In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a Dirichlet L-series is a function of the form


Here χ is a Dirichlet character

Dirichlet character

In number theory, Dirichlet characters are certain arithmetic functions which arise from completely multiplicative characters on the units of \mathbb Z / k \mathbb Z...

 and s a complex variable with real part greater than 1. By analytic continuation

Analytic continuation

In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...

, this function can be extended to a meromorphic function

Meromorphic function

In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function...

 on the whole complex plane

Complex plane

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...

, and is then called a Dirichlet L-function and also denoted L(s, χ).

These functions are named after Johann Peter Gustav Lejeune Dirichlet

Johann Peter Gustav Lejeune Dirichlet

Johann Peter Gustav Lejeune Dirichlet was a German mathematician with deep contributions to number theory , as well as to the theory of Fourier series and other topics in mathematical analysis; he is credited with being one of the first mathematicians to give the modern formal definition of a...

 who introduced them in to prove the theorem on primes in arithmetic progressions

Dirichlet's theorem on arithmetic progressions

In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n ≥ 0. In other words, there are infinitely many primes which are...

 that also bears his name. In the course of the proof, Dirichlet shows that is non-zero at s = 1. Moreover, if χ is principal, then the corresponding Dirichlet L-function has a simple pole at s = 1.

Zeros of the Dirichlet L-functions

If χ is a primitive character with χ(−1) = 1, then the only zeros of L(s,χ) with Re(s) < 0 are at the negative even integers.
If χ is a primitive character with χ(−1) = −1, then the only zeros of L(s,χ) with Re(s) < 0 are at the negative odd integers.

Up to the possible existence of a Siegel zero

Siegel zero

In mathematics, more specifically in the field of analytic number theory, a Siegel zero, named after Carl Ludwig Siegel, is a type of potential counterexample to the generalized Riemann hypothesis, on the zeroes of Dirichlet L-function....

, zero-free regions including and beyond the line Re(s) = 1 similar to that of the Riemann zeta function are known to exist for all Dirichlet L-functions.

Just as the Riemann zeta function is conjectured to obey the Riemann hypothesis

Riemann hypothesis

In mathematics, the Riemann hypothesis, proposed by , is a conjecture about the location of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2...

, so the Dirichlet L-functions are conjectured to obey the generalized Riemann hypothesis

Generalized Riemann hypothesis

The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global L-functions, which are formally similar to the Riemann zeta-function...

.

Euler product

Since a Dirichlet character χ is completely multiplicative, its L-function can also be written as an Euler product

Euler product

In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The name arose from the case of the Riemann zeta-function, where such a product representation was proved by Leonhard Euler.-Definition:...

 in the half-plane of absolute convergence

Absolute convergence

In mathematics, a series of numbers is said to converge absolutely if the sum of the absolute value of the summand or integrand is finite...

:
where the product is over all prime number

Prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

s.

Functional equation

Let us assume that χ is a primitive character to the modulus k. Defining


where Γ denotes the Gamma function

Gamma function

In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...

 and the symbol a is given by


one has the functional equation

Functional equation (L-function)

In mathematics, the L-functions of number theory are expected to have several characteristic properties, one of which is that they satisfy certain functional equations. There is an elaborate theory of what these equations should be, much of which is still conjectural...

Here we wrote τ(χ) for the Gauss sum

Gauss sum

In mathematics, a Gauss sum or Gaussian sum is a particular kind of finite sum of roots of unity, typicallyG := G= \sum \chi\cdot \psi...

Note that |τ(χ)|=k^1/2.

Relation to the Hurwitz zeta-function

The Dirichlet L-functions may be written as a linear combination of the Hurwitz zeta-function at rational values. Fixing an integer k ≥ 1, the Dirichlet L-functions for characters modulo k are linear combinations, with constant coefficients, of the ζ(s,q) where q = m/k and m = 1, 2, ..., k. This means that the Hurwitz zeta-function for rational q has analytic properties that are closely related to the Dirichlet L-functions. Specifically, let be a character modulo k. Then we can write its Dirichlet L-function as


In particular, the Dirichlet L-function of the trivial character (which implies the modulus k is prime) yields the Riemann zeta-function:

See also

• Generalized Riemann hypothesis
  Generalized Riemann hypothesis
  The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global L-functions, which are formally similar to the Riemann zeta-function...
  
  
• L-function
  L-function
  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 L-series for a Dirichlet character are constructed, and their general properties, in most cases...
  
  
• Modularity theorem
• Artin conjecture
• Special values of L-functions
  Special values of L-functions
  In mathematics, the study of special values of L-functions is a subfield of number theory devoted to generalising formulae such as the Leibniz formula for pi, namely...