-functions are certain meromorphic functions generalizing the Riemann zeta function. They are typically defined on parts of the complex plane by a power series expressions – called the -series – which converges in that region, and then meromorphically extended to all of the complex plane by analytic continuation.
The most canonically defined class of examples of L-functions are the Artin L-functions defined for any Galois representation as the Euler products of, essentially, the characteristic polynomials of all the Frobenius homomorphisms acting via .
Most other kinds of L-functions are such as to reproduces these Artin L-functions from more “arithmetic” data:
for 1-dimensional Galois representations (hence for ) Artin reciprocity produces for each a Dirichlet character, or more generally a Hecke character? , and therefrom is built a Dirichlet L-function or Hecke L-function , respectively, which equals the corresponding Artin L-function ;
for general -dimensional Galois representations the conjecture of Langlands correspondence states that there is an automorphic representation corresponding to and an automorphic L-function built from that, which equalso the Artin L-function .
L-functions typically satisfy analogs of all the special properties enjoyed by the Riemann zeta function, such as satisfying a “functional equation” which asserts invariance under modular transformations of the parameter.
The generalized Riemann conjecture is concerned with zeros of the Dedekind zeta function for which the L-series (the Dirichlet L-function) is complicated from the classical Riemann case by the presence of the additional parameter, the Dirichlet character.
Stephen Gelbart, section C starting on p. 14 (190) of An elementary introduction to the Langlands program, Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 177–219 (web)
E. Kowalski, first part of Automorphic forms, L-functions and number theory (March 12–16) Three Introductory lectures (pdf)
Dorian Goldfeld, Joseph Hundley, chapter 2 of Automorphic representations and L-functions for the general linear group, Cambridge Studies in Advanced Mathematics 129, 2011 (pdf)
wikipedia L-function, Dirichlet L-function, special values of L-function, generalized Riemann hypothesis, Artin L-function, equivariant L-function, functional equation), modularity theorem
Terrence Tao‘s blog: Distinguished Lecture Series III: Shou-wu Zhang, “Triple L-series and effective Mordell conjecture”
Some history is in
Last revised on July 18, 2015 at 04:22:29. See the history of this page for a list of all contributions to it.