\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://oeis.org/#1}{\underline{#1}}} \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 Dedekind Sums with Arguments \\ \vskip .1in Near Euler's Number $e$ } \vskip 1cm \large Kurt Girstmair\\ Institut f\"ur Mathematik \\ Universit\"at Innsbruck \\ Technikerstr.\ 13/7 \\ A-6020 Innsbruck \\ Austria \\ \href{mailto:Kurt.Girstmair@uibk.ac.at}{\tt Kurt.Girstmair@uibk.ac.at}\\ \end{center} \def\BE{\begin{equation}} \def\EE{\end{equation}} \def\BD{\begin{displaymath}} \def\ED{\end{displaymath}} \def\BEA{\begin{eqnarray*}} \def\EEA{\end{eqnarray*}} \def\BI{\bibitem} \def\Z{\mathbb Z} \def\Q{\mathbb Q} \def\R{\mathbb R} \def\C{\mathbb C} \def\MB{\mbox} \def\LD{\ldots} \def\BQ{``} \def\EQ{'' } \def\EQP{''} \def\DED{Dedekind } \vskip .2 in \begin{abstract} We study the asymptotic behaviour of the classical Dedekind sums $s(m/n)$ for convergents $m/n$ of $e$, $e^2$, and $(e+1)/(e-1)$, where $e=2.71828\LD$ is Euler's number. Our main tool is the Barkan-Hickerson-Knuth formula, which yields a precise description of what happens in all cases. \end{abstract} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction and results} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \DED sums have quite a number of interesting applications in analytic number theory (modular forms), algebraic number theory (class numbers), lattice point problems and algebraic geometry (for instance \cite{{Ap}, {Me}, {RaGr}, {Ur}}). Let $n$ be a positive integer and $m\in \Z$, $(m,n)=1$. The classical \DED sum $s(m/n)$ is defined by \BD s(m/n)=\sum_{k=1}^n ((k/n))((mk/n)) \ED where $((\LD))$ is the usual sawtooth function (for example, \cite[p.\ 1]{RaGr}). In the present setting it is more natural to work with \BD S(m/n)=12s(m/n) \ED instead. In the previous paper \cite{Gi} we used the Barkan-Hickerson-Knuth-formula to study the asymptotic behaviour of $S(s_k/t_k)$ for the convergents $s_k/t_k$ of a periodic simple continued fraction $\alpha=[a_0,a_1, a_2,\LD]$, i. e., for a quadratic irrational $\alpha$. In this situation two cases are possible: The sequence $S(s_k/t_k)$ either remains bounded with a finite number of cluster points or it essentially behaves like $C\cdot k$ for some constant $C$ depending on $\alpha$. In the latter case $S(s_k/t_k)-C\cdot k$ remains bounded with finitely many cluster points. The former case occurs, for instance, if the period length of $\alpha$ is odd. Since the order of magnitude of $|S(m/n)|$ is $\log^2 n$ on average \cite{GiSch}, quadratic irrationalities produce \DED sums of a considerably smaller size. In fact, the inequality $k\le 2\log t_k/\log 2 + 1$ was already proved in 1841 \cite{Sh}. Accordingly, if $|S(s_k/t_k)|$ is not bounded, we have $|S(s_k/t_k)|=O(\log t_k)$ for a quadratic irrational $\alpha$. Because the structure of the continued fraction expansions of transcendental numbers like $e$ or $e^2$ is similar to that of quadratic irrationals \cite[p.\ 123 ff.]{Pe}, nothing prevents us from applying the Barkan-Hickerson-Knuth-formula ((\ref{2.1}) below) to these cases. It turns out that the asymptotic behaviour of \DED sums is quite similar to the case of quadratic irrationals. Only the case \BQ$S(s_k/t_k)$ bounded\EQ cannot occur, as the said formula shows, since the continued fraction expansions of these numbers have unbounded digits. We shall show \begin{theorem} % Theorem 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \label{t1} For a nonnegative integer $k$ put \BD L(k)=\begin{cases} \frac k3, & \MB{if } k\equiv{0,1,5}\pmod 6; \\ -\frac k3, & \MB{otherwise.}\rule{0pt}{14pt} \end{cases} \ED Then we have, for the convergents $s_k/t_k$ of Euler's number $e$, \BD S(s_k/t_k)-L(k)=O\left(\frac 1k\right)+\begin{cases} e-3, &\MB{ if }k\equiv 3\pmod 6;\\ e-3-\frac 56, &\MB{ if } k\equiv 1\pmod 6; \rule{0pt}{13pt}\\ e-3+\frac 23, &\MB{ if }k\equiv 2\pmod 6; \rule{0pt}{13pt}\\ e-3+\frac 56, &\MB{ if } k\equiv 4\pmod 6; \rule{0pt}{13pt}\\ e-3-\frac 23, &\MB{ if }k\equiv 5\pmod 6. \rule{0pt}{13pt} \end{cases} \ED \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% The continued fraction expansion of $e^2=7.38905\LD$ is more complicated than that of $e$. This has the effect that the analogue of Theorem \ref{t1} also looks more complicated. We obtain \begin{theorem} % Theorem 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \label{t2} For a nonnegative integer $k$ put \BD L(k)=\begin{cases} -\frac{3k}5, & \MB{if }k\equiv {1,2,3}\pmod{10}; \\ \frac{3k}5, & \MB{if } k\equiv{6,7,8} \pmod{10}; \rule{0pt}{14pt}\\ -\frac{6k}5, & \MB{if }k\equiv{0,4}\pmod{10}; \rule{0pt}{14pt}\\ \frac{6k}5, & \MB{if }k\equiv{5,9}\pmod{10}. \rule{0pt}{14pt} \end{cases} \ED Then we have, for the convergents $s_k/t_k$ of the number $e^2$, \BD S(s_k/t_k)-L(k)=O\left(\frac 1k\right)+\begin{cases} e^2-7, &\MB{ if } k\equiv0\pmod{10};\\ e^2-\frac{37}5, & \MB{ if }k\equiv{1}\pmod{10};\rule{0pt}{13pt}\\ e^2-\frac{29}5, &\MB{ if } k\equiv{2}\pmod{10}; \rule{0pt}{13pt}\\ e^2-\frac{31}5+\frac 12, &\MB{ if } k\equiv 3\pmod{10}; \rule{0pt}{13pt}\\ e^2-\frac{16}5, &\MB{ if } k\equiv 4\pmod{10}; \rule{0pt}{13pt}\\ e^2+1, &\MB{ if } k\equiv 5\pmod{10};\rule{0pt}{13pt}\\ e^2+\frac 75, &\MB{ if } k\equiv 6\pmod{10}; \rule{0pt}{13pt}\\ e^2-\frac 15, &\MB{ if } k\equiv 7\pmod{10};\rule{0pt}{13pt}\\ e^2-\frac 45+\frac 12, &\MB{ if } k\equiv 8\pmod{10}; \rule{0pt}{13pt}\\ e^2-\frac{14}5, &\MB{ if } k\equiv 9\pmod{10}. \rule{0pt}{13pt}\\ \end{cases} \ED \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Finally, we consider the case of $e^*=(e+1)/(e-1)$, which is fairly simple. \begin{theorem} % Theorem 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \label{t3} For a nonnegative integer $k$ put \BD L(k)=\begin{cases} -2k, & \MB{ if }k \MB{ is even;} \\ 2k, & \MB{ if }k \MB{ is odd.