$ on overlapped intervals, slightly modifying the definition of $P_n(q)$. The values of $L_n(q)$ at $q = 1$ and $q = -1$ are related to the sum of divisors of $n$ and to the number of integer solutions of the equations $x^2 + xy + y^2 = n$ and $x^2 + 3 y^2 = n$. \end{abstract} \section{Introduction} For a given integer $n \geq 1$, consider the two-sided sequence $$ p_{n,k} = \ln\left(k+\sqrt{k^2 + 2\,n}\right), $$ where $k \in \mathbb{Z}$ and define the intervals $$ \mathcal{P}_{n,k} = \left(p_{n,k}-\ln 2, p_{n,k}\right]. $$ Kassel and Reutenauer \cite{kassel2015counting} introduced the polynomials\footnote{The original definition of $P_n(q)$, which we refer to as \emph{Kassel-Reutenauer polynomials} \cite{kassel2015counting} is rather different, but equivalent, to the one presented here. We preferred to take the logarithm of the divisors in place of the divisors themselves in order to work with intervals $\mathcal{P}_{n,k}$ of constant length.} $$ \frac{P_n(q)}{q^{n-1}} = \sum_{d|n}\sum_{k\in \mathbb{Z}} \boldsymbol{1}_{\mathcal{P}_{n,k}}\left(\ln d\right) \, q^k,$$ where $\boldsymbol{1}_A(x)$ is the characteristic function of the set $A$, i.e., $\boldsymbol{1}_A(x) = 1$ if $x \in A$, otherwise $\boldsymbol{1}_A(x) = 0$. Each polynomial $P_n(q)$ is monic of degree $2n-2$, its coefficients are non-negative integers and it is self-reciprocal \cite{kassel2016complete}. The evaluations of $P_n(q)$ at some complex roots of $1$ have number-theoretical interpretations \cite{kassel2016complete}, e.g., \begin{eqnarray*} \sigma(n) &=& P_n(1), \\ \frac{r_{1,0,1}(n)}{4} &=& P_n(-1), \\ \frac{r_{1,0,2}(n)}{2} &=& \left| P_n\left(\sqrt{-1}\right)\right|, \\ \frac{r_{1,1,1}(n)}{6} &=& \textrm{Re } P_n\left( \frac{-1+\sqrt{-3}}{2}\right), \end{eqnarray*} where $\sigma(n)$, $\frac{r_{1,0,1}(n)}{4}$, $\frac{r_{1,0,2}(n)}{2}$ and $\frac{r_{1,1,1}(n)}{6}$ are multiplicative functions \cite{lorenz1871bidrag} given by \begin{eqnarray*} \sigma(n) &=& \sum_{d|n} d, \\ r_{a,b,c}(n) &=& \# \left\{(x,y)\in \mathbb{Z}^2: \quad a\,x^2 + b\,x\,y + c\,y^2 = n \right\}. \end{eqnarray*} Furthermore, for $q = \frac{1+\sqrt{-3}}{2}$, the same sequence $n \mapsto P_n(q)$ is related to $r_{1,0,1}(n)$ in three ways \cite{kassel2016fourier}, depending on the congruence class of $n$ in $\mathbb{Z}/3\mathbb{Z}$, $$ \left|P_n\left(\frac{1+\sqrt{-3}}{2} \right)\right| = \begin{cases} r_{1,0,1}(n), & \text{if $n \equiv 0 \pmod{3}$;} \\ \frac{1}{4}\,r_{1,0,1}(n), & \text{if $n \equiv 1 \pmod{3}$;} \\ \frac{1}{2}\,r_{1,0,1}(n), & \text{if $n \equiv 2 \pmod{3}$.