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 pi are distinct elements of P, the αi 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 .

Read more about this topic:  Abstract Analytic Number Theory

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