Dirichlet beta function

This article is about the Dirichlet beta function. For other beta functions, see Beta function (disambiguation).
The Dirichlet beta function

In mathematics, the Dirichlet beta function (also known as the Catalan beta function) is a special function, closely related to the Riemann zeta function. It is a particular Dirichlet L-function, the L-function for the alternating character of period four.

Definition

The Dirichlet beta function is defined as

\beta(s) = \sum_{n=0}^\infty \frac{(-1)^n} {(2n+1)^s},

or, equivalently,

\beta(s) = \frac{1}{\Gamma(s)}\int_0^{\infty}\frac{x^{s-1}e^{-x}}{1 + e^{-2x}}\,dx.

In each case, it is assumed that Re(s) > 0.

It is also the simplest example of a series non-directly related to \zeta(s) which can also be factorized as an Euler product, thus leading to the idea of Dirichlet character defining the exact set of Dirichlet series having a factorization over the prime numbers.

At least for Re(s) ≥ 1:

\prod_{\scriptstyle p \, \in\, \mathcal{P}\ \ p \ge 3} \frac{1}{1 -\, \scriptstyle(-1)^{\frac{p-1}{2}} \textstyle p^{-s}} = \sum_{n=0}^\infty (2n+1)^{-s} (-1)^{n} = \beta(s)

Alternatively, the following definition, in terms of the Hurwitz zeta function, is valid in the whole complex s-plane:

\beta(s) = 4^{-s} \left( \zeta\left(s,{1 \over 4}\right)-\zeta\left( s, {3 \over 4}\right) \right). proof

Another equivalent definition, in terms of the Lerch transcendent, is:

\beta(s) = 2^{-s} \Phi\left(-1,s,{{1} \over {2}}\right),

which is once again valid for all complex values of s.

Also the series representation of Dirichlet beta function can be formed in terms of the polygamma function

\beta(s) =\frac{1}{2^s} \sum_{n=0}^\infty\frac{(-1)^{n}}{\left(n+\frac{1}{2}\right)^{s}}=\frac1{(-2)^{2s}(s-1)!}\left[\psi^{(s-1)}\left(\frac{1}{4}\right)-\psi^{(s-1)}\left(\frac{3}{4}\right)\right].

Functional equation

The functional equation extends the beta function to the left side of the complex plane Re(s)<0. It is given by

\beta(1-s)=\left(\frac{\pi}{2}\right)^{-s}\sin\left(\frac{\pi}{2}s\right)\Gamma(s)\beta(s)

where Γ(s) is the gamma function.

Special values

Some special values include:

\beta(0)= \frac{1}{2},
\beta(1)\;=\;\tan^{-1}(1)\;=\;\frac{\pi}{4},
\beta(2)\;=\;G,

where G represents Catalan's constant, and

\beta(3)\;=\;\frac{\pi^3}{32},
\beta(4)\;=\;\frac{1}{768}(\psi_3(\frac{1}{4})-8\pi^4),
\beta(5)\;=\;\frac{5\pi^5}{1536},
\beta(7)\;=\;\frac{61\pi^7}{184320},

where \psi_3(1/4) in the above is an example of the polygamma function. More generally, for any positive integer k:

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

where \!\ E_{n} represent the Euler numbers. For integer k ≥ 0, this extends to:

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

Hence, the function vanishes for all odd negative integral values of the argument.

sapproximate value β(s)OEIS
1/5 0.5737108471859466493572665 A261624
1/4 0.5907230564424947318659591 A261623
1/3 0.6178550888488520660725389 A261622
1/2 0.6676914571896091766586909 A195103
1 0.7853981633974483096156608 A003881
2 0.9159655941772190150546035 A006752
3 0.9689461462593693804836348 A153071
4 0.9889445517411053361084226 A175572
5 0.9961578280770880640063194 A175571
6 0.9986852222184381354416008 A175570
7 0.9995545078905399094963465
8 0.9998499902468296563380671
9 0.9999496841872200898213589
10 0.9999831640261968774055407

There are zeros at -1; -3; -5; -7 etc.

See also

References

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