} \end{cases} $$ For any integer $n \geq 1$, consider the two-sided sequence $$ \ell_{n,k} = \ln\left( \tfrac{3}{2}\,k+\sqrt{\left(\tfrac{3}{2}\,k \right)^2 + 3\,n} \right) $$ and the intervals $$\mathcal{L}_{n,k} = \left(\ell_{n,k}-\ln 3,\ell_{n,k}\right],$$ where $k$ runs over the integers. Define a variation of the polynomials $P_n(q)$ as follows: $$ \frac{L_n(q)}{q^{n-1}} = \sum_{d|n}\sum_{k\in \mathbb{Z}} \boldsymbol{1}_{\mathcal{L}_{n,k}}\left(\ln d\right) \, q^k. $$ For example, in order to compute $L_6(q)$ from the definition, we need to consider the intervals $\left(\ell_{6,k}-\ln 3, \ell_{6,k}\right]$ on the real line and to count the number of values of $\ln d$ inside each interval, where $d$ runs over the divisors of $n$. These data are shown in Figure \ref{Fig:1}, where the numbers $\ell_{6,k}$ are plotted on the line below (the corresponding values of $k$ are labelled) whereas the numbers $\ln d$ are plotted on the line above (the corresponding values of $d$ are labelled). Counting the number of intersections between the horizontal and the vertical lines, we obtain that the coefficients of $\frac{L_{6}(q)}{q^{6-1}}$ are as follows: $$ \frac{L_{6}(q)}{q^{6-1}} = q^{5} + q^{4} + q^{3} + 2 \, q^{2} + 2 \, q + 2\,q^0 + 2\,q^{-1} + 2 \,q^{-2} + q^{-3} + q^{-4} + q^{-5}. $$ \begin{figure} \label{Fig:1} \begin{center} \begin{tikzpicture}[scale=2] \draw[->, thick] (-1.5,0) -- (3.5,0) node[below] {$\ell_{6,k}$}; \draw[->, thick] (-1.5,1.2) -- (3.5,1.2) node[below] {$\ln d$}; \foreach \x/\xtext in {-0.051425544115108225/-6, 0.11050682349650519/-5, 0.298970044167493/-4, 0.5215628360911632/-3, 0.7867069304856741/-2, 1.0986122886681098/-1, 1.4451858789480825/0, 1.791759469228055/1, 2.1036648274104905/2, 2.3688089218050012/3, 2.591401713728671/4, 2.7798649343996593/5, 2.941797302011273/6 } { \draw[dashed] (\x,7/12+\xtext/12) -- (\x,0) node[below] {\xtext}; \draw (\x-1.09861228867,7/12+\xtext/12) -- (\x,7/12+\xtext/12) ; \draw[black, fill = white] (\x-1.09861228867,7/12+\xtext/12) circle (0.03cm) ; \draw[black, fill = black] (\x,7/12+\xtext/12) circle (0.03cm) ; } \draw[->, thick] (-1.5,0) -- (3.5,0); \foreach \x/\xtext in {0.0/1, 0.6931471805599453/2, 1.0986122886681098/3, 1.791759469228055/6} { % \draw[thick] (\x,0) -- (\x,\xtext/12); \draw[black,fill=gray] (\x,1.2) circle (0.03cm); \draw[gray, thick] (\x,0) -- (\x, 1.2) node[black, above] {$\xtext$}; } %\draw (0.2,0.5pt) node[above] {$c$}; %\draw[[-, ultra thick, black] (0,0) -- (0.01,0); %\draw[-), ultra thick, black] (0.19,0) -- (0.2,0); %\fill[opacity = 0.2, blue,rounded corners=1ex] (0,-.16ex) -- (0.2, -.16ex) -- (0.2, .16ex) -- (0,.16ex) -- cycle; \end{tikzpicture} \end{center} \caption{Representation of $L_6(q)$.