Abstract Analytic Number Theory - Arithmetic Semigroups

Arithmetic Semigroups

The fundamental notion involved is that of an arithmetic semigroup, which is a commutative monoid G satisfying the following properties:

• There exists a countable subset (finite or countably infinite) P of G, such that every element a ≠ 1 in G has a unique factorisation of the form

where the p_i are distinct elements of P, the α_i are positive integers, r may depend on a, and two factorisations are considered the same if they differ only by the order of the factors indicated. The elements of P are called the primes of G.

• There exists a real-valued norm mapping on G such that
  1. The total number of elements of norm is finite, for each real .