\chapter{INTRODUCTION} % MUST be CAPITALIZED
\section{The Smoothed Partial Zeta-function}
A real quadratic number field, $k$, is an extension field of $\Q$,
$$\{s + t \sqrt{D} \colon s,t \in \Q \},$$ where $D$ is of a positive
square-free integer. There is only one non-trivial automorphism,
$\sigma : k \longrightarrow k$ $$\sigma \colon s+t \sqrt{D}
\longmapsto s-t \sqrt{D}.$$ If $\alpha$ is an element of $k$, we refer
to $\alpha' = \sigma(\alpha)$ as its conjugate. We define an
embedding of $k$ into the real affine plane $$e: k \hookrightarrow
\R^2 \qquad e(\alpha) = (\alpha,\alpha').$$ If both $\alpha$ and
$\alpha'$ are positive real numbers, we say that the element,
$\alpha$, is {\bf totally positive} and denote this by $\alpha \gg 0$.
We let {\ring} denote the ring of algebraic integers in $k$. For any
integral ideal $\ga \subset \ring$ we may define the {\bf norm} of
this ideal to be the cardinality, $$N(\ga) = \left|\ring / {\ga}
\right|,$$ of the quotient group. The set
$${\ga}' = \{ \alpha' \in k \colon \alpha \in \ga \}$$is an integral
ideal of {\ring} if {\ga} is. Therefore, we have $$(N{\ga}){\ring} =
{\ga}{\ga}'$$ for any integral ideal {\ga}, giving us an alternative
definition of the norm. We generalize the definition of norm of an
ideal to apply to all elements of the field $$ N\alpha =
(\alpha)(\alpha') \quad \forall \alpha \in k $$ and thus norm as
either a function of ideals or of elements is a multiplicative
function.
For a non-zero integral ideal $\gb \subset \ring$, we define the {\bf
inverse} of $\gb$, $${\gb}^{-1} = \{ \beta \in k \colon \beta{\gb}
\subset \ring \}.$$An inverse of an integral ideal contains the
multiplicative identity, and is, thus, no longer an ideal in $\ring$,
but it is still a Noetherian $\ring$-module. We then define a {\bf
fractional ideal}, given $\gb^{-1}$ and $\gf$ to be the ring
$$\ideal = \{ xy \colon x\in \gb^{-1}, y \in \gf\}.$$The set of all
fractional ideals forms a multiplicative group.
The {\bf partial zeta-function}, $$\parzeta =
\sum_{\ga}{1\over\displaystyle \left( N{\ga}\right)^s }$$ is defined
for two relatively prime integral ideals, {\gb} and {\gf}, where the
sum is taken over all integral ideals $\ga \subset \ring$ such that
$$\ga = \alpha \gb \qquad \alpha \gg 0 \quad \alpha \equiv
1\bmod{\gf}.$$As in the case of the Riemann zeta-function, $$\zeta(s)
= \sum_{n \in {\Z}^+} {1 \over\displaystyle n^s}$$or the Dedekind
zeta-function, $$\zeta_{k}(s) = \sum_{\ga \in
\ring}{1\over\displaystyle \left( N{\ga}\right)^s }$$the partial
zeta-function is a function of the complex plane which is absolutely
convergent for Re$(s) > 1$. It has an analytic continuation to the
entire complex plane which is holomorphic except for the simple pole
at $s=1$.
\begin{eqnarray*}
\zeta_{\gothe}(s,{\gb}{\gf}^{-1})& = &\sum_{{\ga}}{1
\over\displaystyle (N{\ga})^s}\qquad\qquad {\ga} \equiv {\ideal}
\bmod{P^+(\ring)}\\ & = &\sum_{x}{1 \over\displaystyle
(N(x{\gb}{\gf}^{-1}))^s} \qquad x \in D \cap e(\ideal)\\ & = &{1
\over\displaystyle (N{\gb})^s} \sum_{x}{1 \over\displaystyle
(N(x{\gf}^{-1}))^s} \qquad x \in D \cap e(\ideal)\\ & = &{1
\over\displaystyle (N{\gb})^s} \left ( {1 \over\displaystyle
m(\gf)} \sum_{x}{1 \over\displaystyle (N(x{\gf}^{-1}))^s}
\right) \qquad x \in \fundom \cap e(\ideal).
\end{eqnarray*}
\section{A Comparison of Algorithms}
When Siegel proved in \cite[revised version]{Sie2} that the
zeta-function {\parzeta} was rational for any rational integer value
of $s$, the statement and method of proof was much more general than
the partial zeta-functions of a real quadratic number field. His
previous work in \cite{Sie1} that led toward this result, however, was
a study of quadratic number fields exclusively as well as the study of
Bernoulli polynomials. The proof and formula for a more general
zeta-function involved the theory of elliptic modular forms. For the
purposes of comparing the number of terms used in the summation, Hayes
\cite{Ha1} provides an equivalent version of Siegel's formula,
simplified for a quadratic number field.
\begin{lemma}(Siegel)
Given a basis ${\rho, \sigma}$ for $\ideal$ and the
integers from the associated matrix Mat$_{\gf}$, we may write an
expression,
\begin{eqnarray*}
\zerozeta & = & {a+d \over\displaystyle 4c} \cdot [B_2(\langle
-cw_0\rangle) + B_2(\langle cz_0 \rangle)]\\
& & - \sum_r B_1 \left( \left \langle {ar \over\displaystyle c} -
w_0 \right \rangle \right)\cdot B_1 \left( \left \langle {r
\over\displaystyle c} + z_0 \right \rangle \right)
\end{eqnarray*}for the partial zeta-function evaluated at $s=0$.
\end{lemma}