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In mathematics, anL-function is ameromorphicfunction on thecomplex plane, associated to one out of several categories ofmathematical objects. AnL-series is aDirichlet series, usuallyconvergent on ahalf-plane, that may give rise to anL-function viaanalytic continuation. TheRiemann zeta function is an example of anL-function, and some important conjectures involvingL-functions are theRiemann hypothesis and itsgeneralizations.
The theory ofL-functions has become a very substantial, and still largelyconjectural, part of contemporaryanalytic number theory. In it, broad generalisations of the Riemann zeta function and theL-series for aDirichlet character are constructed, and their general properties, in most cases still out of reach of proof, are set out in a systematic way. Because of theEuler product formula there is a deep connection betweenL-functions and the theory ofprime numbers.
The mathematical field that studiesL-functions is sometimes calledanalytic theory ofL-functions.
We distinguish at the outset between theL-series, aninfinite series representation (for example theDirichlet series for theRiemann zeta function), and theL-function, the function in the complex plane that is itsanalytic continuation. The general constructions start with anL-series, defined first as aDirichlet series, and then by an expansion as anEuler product indexed by prime numbers. Estimates are required to prove that this converges in some right half-plane of thecomplex numbers. Then one asks whether the function so defined can be analytically continued to the rest of the complex plane (perhaps with somepoles).
It is this (conjectural)meromorphic continuation to the complex plane which is called anL-function. In the classical cases, already, one knows that useful information is contained in the values and behaviour of theL-function at points where the series representation does not converge. The general termL-function here includes many known types of zeta functions. TheSelberg class is an attempt to capture the core properties ofL-functions in a set of axioms, thus encouraging the study of the properties of the class rather than of individual functions.
One can list characteristics of known examples ofL-functions that one would wish to see generalized:
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 zeta function connects through its values at positive even integers (and negative odd integers) to theBernoulli numbers, one looks for an appropriate generalisation of that phenomenon. In that case results have been obtained forp-adicL-functions, which describe certainGalois 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 withrandom matrix theory andquantum chaos are also of interest. The fractal structure of the distributions has been studied usingrescaled range analysis.[2] Theself-similarity of the zero distribution is quite remarkable, and is characterized by a largefractal dimension of 1.9. This rather large fractal dimension is found over zeros covering at least fifteen orders of magnitude for theRiemann zeta function, and also for the zeros of otherL-functions of different orders and conductors.
One of the influential examples, both for the history of the more generalL-functions and as a still-open research problem, is the conjecture developed byBryan Birch andPeter Swinnerton-Dyer in the early part of the 1960s. It applies to anelliptic curveE, and the problem it attempts to solve is the prediction of the rank of the elliptic curve over the rational numbers (or anotherglobal 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 ofL-functions. This was something like a paradigm example of the nascent theory ofL-functions.
This development preceded theLanglands program by a few years, and can be regarded as complementary to it: Langlands' work relates largely toArtinL-functions, which, likeHeckeL-functions, were defined several decades earlier, and toL-functions attached to generalautomorphic representations.
Gradually it became clearer in what sense the construction ofHasse–Weil zeta functions might be made to work to provide validL-functions, in the analytic sense: there should be some input from analysis, which meantautomorphic analysis. The general case now unifies at a conceptual level a number of different research programs.
