Independence of Hecke zeta functions of finite order over normal fields

We study oscillations of the remainder term corresponding to the counting functions of the sets of elements with unique factorization length in semigroups of algebraic numbers such as the semigroup of algebraic integers or totally positive algebraic integers in a given normal field $K$. The results imply existence of oscillations when the exponent of the class group of the semigroup in question is sufficiently large depending on the field's degree. In particular, when $K$ is a quadratic field or a normal cubic field oscillations exist whenever the class group is not isomorphic to $C_2^a \oplus C_3^b \oplus C_4^c$ for nonnegative integers $a, b, c$. The main part of this study is concerned with the problem of multiplicative independence of Hecke zeta functions. We also show that there are infinitely many fields whose Dedekind zeta function has infinitely many nontrivial multiple zeros.