}\rule{0pt}{14pt} \end{cases} \ED Then we have, for the convergents $s_k/t_k$ of $e^*$, \BD S(s_k/t_k)-L(k)=O\left(\frac 1k\right)+\begin{cases} e^*-2, &\MB{ if }k\MB{ is even;}\\ e^*-1, &\MB{ if }k\MB{ is odd.} \rule{0pt}{13pt} \end{cases} \ED \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Proofs} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% We start with the continued fraction expansion $[a_0,a_1,a_2,\LD]$ of an arbitrary irrational number. The numerators and denominators of its convergents $s_k/t_k$ are defined by the recursion formulas \begin{eqnarray} \label{2.0} s_{-2}=0,&& s_{-1}=1,\hspace{5mm} s_k=a_ks_{k-1}+s_{k-2} \,\MB{ and } \nonumber\\ t_{-2}=1,&& t_{-1}=0,\hspace{5mm} t_k=a_kt_{k-1}+t_{k-2},\,\MB{ for } k\ge 0. \end{eqnarray} The Barkan-Hickerson-Knuth formula says that for $k\ge 0$ \BE \label{2.1} S(s_k/t_k)=\sum_{j=1}^k(-1)^{j-1}a_j+\begin{cases} \vspace{5mm} (s_k'+ t_{k-1}')/{t_k'}-3, & \MB{if } k \MB{ is odd}; \\ (s_k'-t_{k-1}')/{t_k'}, & \MB{if } k \MB{ is even;} \end{cases} \EE \cite{Ba}, \cite{Hi}, \cite{Kn}. Here $s_k'$ and $t_k'$ are defined as in (\ref{2.0}), but for the number $[0,a_1,a_2,\LD]$ instead of $[a_0,a_1,a_2,\LD]$. We prove the simplest case first. \begin{proof}[Proof of Theorem \ref{t3}] The digits $a_j$ of the continued fraction expansion of $e^*$ are $a_j=4j+2$, $j=0,1,2,\LD$ \cite[p.\ 124]{Pe}. An easy calculation shows that for $k\ge 0$ \BE \label{2.3} \sum_{j=1}^k(-1)^{j-1}a_j=\begin{cases} -2k, & \MB{ if }k \MB{ is even;} \\ 2k+4, & \MB{ if }k \MB{ is odd.} \end{cases} \EE Now $s_k'/t_k'$ converges against $[0,a_1,a_2,\LD]=e^*-2$, and $|e^*-2-s_k'/t_k'|<1/t_k'^2$ \cite[p.\ 37]{Pe}. We remarked in the Introduction that $k=O(\log t_k')$. Hence we also have $|e^*-2-s_k'/t_k'|=O(1/k)$. Finally, (\ref{2.0}) gives $t_{k-1}'/t_{k}'=t_{k-1}'/(a_kt_{k-1}'+t_{k-2}')\le 1/a_k=O(1/k).$ These observations, together with (\ref{2.1}) and (\ref{2.3}), prove the theorem. \end{proof} \begin{proof}[Proof of Theorem \ref{t1}] In the case of $e=[a_0,a_1,a_2,\LD]$ one easily derives from \cite[p.\ 124]{Pe} that \BD a_j=\begin{cases} 2, & \MB{ if } j=0; \\ 2(j-1)/3+2, &\MB{ if } j\equiv 2\pmod 3;\rule{0pt}{13pt} \\ 1, &\MB{ otherwise}.\rule{0pt}{13pt} \end{cases} \ED An elementary computation with arithmetic series (which is more laborious than that of the proof of Theorem \ref{t3}) yields \BE \label{2.5} \sum_{j=1}^k(-1)^{j-1}a_j=\begin{cases} \frac k3, & \MB{ if } k\equiv 0\pmod 6; \\ -\frac k3+1, & \MB{ if } k\equiv 3\pmod 6;\rule{0pt}{14pt}\\ \frac {k-1}3 +1,& \MB{ if } k\equiv 1\pmod 6;\rule{0pt}{14pt} \\ -\frac {k-1}3, & \MB{ if } k\equiv 4\pmod 6;\rule{0pt}{14pt}\\ -\frac {k-2}3-1,& \MB{ if } k\equiv 2\pmod 6;\rule{0pt}{14pt} \\ \frac {k-2}3+2, & \MB{ if } k\equiv 5\pmod 6.