Jiang, Tidor, Yao, Zhang, and Zhao recently showed that connected bounded degree graphs have sublinear second eigenvalue multiplicity (always referring to the adjacency matrix). This result was a key step in the solution to the problem of equiangular lines with fixed angles. It led to the natural question: what is the maximum second eigenvalue multiplicity of a connected bounded degree 𝑛-vertex graph? The best-known upper bound is 𝑂(𝑛∕ log log 𝑛). The previously known bestknown lower bound is on the order of 𝑛 1∕3 (for infinitely many 𝑛), coming from Cayley graphs on PSL(2, 𝑞).Here we give a construction showing a lower bound of √ 𝑛∕ log 2 𝑛. We also construct Cayley graphs with second eigenvalue multiplicity at least 𝑛 2∕5 − 1. Earlier techniques show that there are at most 𝑂(𝑛∕ log log 𝑛) eigenvalues (counting multiplicities) within 𝑂(1∕ log 𝑛)of the second eigenvalue. We give a construction showing this upper bound on approximate second eigenvalue multiplicity is tight up to a constant factor. This demonstrates a barrier to earlier techniques for upper bounding eigenvalue multiplicities.

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.