Special values of L-functions

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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 π, namely $${\displaystyle 1\,-\,{\frac {1}{3}}\,+\,{\frac {1}{5}}\,-\,{\frac {1}{7}}\,+\,{\frac {1}{9}}\,-\,\cdots \;=\;{\frac {\pi }{4}},\!}$$

by the recognition that expression on the left-hand side is also ${\displaystyle L(1)}$ where ${\displaystyle L(s)}$ is the Dirichlet L-function for the field of Gaussian rational numbers. 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. The factor ${\displaystyle {\tfrac {1}{4}}}$ on the right hand side of the formula corresponds to the fact that this field contains four roots of unity.

There are two families of conjectures, formulated for general classes of L-functions (the very general setting being for L-functions associated to Chow motives over number fields), the division into two reflecting the questions of:

1. how to replace ${\displaystyle \pi }$ in the Leibniz formula by some other "transcendental" number (regardless of whether it is currently possible for transcendental number theory to provide a proof of the transcendence); and
2. how to generalise the rational factor in the formula (class number divided by number of roots of unity) by some algebraic construction of a rational number that will represent the ratio of the L-function value to the "transcendental" factor.

Subsidiary explanations are given for the integer values of ${\displaystyle n}$ for which a formulae of this sort involving ${\displaystyle L(n)}$ can be expected to hold.

The conjectures for (a) are called Beilinson's conjectures, for Alexander Beilinson.^[1][2] The idea is to abstract from the regulator of a number field to some "higher regulator" (the Beilinson regulator), a determinant constructed on a real vector space that comes from algebraic K-theory.

The conjectures for (b) are called the Bloch–Kato conjectures for special values (for Spencer Bloch and Kazuya Kato; this circle of ideas is distinct from the Bloch–Kato conjecture of K-theory, extending the Milnor conjecture, a proof of which was announced in 2009). 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.^[3] In a further extension, the equivariant Tamagawa number conjecture (ETNC) has been formulated, to consolidate the connection of these ideas with Iwasawa theory, and its so-called Main Conjecture.

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

• Brumer–Stark conjecture

• Kings, Guido (2003), "The Bloch–Kato conjecture on special values of L-functions. A survey of known results", Journal de théorie des nombres de Bordeaux, 15 (1): 179–198, doi:10.5802/jtnb.396, ISSN 1246-7405, MR 2019010
• "Beilinson conjectures", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• "K-functor in algebraic geometry", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Mathar, Richard J. (2010), "Table of Dirichlet L-Series and Prime Zeta Modulo Functions for small moduli", arXiv:1008.2547 [math.NT]

• L-funktionen und die Vermutingen von Deligne und Beilinson (L-functions and the conjectures of Deligne and Beilsnson)