\rule{0pt}{14pt} \end{cases} \EE In the same way as in the proof Theorem \ref{t3} we have $s_k'/t_k'\to e-2$ and $|e-2-s_k'/t_k'|=O(1/k)$. If $k\equiv 2$ (mod $3$), we note $t_{k-1}'/t_k'\le 1/a_k=O(1/k)$. If $k\equiv 0$ (mod $3$) and $k\ge 3$, we have \BE \label{2.7} \frac{t_{k-1}'}{t_k'}=\frac{t_{k-1}'}{t_{k-1}'-t_{k-2}'}=\frac 1{1+t_{k-2}'/t_{k-1}'}. \EE Since $t_{k-2}'/t_{k-1}'=O(1/k)$, this shows $t_{k-1}'/t_k'=1+O(1/k)$. If $k\equiv 1$ (mod $3$) and $k\ge 4$, formula (\ref{2.7}) also holds. Together with $t_{k-2}'/t_{k-1}'=1+O(1/k)$, it gives $t_{k-1}'/t_k'=1/2+O(1/k)$. These observations, combined with (\ref{2.1}) and (\ref{2.5}), prove the theorem. \end{proof} \begin{proof}[Proof of Theorem \ref{t2}] The proof follows the above pattern. One obtains from \cite[p.\ 125]{Pe} \BD a_j=\begin{cases} 7, & \MB{ if } j=0; \\ (3j+7)/5, &\MB{ if } j\equiv 1\pmod 5;\rule{0pt}{13pt} \\ (3j+3)/5, &\MB{ if } j\equiv 4\pmod 5;\rule{0pt}{13pt} \\ 12j/5+6, &\MB{ if } j\equiv 0\pmod 5, j>0;\rule{0pt}{13pt} \\ 1, &\MB{ otherwise}.\rule{0pt}{13pt} \end{cases} \ED Further, \BE \label{2.9} \sum_{j=1}^k(-1)^{j-1}a_j=\begin{cases} -\frac {6k}5, & \MB{ if } k\equiv 0\pmod{10}; \\ \frac {6k}5+11, & \MB{ if } k\equiv 5\pmod{10};\rule{0pt}{14pt}\\ -\frac {3(k-1)}5 +2,& \MB{ if } k\equiv 1\pmod{10};\rule{0pt}{14pt} \\ \frac {3(k-1)}5+9, & \MB{ if } k\equiv 6\pmod{10};\rule{0pt}{14pt}\\ -\frac {3(k-2)}5+1, & \MB{ if } k\equiv 2\pmod{10};\rule{0pt}{14pt} \\ \frac {3(k-2)}5+10, & \MB{ if } k\equiv 7\pmod{10};\rule{0pt}{14pt} \\ -\frac {3(k-3)}5+2, & \MB{ if } k\equiv 3\pmod{10};\rule{0pt}{14pt} \\ \frac {3(k-3)}5+9, & \MB{ if } k\equiv 8\pmod{10};\rule{0pt}{14pt} \\ -\frac {6(k-4)}5-1, & \MB{ if } k\equiv 4\pmod{10};\rule{0pt}{14pt} \\ \frac {6(k-4)}5+12, & \MB{ if } k\equiv 9\pmod{10}.\rule{0pt}{14pt} \end{cases} \EE In the same way as in the proof of Theorem \ref{t1} we observe $|e^2-7-s_k'/t_k'|=O(1/k)$ and \BD \frac{t_{k-1}'}{t_k'}=O\left(\frac 1k\right)+ \begin{cases} 0, & \MB{ if } k\equiv {0,1,4}\pmod 5;\\ 1, & \MB{ if } k\equiv 2\pmod 5; \rule{0pt}{13pt}\\ \frac 12, & \MB{ if } k\equiv 3\pmod 5.\rule{0pt}{13pt} \end{cases} \ED Thereby, and by (\ref{2.9}), we obtain the theorem. \end{proof} \begin{remark} 1. It is easy to see that the error term $O(1/k)$ in the theorems cannot be made smaller. Accordingly, the convergence is rather slow, which is a further difference between the present cases and the case of quadratic irrationals. 2. The continued fraction expansions of $e^{2/q}$ and $(e^{2/q}+1)/(e^{2/q}-1$) for integers $q\ge 1$ have a shape similar to that of $e$, $e^2$, and $e^*$ \cite[p.\ 124 f.]