Prime zeta function

The Primzetafunktion is a mathematical function that plays a role in analytic number theory, a branch of mathematics. It is related to the Riemann zeta function. Like many other number-theoretic functions they acquired their meaning over the connection to the primes.

Definition

For a complex number whose real part is greater than 1, the Primzetafunktion is defined by a Dirichlet series that extends over all prime numbers.

Although this representation converges only on the right half-plane shifted by 1, there exists an analytic continuation to the entire right half-plane.

Connection to the Riemann zeta function

There exists a correlation between the Primzetafunktion and the logarithmic Riemann zeta function. This applies to all and expressed formulaic from over:

As a simple proof ability of this compound, the Euler product of zeta function serves. with

Is obtained by taking the logarithm on both sides:

In the last step, the Taylor series of the natural logarithm was applied to the point.

Further illustrations

About a Möbius inversion we obtain the frequently used display:

Where here denotes the Möbius function.

Properties

The Primzetafunktion is a meromorphic function in the whole, with the right half-plane, respectively. It has a square-free positive integer singularities in the form of branch points at

• All locations
• All locations, with any ( non-trivial ) zero of the Riemann zeta function called.

This is in consideration of the display

Significantly, since the log at all points, respectively, and is not defined ( in the sum ).

Since the Riemann zeta function in the so-called critical strip has infinitely many non-trivial zeros, there is a Polverdichtung on the straight line, which can be regarded as a natural boundary of the domain of Primzetafunktion.

Furthermore, if any:

Derivation

The Primzetafunktion is holomorphic throughout. A derivative term is:

For the - th derivative of the following applies:

Antiderivative

A primitive function is given by:

Special values

About other integer values ​​of Primzetafunktion is still not known. Decimal expansions are: