Special values of L-functions


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
by the recognition that expression on the left-hand side is also L where L 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 division into two reflecting the questions of:
Subsidiary explanations are given for the integer values of n for which such formulae L can be expected to hold.
The conjectures for are called Beilinson's conjectures, for Alexander Beilinson. The idea is to abstract from the regulator of a number field to some "higher regulator", a determinant constructed on a real vector space that comes from algebraic K-theory.
The conjectures for are called the Bloch–Kato conjectures for special values . For the sake of greater clarity, they are also called the Tamagawa number conjecture, a name arising via the Birch–Swinnerton-Dyer conjecture and its formulation as an elliptic curve analogue of the Tamagawa number problem for linear algebraic groups. In a further extension, the equivariant Tamagawa number conjecture has been formulated, to consolidate the connection of these ideas with Iwasawa theory, and its so-called Main Conjecture.

Current status

All of these conjectures are known to be true only in special cases.