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Contents Index Dark Mode Prev Up Next \(\def\versionnumber{1.54}
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Section 10.4 Dedekind Sums
This exercise outlines one more nontraditional application of the Residue
Theorem 9.2.2 . Given two positive, relatively prime integers
\(a\) and
\(b\text{,}\) let
\begin{equation*}
f(z) \ := \ \cot (\pi az) \cot (\pi bz) \cot (\pi z) \,\text{.}
\end{equation*}
Choose an
\(\epsilon > 0\) such that the rectangular path
\(\gamma_R\) from
\(1-\epsilon-iR\) to
\(1-\epsilon+iR\) to
\(-\epsilon+iR\) to
\(-\epsilon-iR\) back to
\(1-\epsilon-iR\) does not pass through any of the poles of
\(f\text{.}\)
Compute the residues for the poles of \(f\) inside \(\gamma_R\text{.}\) Hint: Use the periodicity of the cotangent and the fact that
\begin{equation*}
\cot z \ = \ \frac 1 z - \frac 1 3 \, z + \text{ higher-order terms }\text{.}
\end{equation*}
Prove that \(\lim_{R \to \infty} \int_{\gamma_R} f =
-2i\) and deduce that for any \(R>0\)
\begin{equation*}
\int_{\gamma_R} f \ = \ -2i \,\text{.}
\end{equation*}
Define
\begin{equation}
s(a,b) \ := \ \frac{1}{4b} \sum_{k=1}^{b-1} \cot \left(
\frac{ \pi k a }{ b } \right) \cot \left( \frac{ \pi k }{ b
} \right) \text{.}\tag{10.1}
\end{equation}
\begin{equation}
s(a,b) + s(b,a) \ = \ - \frac 1 4 + \frac 1 {12} \left( \frac a
b + \frac 1 {ab} + \frac b a \right) \text{.}\tag{10.2}
\end{equation}
Historical remark. The sum in
(10.1) is called a
Dedekind sum . It first appeared in the study of the
Dedekind \(\eta\) -function
\begin{equation*}
\eta (z) \ = \ \exp \left( \tfrac{ \pi i z }{ 12 } \right) \prod_{k
\geq 1} \left( 1 - \exp ( 2 \pi i k z ) \right)
\end{equation*}
in the 1870’s and has since intrigued mathematicians from such different areas as topology, number theory, and discrete geometry. The
reciprocity law (10.2) is the most important and famous identity of the Dedekind sum. The proof that is outlined here is due to Hans Rademacher (1892–1969).
