Dirichlet beta function

Dirichlet beta function β (s)

The Dirichlet beta function , written with the Greek letter ${\displaystyle \beta }$ , is a special mathematical function that plays a role in analytical number theory , a branch of mathematics. It forms z. B. the basis for the analytical theory of the distribution of prime numbers in the arithmetic sequences and and is related to the Riemann zeta function . ${\displaystyle 4m+1}$ ${\displaystyle 4m+3}$

It was named after the German mathematician Peter Gustav Lejeune Dirichlet (1805-1859).

definition

For a complex number whose real part is greater than 0, the beta function is defined using the Dirichlet series : ${\displaystyle s}$

${\displaystyle \beta (s)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)^{s}}}=1-{\frac {1}{3^{s}}}+{\frac {1}{5^{s}}}-{\frac {1}{7^{s}}}+{\frac {1}{9^{s}}}-+\ldots }$

Although this expression only converges on the right half plane , it represents the basis for all further representations of the beta function. To calculate the beta function for all numbers in the complex plane, one uses its analytical continuation . ${\displaystyle \mathbb {H} =\{s\in \mathbb {C} |\mathrm {Re} \,s>0\}}$

Product presentation

For the beta function there is a product representation that converges for all complexes whose real part is greater than 1. ${\displaystyle s}$

${\displaystyle \beta (s)=\prod _{p\equiv 1\ \mathrm {mod} \ 4}{\frac {1}{1-p^{-s}}}\prod _{p\equiv 3\ \mathrm {mod} \ 4}{\frac {1}{1+p^{-s}}}}$

This implies that all prime numbers of the form (i.e. ) are multiplied. Analog means that it is multiplied over all prime numbers that have the form (i.e. ). ${\displaystyle p\equiv 1\ \mathrm {mod} \ 4}$ ${\displaystyle p=4m+1}$ ${\displaystyle p=5,13,17,...}$ ${\displaystyle p\equiv 3\ \mathrm {mod} \ 4}$ ${\displaystyle p=4m+3}$ ${\displaystyle p=3,7,11,...}$

Functional equation

The functional equation applies to all of them : ${\displaystyle z\in \mathbb {C} }$

${\displaystyle \beta (1-z)=\left({\frac {2}{\pi }}\right)^{z}\sin \left({\tfrac {1}{2}}\pi z\right)\Gamma (z)\beta (z).}$

Here is the gamma function . ${\displaystyle \Gamma (z)}$

It extends the domain of the beta function to the entire complex number plane .

Further representations

The integral representation is obtained via the Mellin transformation of the function : ${\displaystyle f(x)={\frac {1}{e^{x}+e^{-x}}}}$

${\displaystyle \beta (s)={\frac {1}{\Gamma (s)}}\int \limits _{0}^{\infty }{\frac {x^{s-1}}{e^{x}+e^{-x}}}\,\mathrm {d} x,}$

where again denotes the gamma function . ${\displaystyle \Gamma (s)}$

Together with the Hurwitz zeta function , one obtains the relation for all complexes : ${\displaystyle s}$

${\displaystyle \beta (s)=4^{-s}\left(\zeta \left(s,{\tfrac {1}{4}}\right)-\zeta \left(s,{\tfrac {3}{4}}\right)\right).}$

Another equivalent representation for all complexes includes the transcendent Lerchian zeta function and reads: ${\displaystyle s}$ ${\displaystyle \Phi }$

${\displaystyle \beta (s)=2^{-s}\Phi \left(-1,s,{\tfrac {1}{2}}\right).}$

Special values

Some special values ​​of the function are ${\displaystyle \beta }$

${\displaystyle \beta (0)={\tfrac {1}{2}}}$

${\displaystyle \beta (1)=\arctan 1={\frac {\pi }{4}}}$

${\displaystyle \beta (2)=G\ }$

${\displaystyle \beta (3)={\frac {\pi ^{3}}{32}}}$

${\displaystyle \beta (4)={\frac {1}{768}}\left(\psi _{3}({\tfrac {1}{4}})-8\pi ^{4}\right)}$

