Methods and Techniques
The use of arithmetic functions and zeta functions is extensive. The idea is to extend the various arguments and techniques of arithmetic functions and zeta functions in classical analytic number theory to the context of an arbitrary arithmetic semigroup which may satisfy one or more additional axioms. Such a typical axiom is the following, usually called "Axiom A" in the literature:
- Axiom A. There exist positive constants A and, and a constant with, such that
For any arithmetic semigroup which satisfies Axiom A, we have the following abstract prime number theorem:
where πG(x) = total number of elements p in P of norm |p| ≤ x.
Read more about this topic: Abstract Analytic Number Theory
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