  Here we study problems related to the proportions of zeros, especially simple
and distinct zeros on the critical line, of Dedekind zeta functions. We obtain
new bounds on a counting function that measures the discrepancy of the zeta
functions from having all zeros simple. In particular, for quadratic number
fields, we deduce that more than 45% of the zeros are distinct. This extends
work based on Montgomery's pair correlation approach for the Riemann zeta
function. Our optimization problems can be interpreted as interpolants between
the pair correlation bound for the Riemann zeta function and the Cohn-Elkies
sphere packing bound in dimension 1. We compute the bounds through optimization
over Schwartz functions using semidefinite programming and also show how
semidefinite programming can be used to optimize over functions with bounded
support.
