Editing L-function

{{Short description|Meromorphic function on the complex plane}}
{{Expand German|L-Funktion|date=March 2024}}
{{DISPLAYTITLE:''L''-function}}
[[File:Riemann-Zeta-Func.png|right|thumb|300px|The [[Riemann zeta function]] can be thought of as the archetype for all ''L''-functions.<ref>{{cite web |first=Jörn |last=Steuding |title=An Introduction to the Theory of ''L''-functions |work=Preprint |date=June 2005 |url=https://www.scribd.com/document/230217684/An-Introduction-to-the-Theory-of-L-Functions }}</ref>]]

In mathematics, an '''''L''-function''' is a [[meromorphic]] [[Function (mathematics)|function]] on the [[complex plane]], associated to one out of several categories of [[mathematical object]]s. An '''''L''-series''' is a [[Dirichlet series]], usually [[convergence (mathematics)|convergent]] on a [[half-plane]], that may give rise to an ''L''-function via [[analytic continuation]]. The [[Riemann zeta function]] is an example of an ''L''-function, and some important conjectures involving ''L''-functions are the [[Riemann hypothesis]] and its [[Generalized Riemann hypothesis|generalizations]].

The theory of ''L''-functions 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 [[Dirichlet L-function|''L''-series]] 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.  Because of the [[Euler product formula]] there is a deep connection between ''L''-functions and the theory of [[prime number]]s.

The mathematical field that studies ''L''-functions is sometimes called '''analytic theory of ''L''-functions'''.

== Construction ==

We distinguish at the outset between the '''''L''-series''', an [[infinite set|infinite]] series representation (for example the [[Dirichlet series]] for the [[Riemann zeta function]]), and the '''''L''-function''', the function in the complex plane that is its [[analytic continuation]]. The general constructions start with an ''L''-series, 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 half-plane of the [[complex number]]s. Then one asks whether the function so defined can be analytically continued to the rest of the complex plane (perhaps with some [[Pole (complex analysis)|pole]]s).

It is this (conjectural) [[meromorphic]] continuation to the complex plane which is called an ''L''-function. In the classical cases, already, one knows that useful information is contained in the values and behaviour of the ''L''-function at points where the series representation does not converge. The general term ''L''-function here includes many known types of zeta functions. The [[Selberg class]] is an attempt to capture the core properties of ''L''-functions 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 ''L''-functions that one would wish to see generalized:

* location of zeros and poles;
* [[functional equation (L-function)|functional equation]], with respect to some vertical line Re(''s'') = constant;
* interesting values at integers related to quantities from [[algebraic K-theory|algebraic ''K''-theory]].

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 the [[Bernoulli numbers]], one looks for an appropriate generalisation of that phenomenon. In that case results have been obtained for [[p-adic L-function|''p''-adic ''L''-function]]s, which describe certain [[Galois module]]s.

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.<ref name="Shanker">{{cite journal|author=O. Shanker|year=2006|title=Random matrices, generalized zeta functions and self-similarity of zero distributions|journal=J. Phys. A: Math. Gen.|volume=39|issue=45 |pages=13983–13997    | doi = 10.1088/0305-4470/39/45/008|bibcode=2006JPhA...3913983S|s2cid=54958644 }}</ref> The [[self-similarity]] 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 ''L''-functions of different orders and conductors.

== Birch and Swinnerton-Dyer conjecture ==
{{main|Birch and Swinnerton-Dyer conjecture}}

One of the influential examples, both for the history of the more general ''L''-functions and as a still-open research problem, is the conjecture developed by [[Bryan Birch]] and [[Peter Swinnerton-Dyer]] 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 ''L''-functions. This was something like a paradigm example of the nascent theory of ''L''-functions.

== 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 L-function|Artin ''L''-functions]], which, like [[Hecke L-function (disambiguation)|Hecke ''L''-functions]]<!-- intentional link to DAB page -->, were defined several decades earlier, and to ''L''-functions attached to general [[automorphic representation]]s.

Gradually it became clearer in what sense the construction of [[Hasse–Weil zeta function]]s might be made to work to provide valid ''L''-functions, 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==
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*[[Generalized Riemann hypothesis]]
*[[Dirichlet L-function|Dirichlet ''L''-function]]
*[[Automorphic L-function|Automorphic ''L''-function]]
*[[Modularity theorem]]
*[[Artin conjecture (L-functions)|Artin conjecture]]
*[[Special values of L-functions|Special values of ''L''-functions]]
*[[Explicit formulae for L-functions]]
*[[Shimizu L-function|Shimizu ''L''-function]]
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==References==
{{reflist}}

* {{Neukirch ANT}}

==External links==
* {{cite web | url = http://www.lmfdb.org | title = LMFDB, the database of L-functions, modular forms, and related objects }}
*{{SpringerEOM|title=L-function|id=L-function&oldid=19281|last=Lavrik|first=A.F.|mode=cs1}}
;Articles about a breakthrough third degree transcendental L-function
:*{{cite news | url = http://www.physorg.com/news124636003.html | title = Glimpses of a new (mathematical) world | date = March 13, 2008 | work = Physorg.com | department = Mathematics | agency = American Institute of Mathematics}}
:*{{cite news | url = http://www.sciencenews.org/view/generic/id/9542/title/Math_Trek__Creeping_Up_on_Riemann | title = Creeping Up on Riemann | work = Science News | date = April 2, 2008 | first = Julie | last = Rehmeyer | access-date = August 5, 2008 | archive-date = February 16, 2012 | archive-url = https://web.archive.org/web/20120216201232/http://www.sciencenews.org/view/generic/id/9542/title/Math_Trek__Creeping_Up_on_Riemann | url-status = dead }}
:*{{cite news | url = http://www.physorg.com/news137248087.html | title = Hunting the elusive L-function | date = August 6, 2008 | work = Physorg.com | department = Mathematics | agency = University of Bristol }}
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[[Category:Zeta and L-functions|*]]