L-function

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 f {\displaystyle \textstyle f} is associated with a Dirichlet series:

∑ n ∈ N λ ( f , n ) n − s {\displaystyle \sum _{n\in \mathbb {N} }\lambda (f,n)n^{-s}} , which is also called an L-series, and an Euler product:

∏ p ∈ P ( 1 − α 1 ( f , p ) p − s ) − 1 ⋅ ⋯ ⋅ ( 1 − α d ( f , p ) p − s ) − 1 {\displaystyle \prod _{p\in \mathbb {P} }(1-\alpha _{1}(f,p)p^{-s})^{-1}\cdot \cdots \cdot (1-\alpha _{d}(f,p)p^{-s})^{-1}} . Here, λ ( f , n ) ∈ C {\displaystyle \textstyle \lambda (f,n)\in \mathbb {C} } for all natural numbers n ∈ N {\displaystyle \textstyle n\in \mathbb {N} } , and λ ( f , 1 ) = 1 {\displaystyle \textstyle \lambda (f,1)=1} . P {\displaystyle \textstyle \mathbb {P} } denotes the set of all prime numbers. The natural number d ∈ N {\displaystyle \textstyle d\in \mathbb {N} } is called the “degree” of the L-function or the Euler product L ( f , s ) {\displaystyle \textstyle L(f,s)} . For every prime number p {\displaystyle \textstyle p} and every i ∈ { 1 , … , d } {\displaystyle \textstyle i\in \{1,\ldots ,d\}} , we have α i ( f , p ) ∈ C {\displaystyle \textstyle \alpha _{i}(f,p)\in \mathbb {C} } . The complex numbers α i ( f , p ) ∈ C {\displaystyle \textstyle \alpha _{i}(f,p)\in \mathbb {C} } are called local roots or local parameters of L ( f , s ) ∈ C {\displaystyle \textstyle L(f,s)\in \mathbb {C} } at p ∈ P {\displaystyle \textstyle p\in \mathbb {P} } . For a given p ∈ P {\displaystyle p\in \mathbb {P} } , the expression

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