Dirichlet eta function

In analytic number theory, the Dirichlet η - function is a special function that is named after the German mathematician Dirichlet ( 1805-1859 ). It is related to the Riemann function.

It is listed with the small Greek letter eta (); the Dedekind η - function, a modular form is also referred to as.

Definition

The Dirichlet function is for all complex with real part greater than 0 defined by the Dirichlet series:

Although the validity of this expression is limited to complex numbers with positive real part, it forms the basis for all representations of the function. They can proceed analytically to the whole complex plane, thus ensuring a calculation of the feature for all arbitrary.

Euler product

Your number-theoretic significance at the function by their connection to the primes for formulaic by the Euler product

Can be expressed.

Functional equation

Throughout the identity holds:

Connection to the Riemann function

The functional equation between Dirichlet and Riemann function can be obtained from the Dirichletreihendarstellungen both functions. The expression is obtained by adding further transformed to Dirichlet series:

We conclude the relationship:

The reserves in the whole of validity.

Further illustrations

Integral representation

An integral representation for the gamma function and contains all states:

Valid for all is:

Series representation

A throughout convergent series is given by:

Product representation

For all converges the Hadamard product, named after its discoverer Jacques Hadamard:

It extends over all non-trivial zeros of the function and is derived simply from the Hadamard product of the zeta function from.

Values

The following applies:

For natural applies to the Bernoulli numbers

For just arguments the general formula:

Thus, the numerical value of a firm-wide in the form

Write, where and are two positive integers, respectively.

The first values ​​for odd arguments are

Zeros

From the relation

Is easy to conclude that both all in, and in addition the same locations as disappears. These include, for both the so-called " trivial" zeros, so

And the " non-trivial " null points in the strip.

The famous and up to now unproven Riemann Hypothesis states that all non- trivial zeros have real part 1/2.

Derivation

The derivative of the function is a Dirichlet series for again.

A closed form expression can

And are obtained by applying the product rule.

More

The affinities of the Dirichlet to function and the Riemann function can be expressed by the following formula:

Or

The Dirichlet eta function is a special case of Polylogarithmus because it is:

As such it is a special case of lerch between zeta function:

Also, applies