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\begin{flushright}
Andrew Odlyzko \\
November 29, 1976
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\begin{center}
{\large \bf DISCRIMINANT BOUNDS} \\
(description of tables)
\end{center}

\noindent
i) Tables 1 and 3 assume the Generalized Riemann Hypothesis (GRH),
while Tables~2 and 4 are unconditional.
Tables~1 and 2 were derived from Tables~3 and 4, respectively. \\
ii) In Tables~1 and 2, an entry $B$ in the totally complex $D^{1/n}$ column corresponding to $n = n_0$
means that for all fields of degrees $n \ge n_0$, the
discriminant satisfies $D^{1/n} > B$.
An entry $A$ in the totally real $D^{1/n}$ column implies that for all
totally real fields of degrees $n \ge n_0$, we have
$D^{1/n} > A$.
The $b$ entries specify which inequalities in the other
tables were used. \\
iii) In Tables~3 and 4, the notation is as follows.
If $K$ is an algebraic number field with $r_1$ real and $2r_2$ complex
conjugate fields, and $D$ denotes the absolute value of the
discriminant of $K$, then for any $b$ we have
$$
D > A^{r_1} B^{2r_2} e^{f-E}
$$
where $A,B$, and $E$ are given in the table, and
$$
f= 2 \sum_P ~~\sum_{m=1}^\In \df{\log ~NP}{(NP)^{m/2}} F( \log ~NP^m )
$$
where the outer sum is over all the prime ideals of $K$, $N$ is
the norm from $K$ to $Q$, and
$$
F(x) = G(x/b)
$$
in the GRH case, and
$$
F(x) = \df{H(x/b)}{\cosh \frac{x}{2} }
$$
in the unconditional case, where $G(x)$, $H(x)$ are even functions
of $x$ which vanish for $x > 2$, and for
$0 \leq x \leq 2$ are given by
\begin{eqnarray*}
G(x) & = & \left( 1- \df{x}{2} \right)
\cos \df{\pi}{2} x + \df{1}{\pi} \sin
\df{\pi}{2} x \\
 H(x) & = & \df{1}{3} (2-x) \left( 1 + \df{1}{2} \cos \pi x \right) + \df{1}{2 \pi} \sin \pi x ~.
\end{eqnarray*}
The values of $A$ and $B$ are lower estimates; the values of $E$
have been rounded upwards from their true values,
which are $8b/3$ in the unconditional case and
$$
8 \pi^2 b \left( \df{e^{b/2} + e^{-b/2}}{\pi^2 + b^2} \right)^2
$$
in the $GRH$ case. \\
iv) Great care was taken to ensure that these bounds should be
true lower bounds, rather than approximations.
By selecting the parameter $b$ more carefully, utilizing more
precise estimates of integrals, and selecting better kernels,
one can obtain improved lower bounds.
For example, all fields of degrees $\ge 8$ satisfy
$D^{1/n} \ge 5.743$ on the $GRH$,
and $D^{1/n} \ge 5.656$ unconditionally.

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