Does Montgomery's pair correlation conjecture also hold true for Dedekind Zeta function ?

Question

All:

Does Montgomery's pair correlation conjecture also hold true for the zeros for Dedekind Zeta function of an algebraic number field K ?

My understanding is that Montgomery's pair correlation conjecture should hold true for Dirichlet's L-functions. Am I correct here ?

Thank you.

Answer 1

For the usual Dedekind Zeta function, the conjecture is exactly the same since the non-trivial zeros are the same as those of the zeta function.

For Dirichlet Functions, an analogous Montgomery conjecture was proposed by one of his students in a dissertation (unfortunately I can't find a copy online):

https://books.google.com/books/about/Pair_Correlation_of_Zeros_of_Dirichlet_L.html?id=7nxlnQEACAAJ

Answer 2

The student of Hugh Montgomery is probably A.E. Ozluk...

I am not at all an expert on these things, but Iwaniec-Kowalski's "Analytic Number theory" seems to have a good section on pair-correlation and such.

There is a larger perspective in which we can ask about what kind of random matrices' eigenvalue pair-correlation (and higher) "are the same as" some $L$-functions'. Papers by Katz-Sarnak have some heuristics, theorems, and conjectures about this.