An L-function is a meromorphic function on the complex plane, and one out of several categories of mathematical objects studied in analytic number theory and related fields. L-functions share fundamental properties and characteristics with the Riemann zeta function, which serves as the prototypical example of an L-function; therefore, L-functions are generalisations of the Riemann zeta function. Some important conjectures involving L-functions are, consequently, the Riemann hypothesis and its generalisations.
A Dirichlet series, usually convergent on a half-plane, that may give rise to an L-function via analytic continuation, is called an L-series.
Fundamental subclasses of L-functions were built on the work of Leonhard Euler (which is now known as the Riemann zeta function). Most notably, the mathematicians Bernhard Riemann (1826–1866), Richard Dedekind (1831–1916), Erich Hecke (1887–1947) and Emil Artin (1898–1962) investigated the subclasses of L-functions, discovering eponymous L-functions each.
The terms "L-function" and "zeta-function" are often used synonymously due to the fundamentally similar and derivative nature of the work, however, not all zeta-functions are L-functions. Most notably, the Prime zeta function is not an L-function, since they cannot be analytically extended to the entire complex plane.
Definition
As one might infer from the introduction, there is still no general and widely accepted definition of an L-function and its construction. Various constructions and definitions as per various prominent authors can be found as follows.
Iwaniec and Kowalski's Analytic Number Theory, 2004 This definition is abstract and incomplete in the sense that it doesn't specify the arithmetic objects to which he assigns L-functions, nor the exact mechanism of this assignment. However, it includes properties generally expected of L-functions.
The definition is extended and starts by defining 6 preliminary definitions, as follows:
Dirichlet Series and Euler Product The arithmetic object is associated with a Dirichlet series:
,
which is also called an L-series, and an Euler product:
.
Here, for all natural numbers , and . denotes the set of all prime numbers.
The natural number is called the “degree” of the L-function or the Euler product . For every prime number and every , we have .
The complex numbers are called local roots or local parameters of at .
For a given , the expression , that is, the th factor in the Euler product, which is called the Euler factor of at .
Gamma Factor
The object is assigned a so-called gamma factor:
where denotes the gamma function, denotes the automorphic number, and denotes the degree of the L-function mentioned above. The parameters are complex numbers. They are called the local parameters of at infinity, or at the infinite prime point.
Leader (Conductor)
A natural number is also assigned to the object , This is the so-called “leader” or “conductor” of the L-function . The prime numbers that do not divide are called unbranched with respect to the L-function .
Complete L-Function
Using the Dirichlet series, the gamma factor, and the leading coefficient associated with , we can now define the so-called complete L-function of :
Root
Furthermore, the object is associated with a complex number
This complex number is called the root of the L-function .
Dual, Arithmetic Object
Now, the arithmetic object is associated with another arithmetic object (which is not specified in detail within the framework of this abstract definition). It is called the dual of and is denoted by . As in the case of , is also a Dirichlet series , an Euler product
with , a gamma factor , a leading term , and a complete L-function .
If , then is called self-dual, which means nothing other than for all .Iwaniec, Kowalski: Analytic Number Theory. 2004, Chapter 5, Section 1, p. 95.
Conditions to satisfy The objects mentioned above, associated with the arithmetic object , must now satisfy the following conditions so that satisfies the definition of an L-function according to Iwaniec and Kowalski: +Conditions to satisfy for the arithmetic object IndexConditionDescription1Absolute value of local parameters for For every prime number and every , we have .2Values of local parameters for unbranched For all primes that are unramified with respect to , and for all , we have .3Requirements for the local parameters at infinityThe parameters are either real or appear in the form of complex conjugate pairs in the gamma factor . Furthermore, for every . This last condition ensures that has no zeroes in and no poles with . denotes the real part of a complex number.4Absolute convergence of the Dirichlet Series and the Euler productBoth the Dirichlet series and the Euler product associated with must converge absolutely for .5Agreement between the L-function, the Dirichlet series, and the Euler product in a complex half-planeThe L-function, the Dirichlet series, and the Euler product associated with must coincide in the complex half-plane :6Analytic Continuity and Pole PointsIt follows from the conditions that the must satisfy that the Dirichlet series is holomorphic in the half-plane . However, this must also be analytically extendable to a meromorphic function of order 1 on the entire , which has poles at most at and .7Absolute value of the root numberThe root number has the absolute value 1. Thus, must equal 1.8Requirements for the objects associated with the dual of With regard to the dual of , the following must hold: for all :, as well as and .
These imply that in the Dirichlet series associated with , the coefficients are precisely the complex conjugates of the coefficients in the Dirichlet series associated with . The gamma factors and leaders associated with and , respectively, must coincide.9Functional equationThe two complete L-functions associated with and , respectively, satisfy the functional equation
for all .
Iwaniec and Kowalski’s definition reflects the fact that a function that is considered an L-function typically appears as a mapping of the L-function to a mathematical object (e.g., a Dirichlet character or an algebraic number field). Their definition is abstract and incomplete, as it leaves open the question of exactly what those mathematical objects are and how that mapping is to be carried out.
