Andrew Odlyzko
November 29, 1976
DISCRIMINANT BOUNDS
(description of tables)
(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 = n0 means that for all fields of degrees
n >= n0, 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 >= n0, 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 r1 real and 2*r2 complex conjugate
fields, and D denotes the absolute value of the discriminant of K,
then for any b we have
D > A^{r1} B^{2*r2} e^{f-E}
where A, B, and E are given in the table, and
f = 2*sum( {log NP} * F(log NP^m) / (NP)^{m/2}, sum on P and m),
where m runs over the positive integers, and P 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) = H(x/b) / cosh(x/2)
in the unconditional case, where G(x), H(x) are even functions
of x which vanish for x > 2, and for 0 <= x <= 2 are given by
G(x) = ( 1- x/2 ) * cos(pi*x/2) + { sin(pi*x/2) } / pi,
H(x) = (2-x) * {1 + (cos(pi(x))/2} / 3 + (sin(pi*x)) / (2*pi).
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*b*pi^2 * { (exp(b/2) + exp(-b/2)) / (pi^2 + b^2) }^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 >= 8 satisfy
D^{1/n} >= 5.743 on the GRH, and D^{1/n} >= 5.656 unconditionally.