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Bloch-Kato Conjecture On Tamagawa Numbers
In mathematics, the study of special values of L-functions is a subfield of number theory devoted to generalising formulae such as the Leibniz formula for pi, namely :1 \,-\, \frac \,+\, \frac \,-\, \frac \,+\, \frac \,-\, \cdots \;=\; \frac,\! by the recognition that expression on the left-hand side is also ''L''(1) where ''L''(''s'') is the Dirichlet L-function for the Gaussian field. This formula is a special case of the analytic class number formula, and in those terms reads that the Gaussian field has class number 1, and also contains four roots of unity, so accounting for the factor ¼. Conjectures There are two families of conjectures, formulated for general classes of ''L''-functions (the very general setting being for ''L''-functions ''L''(''s'') associated to Chow motives over number fields), the division into two reflecting the questions of: how to replace π in the Leibniz formula by some other "transcendental" number (whether or not it is yet possible fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |