Primzeta function

The primzeta function is a mathematical function that plays a role in analytical number theory , a branch of mathematics . It is related to the Riemann zeta function . Like many other number theoretic functions, it gets its meaning from the connection to the prime numbers .

definition

For a complex number whose real part is greater than 1, the primzeta function is defined using a Dirichlet series that extends over all prime numbers. ${\displaystyle s}$

${\displaystyle P(s)=\sum _{p\ \mathrm {prim} }{\frac {1}{p^{s}}}={\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+{\frac {1}{5^{s}}}+{\frac {1}{7^{s}}}+{\frac {1}{11^{s}}}+\ldots }$

Although this representation only converges on the right half-plane shifted by 1, there is a continuation to the complete right half-plane , which, however, is not meromorphic in all points . ${\displaystyle \mathbb {H} =\{s\in \mathbb {C} \mid \mathrm {Re} \,s>0\}}$

Connection to the Riemann zeta function

There is a connection between the primzeta function and the logarithmic Riemann zeta function. This applies to everyone and is expressed in a formula: ${\displaystyle \mathrm {Re} \,s>0}$

${\displaystyle \log \zeta (s)=\sum _{n=1}^{\infty }{\frac {P(ns)}{n}}=P(s)+{\frac {P(2s)}{2}}+{\frac {P(3s)}{3}}+{\frac {P(4s)}{4}}+\ldots }$.

The Euler product of the zeta function serves as a simple way of proving this connection . With

${\displaystyle \zeta (s)=\prod _{p\ \mathrm {prim} }{\frac {1}{1-p^{-s}}}}$

is obtained by taking the logarithm on both sides:

${\displaystyle \log \zeta (s)=\log \prod _{p\ \mathrm {prim} }{\frac {1}{1-p^{-s}}}=-\sum _{p\ \mathrm {prim} }\log \left(1-{\frac {1}{p^{s}}}\right)=\sum _{n=1}^{\infty }{\frac {P(ns)}{n}}}$.

In the last step, the Taylor series of the natural logarithm was applied around the point . ${\displaystyle x=1}$

Further representations

The frequently used representation is obtained via a Möbius inversion :

${\displaystyle P(s)=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\log \zeta (ns)={\log \zeta (s)-{\frac {1}{2}}\log \zeta (2s)-{\frac {1}{3}}\log \zeta (3s)-{\frac {1}{5}}\log \zeta (5s)+{\frac {1}{6}}\log \zeta (6s)+\ldots }}$,

where the Möbius function is referred to here. ${\displaystyle \mu }$

properties

The primzeta function is a completely holomorphic function . It has a square-free positive integer singularities in the form of branch points at ${\displaystyle \{s\in \mathbb {C} \mid \mathrm {Re} \,s\,>1\}}$ ${\displaystyle K}$

• all places ${\displaystyle s=1/K}$
• all digits , where any ( non-trivial ) zero of the Riemann zeta function denotes. ${\displaystyle s=\rho /K}$ ${\displaystyle \rho }$

This is considering the illustration

${\displaystyle P(s)=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\log \zeta (ns)}$

clear, since the logarithm is not defined in all places or and ( in the sum). ${\displaystyle \textstyle \zeta (K\cdot {\frac {\rho }{K}})=\zeta (\rho )=0}$ ${\displaystyle \textstyle \zeta (K\cdot {\frac {1}{K}})=\zeta (1)=\infty }$ ${\displaystyle \mu (K)\not =0}$ ${\displaystyle n=K}$

Since the Riemann zeta function has an infinite number of non-trivial zeros in the so-called critical strip , there is a compression of singularities on the straight line , which can be seen as the natural limit of the domain of definition of the primzeta function. ${\displaystyle S=\{s\in \mathbb {C} |1>\mathrm {Re} \,s>0\}}$ ${\displaystyle \mathrm {Re} \,s=0}$

Furthermore applies to all : ${\displaystyle t}$

${\displaystyle \lim _{\sigma \to \infty }P(\sigma +\mathrm {i} t)=0}$.

Derivation

The primzeta function is completely holomorphic . A derivative expression is: ${\displaystyle \{s\in \mathbb {C} |\mathrm {Re} \,s>1\}}$

${\displaystyle P'(s)=-\sum _{p\ \mathrm {prim} }{\frac {\log p}{p^{s}}}}$.

The following applies to the -th derivative: ${\displaystyle k}$

${\displaystyle P^{(k)}(s)=(-1)^{k}\sum _{p\ \mathrm {prim} }{\frac {(\log p)^{k}}{p^{s}}}}$.

Indefinite integral

An antiderivative is given by:

${\displaystyle \int P(s)\ \mathrm {d} s=-\sum _{p\ \mathrm {prim} }{\frac {1}{p^{s}\log p}}+C}$.

Special values

As Euler was able to prove, the series of reciprocal values ​​of all prime numbers is divergent. The following applies:

${\displaystyle P(1)={\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{5}}+{\frac {1}{7}}+{\frac {1}{11}}+\ldots =\infty }$.

To this day nothing is known about other integer values ​​of the primzeta function. Decimal expansions are:

${\displaystyle P(2)=0{,}45224\ 74200\ 41065\ 49850\ldots }$(Follow A085548 in OEIS )

${\displaystyle P(3)=0{,}17476\ 26392\ 99443\ 53642\ldots }$(Follow A085541 in OEIS )

${\displaystyle P(4)=0{,}07699\ 31397\ 64246\ 84494\ldots }$(Follow A085964 in OEIS )

Web links

• Eric W. Weisstein: "Prime zeta function" on MathWorld (engl.)

Individual evidence

1. ↑ Komaravolu Chandrasekharan: Introduction to Analytical Number Theory . Springer Verlag, 1965/66, Chapter XI, page 2