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Wikipedia / Dirichlet L-function

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

is a Dirichlet character a complex variable with real part. 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; it is then called a Dirichlet L-function.

These functions are named after Peter Gustav Lejeune Dirichlet who introduced them in 1837 to prove his theorem on primes in arithmetic progressions is principal, then the corresponding Dirichlet L-function has a simple pole. Otherwise, the L-function is entire.

Euler product

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.

Primitive characters

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. which induces it:.) An application of the Euler product gives a simple relationship between the corresponding L-functions:. The formula shows that the L is equal to the L, multiplied by only a finite number of factors.

As a special case, the L can be expressed in terms of the Riemann zeta function:

Functional equation

Dirichlet L-functions satisfy a functional equation.

Let . One way to express the functional equation is as is the gamma function is the Gauss sum. which can be expressed as

are entire functions.

For generalizations, see the article on functional equations of L-functions.

Zeros

.

There are no zeros, there are zeros at certain negative integers:

  • .

  • .

These are called the trivial zeros.

is a complex character. The generalized Riemann hypothesis.

Up to the possible existence of a Siegel zero similar to that of the Riemann zeta function are known to exist for all Dirichlet L, we have

a non-real zero.

Relation to the Hurwitz zeta function

Dirichlet L-functions may be written as linear combinations of the Hurwitz zeta function, Dirichlet L and has analytic properties that are closely related to the Dirichlet L, we can write its Dirichlet L-function as

See also

Notes

References