Order of zeros of Dedekind zeta functions

Abstract

Answering a question of Browkin, we provide a new unconditional proof that the Dedekind zeta function of a number field L has infinitely many nontrivial zeros of multiplicity at least 2 if L has a subfield K for which L/K is a nonabelian Galois extension. We also extend this to zeros of order 3 when Gal(L/K) has an irreducible representation of degree at least 3, as predicted by the Artin holomorphy conjecture.

Additional Information

© 2022 American Mathematical Society. Received by editor(s): July 22, 2021. Received by editor(s) in revised form: February 10, 2022. Published electronically: June 17, 2022. The authors were supported by the National Science Foundation (Grants DMS 2002265 and DMS 205118), National Security Agency (Grant H98230-21-1-0059), the Thomas Jefferson Fund at the University of Virginia, and the Templeton World Charity Foundation. We are deeply grateful to Peter Humphries for supervising this project and to Ken Ono for his valuable suggestions. We would also like to thank Robert Lemke Oliver and Samit Dasgupta for helpfully directing us to the work of Stark.

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