  We study universal quadratic forms over totally real number fields using
Dedekind zeta functions. In particular, we prove an explicit upper bound for
the rank of universal quadratic forms over a given number field $K$, under the
assumption that the codifferent of $K$ is generated by a totally positive
element. Motivated by a possible path to remove that assumption, we also
investigate the smallest number of generators for the positive part of ideals
in totally real numbers fields.
