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Here we study problems related to the proportions of zeros, especially simpleand distinct zeros on the critical line, of Dedekind zeta functions. We obtainnew bounds on a counting function that measures the discrepancy of the zetafunctions from having all zeros simple. In particular, for quadratic numberfields, we deduce that more than 45% of the zeros are distinct. This extendswork based on Montgomery's pair correlation approach for the Riemann zetafunction. Our optimization problems can be interpreted as interpolants betweenthe pair correlation bound for the Riemann zeta function and the Cohn-Elkiessphere packing bound in dimension 1. We compute the bounds through optimizationover Schwartz functions using semidefinite programming and also show howsemidefinite programming can be used to optimize over functions with boundedsupport.

Pair correlation for Dedekind zeta functions of abelian extensions