Let
S
S
be a given set of positive rational primes. Assume that the value of the Dedekind zeta function
ζ
K
\zeta _K
of a number field
K
K
is less than or equal to zero at some real point
β
\beta
in the range
1
2
>
β
>
1
{1\over 2} >\beta >1
. We give explicit lower bounds on the residue at
s
=
1
s=1
of this Dedekind zeta function which depend on
β
\beta
, the absolute value
d
K
d_K
of the discriminant of
K
K
and the behavior in
K
K
of the rational primes
p
∈
S
p\in S
. Now, let
k
k
be a real abelian number field and let
β
\beta
be any real zero of the zeta function of
k
k
. We give an upper bound on the residue at
s
=
1
s=1
of
ζ
k
\zeta _k
which depends on
β
\beta
,
d
k
d_k
and the behavior in
k
k
of the rational primes
p
∈
S
p\in S
. By combining these two results, we obtain lower bounds for the relative class numbers of some normal CM-fields
K
K
which depend on the behavior in
K
K
of the rational primes
p
∈
S
p\in S
. We will then show that these new lower bounds for relative class numbers are of paramount importance for solving, for example, the exponent-two class group problem for the non-normal quartic CM-fields. Finally, we will prove Brauer-Siegel-like results about the asymptotic behavior of relative class numbers of CM-fields.