Atle Selberg in the Amalfi Conference on Analytic Number Theory, 1992
The definition proposed by the Norwegian-American mathematician Atle Selberg in 1989 is independent of other mathematical objects. In a non-abstract, unambiguous definition, he specifies a subset of the set of all Dirichlet series whose elements must satisfy certain properties: absolute convergence of the Dirichlet series, analytic continuation, functional equation, Ramanujan conjectureThe Ramanujan conjecture refers to the coefficients of the Dirichlet series. It states: For any , . Here, the implicit constant in the Landau symbol may depend on . and Euler product. This subset is today referred to as the Selberg class.Atle Selberg: Old and new conjectures and results about a class of Dirichlet-series. In: Enrico Bombieri et al. (eds.): Proceedings of the Amalfi Conference on Analytic Number Theory. 1992, pp. 367–385; Collected Papers. Vol. II, Springer, 1991, pp. 47–63.
The overarching hypothesis and the motivating background for the definition of the Selberg class is the so-called Great Riemann Hypothesis. Applied to the Selberg class, this hypothesis states that no zero of an analytic continuation of a Dirichlet series in the Selberg class has a real part greater than 1/2. In the case of the (supposedly) simplest element of the Selberg class (the Riemann Dirichlet series together with its analytic continuation to the Riemann zeta function), this conjecture corresponds to the Riemann Hypothesis, which to this day has been neither proven nor disproven.
It is against this backdrop that the remaining shortcomings in the definition of the term L-function must be viewed: one would like to define the term "L-function" in such a way that L-functions verifiably satisfy the Great Riemann Hypothesis. Yet, on the other hand, one has not even been able to prove the simplest case (the Riemann Hypothesis for the Riemann zeta function), which could be a sign of a lack of understanding of the Riemann zeta function and thus makes it difficult to provide a clear definition of the generalized concept of a L-function.
Examples
This section provides an overview of basic examples of L-functions.
Riemann Zeta Function
The simplest example of an L-function, and at the same time the fundamental basis for any definition of the term L-function based on the work of Leonhard Euler, is the Riemann zeta function .Neukirch: Algebraic Number Theory. Chapter 7, Section 1, 1992, p. 439 ff.
One of the possible “arithmetic objects” in the sense of the definition approach by Iwaniec and Kowalski, to which this L-function can be assigned, is the field of the rational numbers. Its Dirichlet series
that is
for all , converges absolutely for . Together with its Euler product, which is also absolutely convergent, the following holds for :Neukirch: Algebraic Number Theory. Chapter 7, Section 1, Theorem 1.1, 1992, p. 439.
Since all are real, namely equal to 1, is self-dual. The object dual to is therefore also , so . The degree of the Euler product of the Riemann zeta function is . For its local parameters at , the following holds:
for all . Usually, the following gamma factor is used for the Riemann zeta function:
The local parameter at infinity is therefore 0. The conductor of is , so that the complete Riemann zeta function takes the form
This definition is only valid for , since the Riemann zeta function can be defined via its Dirichlet series or its Euler product only in this half-plane. However, the complete Riemann zeta function has an analytic continuation to a meromorphic function on the entire complex plane. This extension is holomorphic except for two simple poles at and with residues of −1 and 1, respectively.Neukirch: Algebraic Number Theory. Chapter 7, Section 1, Theorem 1.6, 1992, p. 445. If we also denote the continued, complete Riemann zeta function by , then it satisfies the root number
the functional equationNeukirch: Algebraic Number Theory. Chapter 7, Section 1, Theorem 1.6, 1992, p. 445.
Thus, the Riemann zeta function, initially defined only for via its Dirichlet series or Euler product, now has an analytic continuation to a meromorphic function on , which is undefined only at , since it has a simple pole with the residue 1 there. If we retain the notation for the continued Riemann zeta function as well, it satisfies the functional equationNeukirch: Algebraic Number Theory. Chapter 7, Paragraph 1, Corollary 1.7, 1992, p. 446.
The (analytically extended) Riemann zeta function raises one of the most important questions in analytic number theory: the question of the exact location of its so-called non-trivial zeros. These lie in the “critical strip” . The Riemann Hypothesis from 1859—which remains neither proven nor disproven to this day—posits that all non-trivial zeros of the Riemann zeta function have a real part of . A proof of this hypothesis would allow for particularly accurate estimates of the distribution of prime numbers.
Dirichlet L-functions
The closest relatives of the Riemann zeta function are the Dirichlet L-functions, which include the Riemann zeta function as a special case. While in the Dirichlet series associated with the Riemann zeta function all coefficients are equal to 1, in Dirichlet L-functions these are defined using a Dirichlet character. They thus take complex values with absolute value 1 or are equal to 0. Let be an integer, and let be a Dirichlet character modulo :