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\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}\\
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\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 }

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\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, &amp; \MB{if } k\equiv{0,1,5}\pmod 6; \\
                 -\frac k3, &amp; \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,                 &amp;\MB{ if }k\equiv 3\pmod 6;\\
                                 e-3-\frac 56, &amp;\MB{ if }       k\equiv 1\pmod 6; \rule{0pt}{13pt}\\
                                 e-3+\frac 23,        &amp;\MB{ if }k\equiv 2\pmod 6; \rule{0pt}{13pt}\\
                                 e-3+\frac 56, &amp;\MB{ if }       k\equiv 4\pmod 6; \rule{0pt}{13pt}\\
                                 e-3-\frac 23,        &amp;\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, &amp; \MB{if }k\equiv {1,2,3}\pmod{10}; \\
                 \frac{3k}5, &amp; \MB{if } k\equiv{6,7,8} \pmod{10}; \rule{0pt}{14pt}\\
                 -\frac{6k}5, &amp; \MB{if }k\equiv{0,4}\pmod{10}; \rule{0pt}{14pt}\\
                  \frac{6k}5, &amp; \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,                   &amp;\MB{ if } k\equiv0\pmod{10};\\
                                 e^2-\frac{37}5,          &amp; \MB{ if }k\equiv{1}\pmod{10};\rule{0pt}{13pt}\\
                                 e^2-\frac{29}5,          &amp;\MB{ if } k\equiv{2}\pmod{10}; \rule{0pt}{13pt}\\
                                 e^2-\frac{31}5+\frac 12, &amp;\MB{ if } k\equiv 3\pmod{10}; \rule{0pt}{13pt}\\
                                 e^2-\frac{16}5,          &amp;\MB{ if } k\equiv 4\pmod{10}; \rule{0pt}{13pt}\\
                                 e^2+1,                   &amp;\MB{ if } k\equiv 5\pmod{10};\rule{0pt}{13pt}\\
                                 e^2+\frac 75,            &amp;\MB{ if } k\equiv 6\pmod{10}; \rule{0pt}{13pt}\\
                                 e^2-\frac 15,            &amp;\MB{ if } k\equiv 7\pmod{10};\rule{0pt}{13pt}\\
                                 e^2-\frac 45+\frac 12,   &amp;\MB{ if } k\equiv 8\pmod{10}; \rule{0pt}{13pt}\\
                                 e^2-\frac{14}5,        &amp;\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, &amp; \MB{ if }k \MB{ is even;} \\
                  2k, &amp; \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,                &amp;\MB{ if }k\MB{ is even;}\\
                                 e^*-1,                &amp;\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,&amp;&amp; s_{-1}=1,\hspace{5mm} s_k=a_ks_{k-1}+s_{k-2} \,\MB{ and } \nonumber\\
   t_{-2}=1,&amp;&amp; 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, &amp; \MB{if } k \MB{ is odd}; \\
                                                               (s_k'-t_{k-1}')/{t_k'},  &amp; \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,  &amp; \MB{ if }k \MB{ is even;} \\
                      2k+4, &amp; \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 $|e^*-2-s_k'/t_k'|&lt;1/t_k'^2$ \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,          &amp; \MB{ if } j=0; \\
               2(j-1)/3+2, &amp;\MB{ if }  j\equiv 2\pmod 3;\rule{0pt}{13pt}  \\
               1,          &amp;\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,       &amp; \MB{ if } k\equiv 0\pmod 6; \\
                      -\frac k3+1,    &amp; \MB{ if } k\equiv 3\pmod 6;\rule{0pt}{14pt}\\
                      \frac {k-1}3 +1,&amp; \MB{ if } k\equiv 1\pmod 6;\rule{0pt}{14pt} \\
                      -\frac {k-1}3,  &amp; \MB{ if } k\equiv 4\pmod 6;\rule{0pt}{14pt}\\
                      -\frac {k-2}3-1,&amp; \MB{ if } k\equiv 2\pmod 6;\rule{0pt}{14pt} \\
                     \frac {k-2}3+2,  &amp; \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,          &amp; \MB{ if } j=0; \\
               (3j+7)/5,   &amp;\MB{ if }  j\equiv 1\pmod 5;\rule{0pt}{13pt}  \\
               (3j+3)/5,   &amp;\MB{ if }  j\equiv 4\pmod 5;\rule{0pt}{13pt}  \\
               12j/5+6,    &amp;\MB{ if }  j\equiv 0\pmod 5, j&gt;0;\rule{0pt}{13pt}  \\
               1,          &amp;\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,       &amp; \MB{ if } k\equiv 0\pmod{10}; \\
                      \frac {6k}5+11,     &amp; \MB{ if } k\equiv 5\pmod{10};\rule{0pt}{14pt}\\
                      -\frac {3(k-1)}5 +2,&amp; \MB{ if } k\equiv 1\pmod{10};\rule{0pt}{14pt} \\
                       \frac {3(k-1)}5+9, &amp; \MB{ if } k\equiv 6\pmod{10};\rule{0pt}{14pt}\\
                      -\frac {3(k-2)}5+1, &amp; \MB{ if } k\equiv 2\pmod{10};\rule{0pt}{14pt} \\
                      \frac {3(k-2)}5+10, &amp; \MB{ if } k\equiv 7\pmod{10};\rule{0pt}{14pt} \\
                      -\frac {3(k-3)}5+2, &amp; \MB{ if } k\equiv 3\pmod{10};\rule{0pt}{14pt} \\
                      \frac {3(k-3)}5+9,  &amp; \MB{ if } k\equiv 8\pmod{10};\rule{0pt}{14pt} \\
                      -\frac {6(k-4)}5-1, &amp; \MB{ if } k\equiv 4\pmod{10};\rule{0pt}{14pt} \\
                      \frac {6(k-4)}5+12, &amp; \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,        &amp; \MB{ if } k\equiv {0,1,4}\pmod 5;\\
                      1,        &amp; \MB{ if } k\equiv 2\pmod 5; \rule{0pt}{13pt}\\
                      \frac 12, &amp; \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
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\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
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\vspace*{+.1in}
\noindent
Received  May 8 2012;
revised version received   May 31 2012.
Published in {\it Journal of Integer Sequences}, June 12 2012.

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\noindent
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