} \end{figure} Like $P_n(q)$, the polynomial $L_n(q)$ is monic of degree $2\,n-2$, self-reciprocal and its coefficients are non-negative integers. The aim of this paper is to express the multiplicative functions \cite[p.\ 421]{lorenz1871bidrag} $\frac{r_{1,1,1}(n)}{6}$ and $\frac{r_{1,0,3}(n)}{2}$ in terms of the evaluations of $L_n(q)$ at roots of the unity. More precisely, we will prove the following result. \begin{theorem} \label{Theo349jr9834j9} For each $n \geq 1$, \begin{eqnarray} \seqnum{A002324}(n) &:=& \frac{r_{1,1,1}(n)}{6} = 4\,\sigma(n) - 3\,L_n\left( 1\right), \label{Idenr8934jr9j349}\\ \seqnum{A096936}(n) &:=& \frac{r_{1,0,3}(n)}{2} = L_n\left( -1\right).\label{Idenr438j98j43sdf} \end{eqnarray} \label{one} \end{theorem} \section{Auxiliary results for the first identity of Theorem~\ref{one}} For any $n \geq 1$, we will use the notation $$ d_{a,m}(n) := \#\left\{d|n: \quad d \equiv a \pmod{m} \right\}. $$ We will use the following well-known result \cite{hirschhorn2001three}. \begin{lemma}\label{Lemur439r9834hr83} For all integers $n \geq 1$, $$ \frac{r_{1,1,1}(n)}{6} = d_{1,3}(n)-d_{2,3}(n). $$ \end{lemma} \begin{lemma}\label{Lem493j9rj49r3r} For any integer $n \geq 1$, \begin{eqnarray*} 3\,\left\lceil 3^{-1}\,n \right\rceil - n &=& \begin{cases} 0, & \text{if $n\equiv 0 \pmod{3}$;} \\ 2, & \text{if $n\equiv 1 \pmod{3}$;} \\ 1, & \text{if $n\equiv 2 \pmod{3}$.} \end{cases} \\ n - 3\,\left\lfloor 3^{-1}\,n \right\rfloor &=& \begin{cases} 0, & \text{if $n\equiv 0 \pmod{3}$;} \\ 1, & \text{if $n\equiv 1 \pmod{3}$;} \\ 2, & \text{if $n\equiv 2 \pmod{3}$.} \end{cases} \end{eqnarray*} \end{lemma} \begin{proof} It is enough to evaluate $3\,\left\lceil 3^{-1}\,n \right\rceil - n$ and $3\,\left\lceil 3^{-1}\,n \right\rceil - n$ at $n = 3k+r$, for $k \in \mathbb{Z}$ and $r \in \{0, 1, 2\}$. \end{proof} \begin{lemma}\label{Lem98ej398jr934jf9ww} For any pair of integers $n \geq 1$ and $k$, the inequalities $$ \ell_{n,k} - \ln 3 < \ln d \leq \ell_{n,k} $$ hold if and only if the inequalities $$ 3^{-1}\,d - \frac{n}{d}\leq k < d - 3^{-1}\,\frac{n}{d} $$ hold. \end{lemma} \begin{proof} The inequalities $$ \ell_{n,k} - \ln 3 < \ln d \leq \ell_{n,k} $$ are equivalent to $$ \ln d \leq \ell_{n,k} < \ln d + \ln 3. $$ Applying the strictly increasing function $x \mapsto \frac{e^x}{3} - n \, e^{-x}$ to the last inequalities we obtain the following equivalent inequalities $$ 3^{-1}\,d - \frac{n}{d}\leq k < d - 3^{-1}\,\frac{n}{d}. $$ Indeed, $\frac{e^{\ln d}}{3} - n \, e^{-\ln d} = 3^{-1}\,d - \frac{n}{d}$, $\frac{e^{\ln d + \ln 3}}{3} - n \, e^{-\left(\ln d + \ln 3\right)} = d - 3^{-1}\frac{n}{d}$ and \begin{eqnarray*} \frac{e^{\ell_{n,k}}}{3} - n \, e^{-\ell_{n,k}} &=& \frac{\tfrac{3}{2}\,k+\sqrt{\left(\tfrac{3}{2}\,k \right)^2 + 3\,n}}{3} - \frac{n}{\tfrac{3}{2}\,k+\sqrt{\left(\tfrac{3}{2}\,k \right)^2 + 3\,n}} \\ &=& \frac{\tfrac{3}{2}\,k+\sqrt{\left(\tfrac{3}{2}\,k \right)^2 + 3\,n}}{3} + \frac{\tfrac{3}{2}\,k-\sqrt{\left(\tfrac{3}{2}\,k \right)^2 + 3\,n}}{3} \\ &=& k. \end{eqnarray*} So the lemma is proved. \end{proof} \begin{lemma}\label{lem4u98j39j9} Let $n \geq 1$ be an integer. For all $d | n$, $$ \sum_{k\in \mathbb{Z}} \boldsymbol{1}_{\mathcal{L}_{n,k}}\left(\ln d\right) = \left\lceil d - 3^{-1}\,\frac{n}{d} \right\rceil - \left\lceil 3^{-1} \, d - \frac{n}{d}\right\rceil. $$ \end{lemma} \begin{proof} For all integers $n \geq 1$ and $k$, we have that \begin{eqnarray*} & & \sum_{k\in \mathbb{Z}} \boldsymbol{1}_{\mathcal{L}_{n,k}}\left(\ln d\right) \\ &=& \# \left\{k \in \mathbb{Z}: \quad \ell_{n,k} - \ln 3 < \ln d \leq \ell_{n,k} \right\} \\ &=& \# \left\{k \in \mathbb{Z}: \quad 3^{-1}\,d - \frac{n}{d}\leq k < d - 3^{-1}\,\frac{n}{d}\right\} \quad (\textrm{Lemma } \ref{Lem98ej398jr934jf9ww}) \\ &=& \# \left\{k \in \mathbb{Z}: \quad \left\lceil 3^{-1}\,d - \frac{n}{d} \right\rceil \leq k < \left\lceil d - 3^{-1}\,\frac{n}{d}\right\rceil\right\} \\ &=& \left\lceil d - 3^{-1}\,\frac{n}{d} \right\rceil - \left\lceil 3^{-1} \, d - \frac{n}{d}\right\rceil. \end{eqnarray*} So the lemma is proved. \end{proof} \section{Auxiliary results for the second identity of Theorem~\ref{one}} We will use the following well-known result \cite[p.\ 421]{lorenz1871bidrag}. \begin{lemma}\label{Lemr89j983jr9} The function $\frac{r_{1,0,3}(n)}{2}$ is multiplicative. \end{lemma} We will use the following well-known result \cite{hirschhorn2001three}. \begin{lemma}\label{Lemurf93wenuf9h2938} For all integers $n \geq 1$, $$ \frac{r_{1,0,3}(n)}{2} = d_{1,3}(n)-d_{2,3}(n) + 2\left(d_{4,12}(n)-d_{8,12}(n) \right). $$ \end{lemma} Recall that the nonprincipal Dirichlet character mod $3$ is the $3$-periodic arithmetic function $\chi_3(n)$ given by $\chi_3(0) = 0$, $\chi_3(1) = 1$ and $\chi_3(2) = -1$. \begin{lemma}\label{Lemrj893j9jewrefef54f} For all $n \geq 1$, $$ \frac{\left( -1\right)^{\left\lfloor 3^{-1}\,n \right\rfloor} - \left( -1\right)^{\left\lceil 3^{-1} \, n \right\rceil}}{2} = (-1)^{n-1}\,\chi_3(n). $$ \end{lemma} \begin{proof} It is enough to substitute $n = 3k+r$, with $k\in \mathbb{Z}$ and $r \in \{0, 1, 2\}$, in both sides in order to check that they are equal. \end{proof} \begin{lemma}\label{Lem7g76g78iuho8h9h98} For all $n \geq 1$, $$ \sum_{d | n} \left( -1\right)^{\frac{n}{d}-1}\, \left(-1\right)^{d-1}\,\chi_3(d) = \left( -1\right)^{n-1}\,\frac{r_{1,0,3}(n)}{2}. $$ \end{lemma} \begin{proof} By Lemma \ref{Lemr89j983jr9}, the function $ \frac{r_{1,0,3}(n)}{2}$ is multiplicative. Also, it is easy to check that the functions $\left( -1\right)^{n-1}$ and $\chi_3(n)$ are multiplicative. So the functions $f(n) = \left( -1\right)^{n-1}\,\frac{r_{1,0,3}(n)}{2}$ and $\left( -1\right)^{n-1} \, \chi_3(n)$ are multiplicative, because the multiplicative property is preserved by ordinary product. The function $g(n) = \sum_{d | n} \left( -1\right)^{\frac{n}{d}-1}\, \left(-1\right)^{d-1}\,\chi_3(d)$ is multiplicative, because Dirichlet convolution preserves the multiplicative property. So it is enough to prove that $f\left(p^k\right) = g\left(p^k\right)$ for each prime power $p^k$. Considering the case $p = 2$. The following elementary equivalences hold for any integer $m \geq 0$, \begin{eqnarray*} 2^{m} \equiv 1 \pmod{3} &\Longleftrightarrow & m \equiv 0 \pmod{2}, \\ 2^{m} \equiv 2 \pmod{3} &\Longleftrightarrow & m \equiv 1 \pmod{2}, \\ 2^{m} \equiv 4 \pmod{12} &\Longleftrightarrow & m \equiv 0 \pmod{2} \textrm{ and } m \neq 0, \\ 2^{m} \equiv 8 \pmod{12} &\Longleftrightarrow & m \equiv 1 \pmod{2}\textrm{ and } m \neq 1. \end{eqnarray*} So, for each integer $k \geq 1$, \begin{eqnarray*} d_{1,3}\left( 2^k\right) &=& \# [0,k] \cap 2\mathbb{Z} = \left\lfloor\frac{k}{2}\right\rfloor + 1, \\ d_{2,3}\left( 2^k\right) &=& \# [1,k] \cap \left(2\mathbb{Z}+1\right) = \left\lceil\frac{k}{2}\right\rceil, \\ d_{4,12}\left( 2^k\right) &=& \# [2,k] \cap 2\mathbb{Z} = \left\lfloor\frac{k}{2}\right\rfloor, \\ d_{8,12}\left( 2^k\right) &=& \# [3,k] \cap \left(2\mathbb{Z}+1\right) = \left\lceil\frac{k}{2}\right\rceil - 1. \end{eqnarray*} For any $k \geq 1$, it follows that \begin{eqnarray*} g\left( 2^k\right) &=& \sum_{j=0}^k \left( -1\right)^{2^{k-j}-1}\, \left(-1\right)^{2^j-1}\,\chi_3\left(2^j \right) \\ &=& \sum_{j=0}^k \left( -1\right)^{2^{k-j}-1}\, \left(-1\right)^{2^j-1}\,\left(-1 \right)^j \\ &=& -1 - \left(-1 \right)^k + \sum_{j=1}^{k-1} \left(-1 \right)^j \\ &=& -1 - \left(-1 \right)^k + \frac{-1 - \left(-1 \right)^k}{2} \\ &=& -3\,\frac{1 + \left(-1 \right)^k}{2} \\ &=& -3 \,\left( 1+ \left\lfloor\frac{k}{2}\right\rfloor -\left\lceil\frac{k}{2}\right\rceil \right) \\ &=& -\left(\left( \left\lfloor\frac{k}{2}\right\rfloor + 1\right)-\left\lceil \frac{k}{2}\right\rceil + 2\left(\left\lfloor\frac{k}{2}\right\rfloor-\left( \left\lceil\frac{k}{2}\right\rceil - 1\right) \right) \right) \\ &=& \left(-1 \right)^{2^k - 1} \, \left(d_{1,3}\left( 2^k\right)-d_{2,3}\left( 2^k\right) + 2\left(d_{4,12}\left( 2^k\right)-d_{8,12}\left( 2^k\right) \right) \right) \\ &=& f\left( 2^k\right) \quad (\textrm{Lemma } \ref{Lemurf93wenuf9h2938}). \end{eqnarray*} Let $p$ and $k \geq 1$ be an odd prime and an integer respectively. Noticing