One classical way to extend the Prime Number
Theorem and related results from the integers to
the ring of
integers of an algebraic number field
is replacing the notion "prime number" by the
notion "prime ideal."
Yet, also problems of the following form received
considerable attention (since the 1960s).
Given some
number field K with ring of
algebraic integers R and A ⊆ R
a subset defined via some arithmetical constraint
(fulfilling certain conditions), e.g., the set of
all irreducibles in R, or the set of all
elements having an essentially
unique factorization into irreducibles in R. Let
A(x) denote the number of pairwise
non-associated elements in
A of total norm at most x.
Determine the asymptotics of A(x).
It is known, by the work of Rémond,
Narkiewicz, Halter-Koch and others, that the
order of magnitude of A(x)
is equal to
(x/(Log x)B)(Log Log x)C
where B and C depend on the ideal class group
of R only, namely they
are determined by the
solution to a certain zero-sum problem over the
class group. However, for many types
of groups these zero-sum problems are open.
First, we summarize why zero-sum problems over
the class group arise in this context, and we
recall the precise
nature of these zero-sum
problems (for certain sets A).
Then, we present recent results on these zero-sum
problems, including results on the generalized
Davenport
constants, thus determining or
estimating the numerical value of these constants
B and C in certain cases.