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Concept# Dirichlet L-function

Summary

In mathematics, a Dirichlet L-series is a function of the form
where is a Dirichlet character and s a complex variable with real part greater than 1. It is a special case of a Dirichlet series. By analytic continuation, it can be extended to a meromorphic function on the whole complex plane, and is then called a Dirichlet L-function and also denoted L(s, χ).
These functions are named after Peter Gustav Lejeune Dirichlet who introduced them in to prove the theorem on primes in arithmetic progressions that also bears his name. In the course of the proof, Dirichlet shows that L(s, χ) is non-zero at s = 1. Moreover, if χ is principal, then the corresponding Dirichlet L-function has a simple pole at s = 1. Otherwise, the L-function is entire.
Since a Dirichlet character χ is completely multiplicative, its L-function can also be written as an Euler product in the half-plane of absolute convergence:
where the product is over all prime numbers.
Results about L-functions are often stated more simply if the character is assumed to be primitive, although the results typically can be extended to imprimitive characters with minor complications. This is because of the relationship between a imprimitive character and the primitive character which induces it:
(Here, q is the modulus of χ.) An application of the Euler product gives a simple relationship between the corresponding L-functions:
(This formula holds for all s, by analytic continuation, even though the Euler product is only valid when Re(s) > 1.) The formula shows that the L-function of χ is equal to the L-function of the primitive character which induces χ, multiplied by only a finite number of factors.
As a special case, the L-function of the principal character modulo q can be expressed in terms of the Riemann zeta function:
Dirichlet L-functions satisfy a functional equation, which provides a way to analytically continue them throughout the complex plane. The functional equation relates the value of to the value of . Let χ be a primitive character modulo q, where q > 1.

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