that $\left(-1\right)^{p^j-1} = 1$ for all $0 \leq j \leq k$. Also, $d_{4,12}\left( p^k\right) = d_{8,12}\left( p^k\right) = 0$, because $p^k$ has not even divisor. So, for any $k \geq 1$, \begin{eqnarray*} g\left( p^k\right) &=& \sum_{j=0}^k \left( -1\right)^{p^{k-j}-1}\, \left(-1\right)^{p^j-1}\,\chi_3\left(p^j \right) \\ &=& \sum_{j=0}^k \chi_3\left(p^j \right) \\ &=& d_{1,3}\left( p^k\right)-d_{2,3}\left( p^k\right) \\ &=& \left( -1\right)^{p^k - 1} \,\left( d_{1,3}\left( p^k\right)-d_{2,3}\left( p^k\right) + 2\left(d_{4,12}\left( p^k\right)-d_{8,12}\left( p^k\right) \right) \right) \\ &=& f\left( p^k\right) \quad (\textrm{Lemma } \ref{Lemurf93wenuf9h2938}). \end{eqnarray*} Therefore, $f(n) = g(n)$ for all $n \geq 1$. \end{proof} \begin{lemma}\label{Lemm9348fj9438jhf9h39w} For each $d|n$, $$ \sum_{k\in \mathbb{Z}} \boldsymbol{1}_{\mathcal{L}_{n,k}}\left(\ln d\right) \, \left( -1\right)^k = \frac{1}{2} \, \left(\left( -1\right)^{\left\lceil 3^{-1} \, d - \frac{n}{d}\right\rceil} - \left( -1\right)^{\left\lceil d - 3^{-1}\,\frac{n}{d} \right\rceil} \right). $$ \end{lemma} \begin{proof} For any integer $n \geq 1$ and any $d|n$, \begin{eqnarray*} \sum_{k\in \mathbb{Z}} \boldsymbol{1}_{\mathcal{L}_{n,k}}\left(\ln d\right) \, \left( -1\right)^k &=& \sum_{3^{-1}\,d - \frac{n}{d}\leq k < d - 3^{-1}\,\frac{n}{d}} \left( -1\right)^k \quad (\textrm{Lemma } \ref{Lem98ej398jr934jf9ww}) \\ &=& \sum_{\lceil 3^{-1}\,d - \frac{n}{d} \rceil \leq k < \lceil d - 3^{-1}\,\frac{n}{d}\rceil} \left( -1\right)^k. \end{eqnarray*} Substituting $a = \lceil 3^{-1}\,d - \frac{n}{d} \rceil$, $b = \lceil d - 3^{-1}\,\frac{n}{d}\rceil$ and $q = -1$ in the geometric sum $$ \sum_{a \leq k < b} q^k = \frac{q^a - q^b}{1-q} $$ we obtain \begin{eqnarray*} \sum_{k\in \mathbb{Z}} \boldsymbol{1}_{\mathcal{L}_{n,k}}\left(\ln d\right) \, \left( -1\right)^k &=& \frac{\left( -1\right)^{\left\lceil 3^{-1} \, d - \frac{n}{d}\right\rceil} - \left( -1\right)^{\left\lceil d - 3^{-1}\,\frac{n}{d} \right\rceil}}{1 - (-1)} \\ &=& \frac{1}{2} \, \left(\left( -1\right)^{\left\lceil 3^{-1} \, d - \frac{n}{d}\right\rceil} - \left( -1\right)^{\left\lceil d - 3^{-1}\,\frac{n}{d} \right\rceil} \right). \end{eqnarray*} So the lemma is proved. \end{proof} \section{Proof of the main result} We proceed now with the proof of the main result of this paper. \begin{proof}[Proof of Theorem \ref{Theo349jr9834j9}] Identity \eqref{Idenr8934jr9j349} follows from the following transformations, \begin{eqnarray*} L_n(1) &=& \sum_{d | n} \sum_{k\in \mathbb{Z}} \boldsymbol{1}_{\mathcal{L}_{n,k}}\left(\ln d\right) \\ &=& \sum_{d | n} \left( \left\lceil d - 3^{-1}\,\frac{n}{d} \right\rceil - \left\lceil 