${\displaystyle \beta (5)={\frac {5\pi ^{5}}{1536}}}$

${\displaystyle \beta (7)={\frac {61\pi ^{7}}{184320}}}$

The Catalan constant denotes and is the third polygamma function . ${\displaystyle G}$ ${\displaystyle \psi _{3}(z)}$

In general, the following applies to positive whole numbers : ${\displaystyle k\geq 0}$

${\displaystyle \beta (2k+1)={{({-1})^{k}}{E_{2k}}{\pi ^{2k+1}} \over {4^{k+1}}(2k)!},}$

where the -th is Euler number . In the case applies ${\displaystyle E_{n}}$ ${\displaystyle n}$ ${\displaystyle k\leq 0}$

${\displaystyle \beta (k)={{E_{-k}} \over {2}}.}$

The following applies in particular to natural : ${\displaystyle k}$

${\displaystyle \!\ \beta (-2k-1)=0.}$

Derivation

A derivative expression for all is given by: ${\displaystyle \mathrm {Re} \,s>0}$

${\displaystyle \beta ^{\prime }(s)=\sum _{n=1}^{\infty }(-1)^{n-1}{\frac {\ln(2n+1)}{(2n+1)^{s}}}.}$

Special values ​​of the derivative function are:

${\displaystyle \beta ^{\prime }(-1)={\frac {2G}{\pi }}=0{,}583121\ldots }$

${\displaystyle \beta ^{\prime }(0)=\ln {\frac {\Gamma ^{2}(1/4)}{2\pi {\sqrt {2}}}}=0{,}391594\ldots }$

${\displaystyle \beta ^{\prime }(1)={\frac {\pi }{4}}\left(\gamma +2\ln 2+3\ln \pi -4\ln \Gamma ({\tfrac {1}{4}})\right)=0{,}192901\ldots }$

(see sequence A113847 in OEIS and sequence A078127 in OEIS with the Euler-Mascheroni constant ). ${\displaystyle \gamma }$

In addition, for positive integers : ${\displaystyle n}$

${\displaystyle \sum _{k=1}^{\infty }\ln {\frac {(4k+1)^{1/(4k+1)^{n}}}{(4k-1)^{1/(4k-1)^{n}}}}=-\beta ^{\prime }(n).}$

additional

Rivoal and Zudilin proved in 2003 that at least one of the values , , , , and irrational is. ${\displaystyle \beta (2)}$ ${\displaystyle \beta (4)}$ ${\displaystyle \beta (6)}$ ${\displaystyle \beta (8)}$ ${\displaystyle \beta (10)}$ ${\displaystyle \beta (12)}$

In addition, Guillera and Sondow proved the following formula in 2005:

${\displaystyle \int \limits _{0}^{1}\int \limits _{0}^{1}{\frac {[-\ln(xy)]^{s}}{1+x^{2}y^{2}}}\mathrm {d} x\mathrm {d} y=\Gamma (s+2)\beta (s+2)}$

Web links

• Eric W. Weisstein : Dirichlet Beta Function . In: MathWorld (English).

Individual evidence

1. ↑ Godfrey Harold Hardy , EM Wright: Introduction to Number Theory . R. Oldenbourg, Munich 1958, p. 292 .
2. ^ Arxiv: Prime Number Races
3. ↑ Tanguy Rivoal, Wadim Zudilin: Diophantine properties of numbers related to Catalan's constant. In: Mathematische Annalen , Vol. 326 (2003), number 4, pages 705-721, ISSN  0025-5831 ; see. PDF of the Mathematical Institute of the University of Cologne
4. ↑ Jesús Guillera, Jonathan Sondow: Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent . In: Ramanujan Journal. An international journal devoted to the areas of mathematics , vol. 16 (2008), number 3, pages 247-270, ISSN  1382-4090 ; see. in arxiv