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.

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:

\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 $p \equiv1 mod 4$ are the primes of the form $4 n +1$ (5,13,17,...) and $p \equiv3 mod 4$ are the primes of the form $4 n +3$ (3,7,11,...). This can be written compactly as

\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:

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

Also it was derived by Malmsten in 1842 that

\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.

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.

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