Dirichlet series

Dirichlet, Peter Gustav Lejeune Dirichlet named after, are rows that are used in analytic number theory to investigate number-theoretic functions with methods from the analysis, especially the theory of functions. Many open number-theoretic problems have become through this context of " approximate solution " available ( through assessments) about questions concerning the distribution of prime numbers.

Convergent Dirichlet series are analytic functions as well as detached from number-theoretic problems interesting as an object of study, since they are closely related to power series and allow a similar "natural" representation of analytic functions.

• 4.1 Riemann zeta function
• 4.2 Dirichlet series of the splitting function
• 4.3 Dirichlet series of the Möbius function
• 4.4 Dirichlet L- series
• 4.5 Dirichlet series of Mangoldt function
• 4.6 Dirichlet lambda function
• 4.7 Dirichlet series of Euler's φ - function
• 4.8 Dirichlet series of the generalized divisor sum function

Definition and formal properties

A Dirichlet series is a series of the form

This series converges absolutely for certain coefficient sequences f (n) and complex numbers see The product of two such absolutely convergent Dirichlet series is again an absolutely convergent Dirichlet series, the coefficients are obtained by convolution of the coefficient sequences as number-theoretic functions. This corresponds to the multiplication of absolutely convergent Dirichlet convolution of their coefficients.

Occasionally you can find in the literature ( for example in Zagier ), the more general definition

With once again, that the first definition, with one obtains

So an ordinary power series.

The space of formal Dirichlet series is provided with a multiplication by transfers valid for absolutely convergent series multiplication rule to arbitrary (even nichtkonvergente ) Dirichlet series ( for this construction compare also the analogous concept formation formal power series ).

This will be the space of formal Dirichlet series with the pointwise addition, scalar multiplication and convolution isomorphic ( as a ring and algebra) to the number-theoretic functions and inherits all the structural properties of this space.

The isomorphism assigns to each number-theoretic function f (n) to the formal Dirichlet series whose coefficients episode it is. This Dirichlet series F (s) is then called the Dirichlet series generated by f (n).

Convergent Dirichlet series

For each Dirichlet series which converges somewhere but not everywhere, there exists a real number such that the series converges in the half-plane ( the real part of ) and diverges in the half-plane. About the behavior on the line can not make any general statement. If the Dirichlet series converges everywhere or nowhere, or is set and is called in all cases of convergence of the Dirichlet series.

Similarly, how to calculate the radius of convergence in the case of power series, one can also, in the case of Dirichlet determine the of convergence with a limit superior of its coefficient sequence, the following applies:

Is divergent, then

However, if convergent, so is

Analytical properties

In its convergence half-plane, the Dirichlet series is compact convergent and there it represents a holomorphic function

The derivatives of the thus determined holomorphic function can be obtained by termwise differentiation. Your -th derivative is the Dirichlet series

Euler products

Dirichlet series with multiplicative number-theoretic functions as coefficients can be represented as Euler product. Is a multiplicative number-theoretic function and Dirichlet series converges generated by their F (s) for the complex number s absolute, then applies

In the case of a completely multiplicative function, this product simplifies to

This infinite product over all primes are called Euler products. The value of these products is defined as a limit of the sequence of finite products formed by extending the product of primes only below a limit N.

Important Dirichlet series

Riemann zeta function

The most famous Dirichlet series is the Riemann zeta function:

It is the number-theoretic one function produces ( with for all n). Since this function is completely multiplicative, the zeta function has the Euler product representation

Dirichlet series of the splitting function

The divisor function ( also more accurate divider number function) that assigns a natural number, the number of its positive divisors is the " convolution square " of the 1 function.

Its associated Dirchletreihe is therefore the square of the zeta function:

Dirichlet series of the Möbius function

The Möbius function is multiplicative with for. So the Dirichlet series generated by their M (s) has the Euler product

The relation M (s) · ζ (s) = 1 is transferred to the corresponding number-theoretic functions, meaning there:

Dirichlet L- series

The also introduced by Dirichlet L- series

Are generated by a Dirichlet character. These series play an important role in the proof of Dirichlet's theorem on primes in arithmetic existence of infinitely many progressions. Since Dirichletcharaktere are completely multiplicative, one can represent the L- series as Euler products

And for the main character modulo k applies:

The L- series generalize the Riemann zeta function. → About the zeros of L- series, there are to this day unproven generalized Riemann hypothesis.

Dirichlet series of Mangoldt function

The Mangold Czech function Λ ( n ) plays a role in the proof of the prime number theorem. This number-theoretic function is defined as

The Dirichlet series generated by it can be expressed by the zeta function:

Dirichlet lambda function

The Dirichlet lambda function is the L- series by

They can be represented by the Riemann zeta function as

It can be calculated in closed form to the places where this is possible for the zeta - function, that is just for positive numbers following relationship exists with the Dirichlet eta function:

Dirichlet series of Euler's φ - function

The Euler φ - function is multiplicative with

Is the Euler product of Dirichlet series generated by it

Dirichlet series of the generalized divisor sum function

The generalized divisor sum function σk ( n ) is multiplicative and is for prime powers

Therefore, the Dirichlet series of σk has the Euler product representation: