The precise construction is rather technical, but greatly simplifies many ideas in complex function theory. Riemann zeta function The Dirichlet series . The zeta function has many surprising analytic and number theoretic properties, and has been one of the central objects of study in analytic number theory. Many of its properties can be generalized to L-functions . ring A mathematical system that has two operations, usually called addition and multiplication. A ring is an abelian group with respect to addition.
Multiplication is associative and distributive with respect to addition.
A subgroup of the general linear group consisting of matrices having determinant 1. R. Taniyama-Shimura conjecture The conjecture that every elliptic curve is actually a modular elliptic curve . A set together with a collection of subsets, called open sets that satisfy certain axioms. Examples of vector spaces include n-tuples of elements of a field and various classes of analytic functions . A vector space is the fundamental object of study in linear algebra.