{Pe}. The same holds for the the numbers $\tan(1/q)$. Therefore, similar theorems about \DED sums can be expected for the convergents of these numbers. 3. Due to a theorem of Hurwitz \cite[p.\ 119]{Pe} one may even hope for similar results for the numbers \BD \frac{ae^{2/q}+b}{ce^{2/q}+d}, \ED where the integer $q$ is $\ge 1$ and $a,b,c,d\in\Z$ are such that $ad-bc\ne 0$. It seems, however, that not all continued fraction expansions of these numbers are explicitly known. 4. The continued fraction expansions of the numbers \BD \sum_{j=0}^{\infty}{b^{-2^j}},\: b\in \Z, b\ge 3, \ED are also known \cite{Sh2}. They are, however, much more involved than those considered here. Accordingly, the asymptotic behaviour of the corresponding \DED sums seems to be far more complicated. \end{remark} \begin{thebibliography}{99} \BI{Ap} T. M. Apostol, {\em Modular Functions and Dirichlet Series in Number Theory}, Springer, 1976. \BI{Ba} Ph. Barkan, Sur les sommes de Dedekind et les fractions continues finies, {\em C. R. Acad. Sci. Paris S\'er. A-B} {\bf 284} (1977) A923--A926. \BI{Gi} K. Girstmair, \DED sums in the vicinity of quadratic irrationals, {\em J. Number Th.} {\bf 132} (2012), 1788--1792. \BI{GiSch} K. Girstmair and J. Schoi{\ss}engeier, On the arithmetic mean of \DED sums, {\em Acta Arith.} {\bf 116} (2005), 189--198. \BI{Hi} D. Hickerson, Continued fractions and density results for Dedekind sums, {\em J. Reine Angew. Math.} {\bf 290} (1977), 113--116. \BI{Kn} D. E. Knuth, Notes on generalized Dedekind sums, {\em Acta Arith.} {\bf 33} (1977), 297--325. \BI{Me} C. Meyer, {\em Die Berechnung der Klassenzahl Abelscher K\"orper \"uber quadratischen Zahlk\"orpern}, Akademie-Verlag, 1957. \BI{Pe} O. Perron, {\em Die Lehre von den Kettenbr\"uchen}, vol. I (3rd ed.), Teubner, 1954. \BI{RaGr} H. Rademacher and E. Grosswald, {\em \DED Sums}, Mathematical Association of America, 1972. \BI{Sh2} J. Shallit, Simple continued fractions for some irrational numbers, {\em J. Number Th.} {\bf 11} (1979), 209--217. \BI{Sh} J. Shallit, Origins of the analysis of the Euclidean algorithm, {\em Hist. Math} {\bf 21} (1994), 401--419. \BI{Ur} G. Urz\'ua, Arrangements of curves and algebraic surfaces, {\em J. Algebraic Geom.} {\bf 19} (2010), 335--365. \end{thebibliography}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bigskip \hrule \bigskip \noindent 2010 {\it Mathematics Subject Classification}: Primary 11F20; Secondary 11A55. \noindent \emph{Keywords: } asymptotic behaviour of Dedekind sums, continued fraction expansions of transcendental numbers. \bigskip \hrule \bigskip \vspace*{+.1in} \noindent Received May 8 2012; revised version received May 31 2012. Published in {\it Journal of Integer Sequences}, June 12 2012. \bigskip \hrule \bigskip \noindent Return to \htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}. \vskip .1in \end{document}