Abstract Analytic Number Theory - Examples

Examples

• The prototypical example of an arithmetic semigroup is the multiplicative semigroup of positive integers G = Z+ = {1, 2, 3, ...}, with subset of rational primes P = {2, 3, 5, ...}. Here, the norm of an integer is simply, so that, the greatest integer not exceeding x.
• If K is an algebraic number field, i.e. a finite extension of the field of rational numbers Q, then the set G of all nonzero ideals in the ring of integers O_K of K forms an arithmetic semigroup with identity element O_K and the norm of an ideal I is given by the cardinality of the quotient ring O_K/I. In this case, the appropriate generalisation of the prime number theorem is the Landau prime ideal theorem, which describes the asymptotic distribution of the ideals in O_K.
• Various arithmetical categories which satisfy a theorem of Krull-Schmidt type can be considered. In all these cases, the elements of G are isomorphism classes in an appropriate category, and P consists of all isomorphism classes of indecomposable objects, i.e. objects which cannot be decomposed as a direct product of nonzero objects. Some typical examples are the following.
  • The category of all finite abelian groups under the usual direct product operation and norm mapping . The indecomposable objects are the cyclic groups of prime power order.
  • The category of all compact simply-connected globally symmetric Riemannian manifolds under the Riemannian product of manifolds and norm mapping, where c > 1 is fixed, and dim M denotes the manifold dimension of M. The indecomposable objects are the compact simply-connected irreducible symmetric spaces.
  • The category of all pseudometrisable finite topological spaces under the topological sum and norm mapping . The indecomposable objects are the connected spaces.