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
- [math]\displaystyle{ \beta(s) = \sum_{n=0}^\infty \frac{(-1)^n} {(2n+1)^s}, }[/math]
or, equivalently,
- [math]\displaystyle{ \beta(s) = \frac{1}{\Gamma(s)}\int_0^{\infty}\frac{x^{s-1}e^{-x}}{1 + e^{-2x}}\,dx. }[/math]
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:[1]
- [math]\displaystyle{ \beta(s) = 4^{-s} \left( \zeta\left(s,{1 \over 4}\right)-\zeta\left( s, {3 \over 4}\right) \right). }[/math]
Another equivalent definition, in terms of the Lerch transcendent, is:
- [math]\displaystyle{ \beta(s) = 2^{-s} \Phi\left(-1,s,{{1} \over {2}}\right), }[/math]
which is once again valid for all complex values of s.
The Dirichlet beta function can also be written in terms of the polylogarithm function:
- [math]\displaystyle{ \beta(s) = \frac{i}{2} \left(\text{Li}_s(-i)-\text{Li}_s(i)\right). }[/math]
Also the series representation of Dirichlet beta function can be formed in terms of the polygamma function
- [math]\displaystyle{ \beta(s) =\frac{1}{2^s} \sum_{n=0}^\infty\frac{(-1)^{n}}{\left(n+\frac{1}{2}\right)^{s}}=\frac1{(-4)^s(s-1)!}\left[\psi^{(s-1)}\left(\frac{1}{4}\right)-\psi^{(s-1)}\left(\frac{3}{4}\right)\right] }[/math]
but this formula is only valid at positive integer values of [math]\displaystyle{ s }[/math].
Euler product formula
It is also the simplest example of a series non-directly related to [math]\displaystyle{ \zeta(s) }[/math] 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:
- [math]\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}} }[/math]
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
- [math]\displaystyle{ \beta(s) = \prod_{p\gt 2\atop p \text{ prime}} \frac{1}{1 -\, \scriptstyle(-1)^{\frac{p-1}{2}} \textstyle p^{-s}}. }[/math]
Functional equation
The functional equation extends the beta function to the left side of the complex plane Re(s) ≤ 0. It is given by
- [math]\displaystyle{ \beta(1-s)=\left(\frac{\pi}{2}\right)^{-s}\sin\left(\frac{\pi}{2}s\right)\Gamma(s)\beta(s) }[/math]
where Γ(s) is the gamma function. It was conjectured by Euler in 1749 and proved by Malmsten in 1842 (see Blagouchine, 2014).
Special values
Some special values include:
- [math]\displaystyle{ \beta(0)= \frac{1}{2}, }[/math]
- [math]\displaystyle{ \beta(1)\;=\;\arctan(1)\;=\;\frac{\pi}{4}, }[/math]
- [math]\displaystyle{ \beta(2)\;=\;G, }[/math]
where G represents Catalan's constant, and
- [math]\displaystyle{ \beta(3)\;=\;\frac{\pi^3}{32}, }[/math]
- [math]\displaystyle{ \beta(4)\;=\;\frac{1}{768}\left(\psi_3\left(\frac{1}{4}\right)-8\pi^4\right), }[/math]
- [math]\displaystyle{ \beta(5)\;=\;\frac{5\pi^5}{1536}, }[/math]
- [math]\displaystyle{ \beta(7)\;=\;\frac{61\pi^7}{184320}, }[/math]
where [math]\displaystyle{ \psi_3(1/4) }[/math] in the above is an example of the polygamma function.
Hence, the function vanishes for all odd negative integral values of the argument.
For every positive integer k:
- [math]\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}, }[/math][citation needed]
where [math]\displaystyle{ A_{k} }[/math] is the Euler zigzag number.
Also it was derived by Malmsten in 1842 (see Blagouchine, 2014) that
- [math]\displaystyle{ \beta'(1)=\sum_{n=1}^{\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) }[/math]
| 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
References
- ↑ Dirichlet Beta – Hurwitz zeta relation, Engineering Mathematics
- Blagouchine, I. V. (2014). "Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results". Ramanujan J. 35 (1): 21–110. doi:10.1007/s11139-013-9528-5. https://www.researchgate.net/publication/257381156_Rediscovery_of_Malmsten's_integrals_their_evaluation_by_contour_integration_methods_and_some_related_results.
- Glasser, M. L. (1972). "The evaluation of lattice sums. I. Analytic procedures". J. Math. Phys. 14 (3): 409. doi:10.1063/1.1666331. Bibcode: 1973JMP....14..409G.
- J. Spanier and K. B. Oldham, An Atlas of Functions, (1987) Hemisphere, New York.
- Weisstein, Eric W.. "Dirichlet Beta Function". http://mathworld.wolfram.com/DirichletBetaFunction.html.
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