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.
Answering a question of Browkin, we provide a new unconditional proof that the Dedekind zeta function of a number field
L
L
has infinitely many nontrivial zeros of multiplicity at least 2 if
L
L
has a subfield
K
K
for which
L
/
K
L/K
is a nonabelian Galois extension. We also extend this to zeros of order 3 when
G
a
l
(
L
/
K
)
Gal(L/K)
has an irreducible representation of degree at least 3, as predicted by the Artin holomorphy conjecture.
We introduce a number field analogue of the Mertens conjecture and demonstrate its falsity for all but finitely many number fields of any given degree. We establish the existence of a logarithmic limiting distribution for the analogous Mertens function, expanding upon work of Ng. Finally, we explore properties of the generalized Mertens function of certain dicyclic number fields as consequences of Artin factorization.
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