Dirichlet beta function

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.

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

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

Euler product formula

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:

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

where p≡1 mod 4 are the primes of the form 4n+1 (5,13,17,...) and p≡3 mod 4 are the primes of the form 4n+3 (3,7,11,...). This can be written compactly as

${\displaystyle \beta (s)=\prod _{p\geq 3 \atop p{\text{ prime}}}{\frac {1}{1-\,\scriptstyle (-1)^{\frac {p-1}{2}}\textstyle p^{-s}}}.}$

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.

For every positive integer k:

${\displaystyle \beta (2k)={\frac {1}{2(2k-1)!}}\sum _{m=0}^{\infty }\left(\left(\sum _{l=0}^{k-1}{\binom {2k-1}{2l}}{\frac {(-1)^{l}A_{2k-2l-1}}{2l+2m+1}}\right)-{\frac {(-1)^{k-1}}{2m+2k}}\right){\frac {A_{2m}}{(2m)!}}{\left({\frac {\pi }{2}}\right)}^{2m+2k},}$

where ${\displaystyle A_{k}}$ is the Euler zigzag number.

Also it was derived by Malmsten in 1842 that

${\displaystyle \beta '(1)=\sum _{n=0}^{\infty }(-1)^{n+1}{\frac {\ln(2n+1)}{2n+1}}\,=\,{\frac {\pi }{4}}{\big (}\gamma -\ln \pi )+\pi \ln \Gamma \left({\frac {3}{4}}\right)}$

s	approximate 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

• Hurwitz zeta function

References

• Glasser, M. L. (1972). "The evaluation of lattice sums. I. Analytic procedures". J. Math. Phys. 14 (3): 409. Bibcode:1973JMP....14..409G. doi:10.1063/1.1666331.
• J. Spanier and K. B. Oldham, An Atlas of Functions, (1987) Hemisphere, New York.
• Weisstein, Eric W. "Dirichlet Beta Function". MathWorld.