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
Famous quotes containing the words methods and/or techniques:
“A woman might claim to retain some of the child’s faculties, although very limited and defused, simply because she has not been encouraged to learn methods of thought and develop a disciplined mind. As long as education remains largely induction ignorance will retain these advantages over learning and it is time that women impudently put them to work.”
—Germaine Greer (b. 1939)
“It is easy to lose confidence in our natural ability to raise children. The true techniques for raising children are simple: Be with them, play with them, talk to them. You are not squandering their time no matter what the latest child development books say about “purposeful play” and “cognitive learning skills.””
—Neil Kurshan (20th century)