Lfunction
In mathematics, an Lfunction is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An Lseries is a Dirichlet series, usually convergent on a halfplane, that may give rise to an Lfunction via analytic continuation.
The theory of Lfunctions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the Lseries for a Dirichlet character are constructed, and their general properties, in most cases still out of reach of proof, are set out in a systematic way.
Contents
Construction
We distinguish at the outset between the Lseries, an infinite series representation (for example the Dirichlet series for the Riemann zeta function), and the Lfunction, the function in the complex plane that is its analytic continuation. The general constructions start with an Lseries, defined first as a Dirichlet series, and then by an expansion as an Euler product indexed by prime numbers. Estimates are required to prove that this converges in some right halfplane of the complex numbers. Then one asks whether the function so defined can be analytically continued to the rest of the complex plane (perhaps with some poles).
It is this (conjectural) meromorphic continuation to the complex plane which is called an Lfunction. In the classical cases, already, one knows that useful information is contained in the values and behaviour of the Lfunction at points where the series representation does not converge. The general term Lfunction here includes many known types of zetafunctions. The Selberg class is an attempt to capture the core properties of Lfunctions in a set of axioms, thus encouraging the study of the properties of the class rather than of individual functions.
Conjectural information
One can list characteristics of known examples of Lfunctions that one would wish to see generalized:
 location of zeros and poles;
 functional equation, with respect to some vertical line Re (s) = constant;
 interesting values at integers related to quantities from algebraic Ktheory
Detailed work has produced a large body of plausible conjectures, for example about the exact type of functional equation that should apply. Since the Riemann zetafunction connects through its values at positive even integers (and negative odd integers) to the Bernoulli numbers, one looks for an appropriate generalisation of that phenomenon. In that case results have been obtained for padic Lfunctions, which describe certain Galois modules.
The statistics of the zero distributions are of interest because of their connection to problems like the Generalized Riemann hypothesis, distribution of prime numbers, etc. The connections with random matrix theory and quantum chaos are also of interest. The fractal structure of the distributions has been studied using rescaled range analysis.^{[2]} The selfsimilarity of the zero distribution is quite remarkable, and is characterized by a large fractal dimension of 1.9. This rather large fractal dimension is found over zeros covering at least fifteen orders of magnitude for the Riemann zeta function, and also for the zeros of other Lfunctions of different orders and conductors.
Birch and SwinnertonDyer conjecture
One of the influential examples, both for the history of the more general Lfunctions and as a stillopen research problem, is the conjecture developed by Bryan Birch and Peter SwinnertonDyer in the early part of the 1960s. It applies to an elliptic curve E, and the problem it attempts to solve is the prediction of the rank of the elliptic curve over the rational numbers (or another global field): i.e. the number of free generators of its group of rational points. Much previous work in the area began to be unified around a better knowledge of Lfunctions. This was something like a paradigm example of the nascent theory of Lfunctions.
Rise of the general theory
This development preceded the Langlands program by a few years, and can be regarded as complementary to it: Langlands' work relates largely to Artin Lfunctions, which, like Hecke Lfunctions, were defined several decades earlier, and to Lfunctions attached to general automorphic representations.
Gradually it became clearer in what sense the construction of Hasse–Weil zetafunctions might be made to work to provide valid Lfunctions, in the analytic sense: there should be some input from analysis, which meant automorphic analysis. The general case now unifies at a conceptual level a number of different research programs.
See also
 Generalized Riemann hypothesis
 Dirichlet Lfunction
 Automorphic Lfunction
 Modularity theorem
 Artin conjecture
 Special values of Lfunctions
 Shimizu Lfunction
References
 ^ Jorn Steuding, An Introduction to the Theory of Lfunctions, Preprint, 2005/06
 ^ O. Shanker (2006). "Random matrices, generalized zeta functions and selfsimilarity of zero distributions". J. Phys. A: Math. Gen. 39 (45): 13983–13997. Bibcode:2006JPhA...3913983S. doi:10.1088/03054470/39/45/008.
 Neukirch, Jürgen (1999). Algebraic Number Theory. Grundlehren der mathematischen Wissenschaften. 322. Berlin: SpringerVerlag. ISBN 9783540653998. MR 1697859. Zbl 0956.11021.
External links
 "LMFDB, the database of Lfunctions, modular forms, and related objects".
 Lavrik, A.F. (2001) [1994], "Lfunction", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 9781556080104
 Articles about a breakthrough third degree transcendental Lfunction

 "Glimpses of a new (mathematical) world". Mathematics. Physorg.com. American Institute of Mathematics. March 13, 2008.
 Rehmeyer, Julie (April 2, 2008). "Creeping Up on Riemann". Science News.
 "Hunting the elusive Lfunction". Mathematics. Physorg.com. University of Bristol. August 6, 2008.