3^{-1} \, d - \frac{n}{d}\right\rceil \right) \quad (\textrm{Lemma }\ref{lem4u98j39j9}) \\ &=& \sum_{d | n} \left( d + \frac{n}{d} \right) + \sum_{d | n}\left\lceil -3^{-1}\,\frac{n}{d} \right\rceil - \sum_{d | n} \left\lceil 3^{-1}d \right\rceil \\ &=& \sum_{d | n} \left( d + \frac{n}{d} \right) - \sum_{d | n} \left\lfloor 3^{-1}\,\frac{n}{d} \right\rfloor - \sum_{d | n} \left\lceil 3^{-1}d \right\rceil \end{eqnarray*} \begin{eqnarray*} &=& \frac{2}{3}\sum_{d | n} \left( d + \frac{n}{d} \right) + \frac{1}{3}\sum_{d | n} \left(\frac{n}{d} - 3\,\left\lfloor 3^{-1}\,\frac{n}{d} \right\rfloor \right) - \frac{1}{3}\sum_{d | n} \left( 3\,\left\lceil 3^{-1}d \right\rceil - d \right) \\ &=& \frac{4\sigma(n)}{3} + \frac{d_{1,3}(n) + 2d_{2,3}(n) }{3} - \frac{2d_{1,3}(n) + d_{2,3}(n)}{3} \quad (\textrm{Lemma } \ref{Lem493j9rj49r3r}) \\ &=& \frac{4\sigma(n)}{3} - \frac{d_{1,3}(n) - d_{2,3}(n)}{3} \\ &=& \frac{4}{3}\,\sigma(n) - \frac{1}{3} \, \frac{r_{1,1,1}(n)}{6} \quad (\textrm{Lemma } \ref{Lemur439r9834hr83}). \end{eqnarray*} Identity \eqref{Idenr438j98j43sdf} follows from the following transformations, \begin{eqnarray*} \frac{L_n\left( -1\right)}{\left( -1\right)^{n-1}} &=& \sum_{d | n} \sum_{k\in \mathbb{Z}} \boldsymbol{1}_{\mathcal{L}_{n,k}}\left(\ln d\right) \, \left( -1\right)^k \\ &=& \sum_{d | n} \frac{1}{2} \, \left(\left( -1\right)^{\left\lceil 3^{-1} \, d - \frac{n}{d}\right\rceil} - \left( -1\right)^{\left\lceil \frac{n}{d} - 3^{-1}\,d \right\rceil} \right)\quad (\textrm{Lemma } \ref{Lemm9348fj9438jhf9h39w}) \\ &=& \sum_{d | n} \frac{1}{2} \, \left(\left( -1\right)^{\left\lceil 3^{-1} \, d \right\rceil - \frac{n}{d}} - \left( -1\right)^{\frac{n}{d} - \left\lfloor 3^{-1}\,d \right\rfloor} \right) \\ &=& \sum_{d | n} \left( -1\right)^{\frac{n}{d}-1}\, \frac{\left( -1\right)^{\left\lfloor 3^{-1}\,d \right\rfloor} - \left( -1\right)^{\left\lceil 3^{-1} \, d \right\rceil}}{2} \\ &=& \sum_{d | n} \left( -1\right)^{\frac{n}{d}-1}\, \left(-1\right)^{d-1}\,\chi_3(d) \quad (\textrm{Lemma } \ref{Lemrj893j9jewrefef54f}) \\ &=& \left( -1\right)^{n-1}\,\frac{r_{1,0,3}(n)}{2} \quad (\textrm{Lemma } \ref{Lem7g76g78iuho8h9h98}). \end{eqnarray*} So the theorem is proved. \end{proof} \section{Final remarks} \begin{enumerate} \item Let $k$ be a field and $\mathcal{R}$ be a $k$-algebra. The \emph{codimension} of an ideal $I$ of $\mathcal{R}$ is the dimension of the quotient $\mathcal{R}/I$ as a vector space over $k$. We let $\mathbb{Z}\oplus \mathbb{Z}$ denote the free abelian group of rank $2$. Let $k = \mathbb{F}_q$ be the finite field with $q$ elements and $\mathcal{R} = \mathbb{F}_q\left[\mathbb{Z}\oplus \mathbb{Z}\right]$ be its group algebra. Kassel and Reutenauer \cite{kassel2015counting} proved that, for any prime power $q$, the number of ideals of codimension $n \geq 1$ of $\mathbb{F}_q\left[\mathbb{Z}\oplus \mathbb{Z}\right]$ is $\left( q-1\right)^2 \, P_n(q)$. So it is natural to look for connections between the values of $L_n(q)$, when $q$ is a prime power, and the algebraic structures related to $\mathbb{F}_q$. \item The polynomials $P_n(q)$ are generated by the product \cite{kassel2016complete} $$ \prod_{m \geq 1}\frac{\left(1-t^m\right)^2}{\left(1-q\, t^m\right)\left(1-q^{-1}\, t^m\right)} = 1 + \left( q + q^{-1} - 2\right) \,\sum_{n=1}^{\infty} \frac{P_n(q)}{q^{n-1}} \, t^n. $$ It would be interesting to find a similar generating function for $L_n(q)$. \end{enumerate} \section{Acknowledgments} The author deeply thanks Prof. S. Brlek and Prof. C. Reutenauer for reading carefully the paper and for providing useful comments. The author gratefully acknowledges the referee. \begin{thebibliography}{9} \bibitem{hirschhorn2001three} M. D. Hirschhorn, Three classical results on representations of a number, in D. Foata and G.-N. Han, eds., \emph{The Andrews Festschrift}, Springer, 2001, pp.~159--165. \bibitem{kassel2015counting} C. Kassel and C. Reutenauer, Counting the ideals of given codimension of the algebra of Laurent polynomials in two variables, \emph{Michigan Math. J.}, to appear. \bibitem{kassel2016complete} C. Kassel and C. Reutenauer, Complete determination of the zeta function of the Hilbert scheme of $ n $ points on a two-dimensional torus, \emph{Ramanujan J.}, to appear. \bibitem{kassel2016fourier} C. Kassel and C. Reutenauer, The Fourier expansion of $\eta(z)$ $\eta(2z)$ $\eta(3z)$ $/$ $\eta(6z)$, {\em Archiv Math.} {\bf 108} (2017), 453--463. \bibitem{lorenz1871bidrag} L. V. Lorenz, \emph{\OE{}uvres scientifiques de L. Lorenz}, Vol.~2, Lehmann \& Stage, 1899. \bibitem{sloane} N. J. A. Sloane et al., The On-Line Encyclopedia of Integer Sequences, available at \url{http://oeis.org}. \end{thebibliography} \bigskip \hrule \bigskip \noindent 2010 {\it Mathematics Subject Classification}: Primary 11A25; Secondary 11A07, 11A05. \noindent \emph{Keywords: } multiplicative function, representation by quadratic form, polynomial, divisor. \bigskip \hrule \bigskip \noindent (Concerned with sequences \seqnum{A002324} and \seqnum{A096936}.) \bigskip \hrule \bigskip \vspace*{+.1in} \noindent Received September 16 2017; revised versions received October 13 2017; October 21 2017. Published in {\it Journal of Integer Sequences}, October 29 2017. \bigskip \hrule \bigskip \noindent Return to \htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}. \vskip .1in \end{document}