Dirichlet beta function
Special mathematical function
title: "Dirichlet beta function" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["zeta-and-l-functions"] description: "Special mathematical function" topic_path: "general/zeta-and-l-functions" source: "https://en.wikipedia.org/wiki/Dirichlet_beta_function" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
::summary Special mathematical function ::
::callout[type=note] the Dirichlet beta function ::
::figure[src="https://upload.wikimedia.org/wikipedia/commons/e/e3/Mplwp_dirichlet_beta.svg" caption="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).
Another equivalent definition, in terms of the Lerch transcendent, is:
:\beta(s) = 2^{-s} \Phi\left(-1,s,\right),
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:
:\beta(s) = \frac{i}{2} \left(\text{Li}_s(-i)-\text{Li}_s(i)\right).
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{(-4)^s(s-1)!}\left[\psi^{(s-1)}\left(\frac{1}{4}\right)-\psi^{(s-1)}\left(\frac{3}{4}\right)\right]
but this formula is only valid at positive integer values of s.
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 are the primes of the form (5,13,17,...) and are the primes of the form (3,7,11,...). This can be written compactly as
:\beta(s) = \prod_{p2\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 \Gamma(s) is the gamma function. It was conjectured by Euler in 1749 and proved by Malmsten in 1842.
Specific values
Positive integers
For every odd positive integer 2n+1, the following equation holds: :\beta(2n+1);=;\frac{(-1)^n E_{2n}}{2(2n)!}\left(\frac\pi2\right)^{2n+1} where E_n is the n-th Euler Number. This yields:
:\beta(1);=;\frac{\pi}{4}, :\beta(3);=;\frac{\pi^3}{32}, :\beta(5);=;\frac{5\pi^5}{1536}, :\beta(7);=;\frac{61\pi^7}{184320} For the values of the Dirichlet beta function at even positive integers no elementary closed form is known, and no method has yet been found for determining the arithmetic nature of even beta values (similarly to the Riemann zeta function at odd integers greater than 3). The number \beta(2)=G is known as Catalan's constant.
It has been proven that infinitely many numbers of the form \beta(2n) and at least one of the numbers \beta(2), \beta(4), \beta(6), ..., \beta(12) are irrational.
The even beta values may be given in terms of the polygamma functions and the Bernoulli numbers:
: \beta(2n)=\frac{\psi^{(2n-1)}(1/4)}{4^{2n-1}(2n)!}n - \frac{\pi^{2n}(2^{2n}-1)|B_{2n}|}{2(2n)!} We can also express the beta function for positive n in terms of the inverse tangent integral: :\beta(n)=\text{Ti}_n(1) :\beta(1)=\arctan(1)
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.
::data[format=table]
| s | approximate value β(s) | OEIS |
|---|---|---|
| 1 | 0.7853981633974483096156608 | |
| 2 | 0.9159655941772190150546035 | |
| 3 | 0.9689461462593693804836348 | |
| 4 | 0.9889445517411053361084226 | |
| 5 | 0.9961578280770880640063194 | |
| 6 | 0.9986852222184381354416008 | |
| 7 | 0.9995545078905399094963465 | |
| 8 | 0.9998499902468296563380671 | |
| 9 | 0.9999496841872200898213589 | |
| :: |
Negative integers
For negative odd integers, the function is zero: :\beta(-2n-1);=;0 For every negative even integer it holds: :\beta(-2n);=;\frac12E_{2n}. It further is: :\beta(0);=; \frac{1}{2} .
Derivative
We have:
\beta'(-1)=\frac{2G}\pi
\beta'(0)=2\ln\Gamma(\tfrac14)-\ln\pi-\tfrac32\ln2=\ln\tfrac\varpi{\sqrt\pi}
\beta'(1)=\tfrac\pi4(\gamma+2\ln2+3\ln\pi-4\ln\Gamma(\tfrac14))=\tfrac\pi4(\gamma-\ln2+2\ln\tfrac\pi\varpi)
with \gamma being Euler's constant, \varpi being the Lemniscate constant and G being Catalan's constant. The last identity was derived by Malmsten in 1842.
References
- {{cite journal |first1=M. L. |last1=Glasser |title=The evaluation of lattice sums. I. Analytic procedures |year=1972 |journal=J. Math. Phys. |doi=10.1063/1.1666331 |volume=14 |issue=3 |page=409 |bibcode=1973JMP....14..409G
- J. Spanier and K. B. Oldham, An Atlas of Functions, (1987) Hemisphere, New York.
References
- [http://engineeringandmathematics.blogspot.co.nz/2012/09/dirichlet-beta-hurwitz-zeta-relation.html Dirichlet Beta – Hurwitz zeta relation], Engineering Mathematics
- Weisstein, Eric W.. "Dirichlet Beta Function".
- (2003). "Diophantine properties of numbers related to Catalan's constant". Mathematische Annalen.
- Zudilin, Wadim. (2019). "Arithmetic of Catalan's constant and its relatives". Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg.
- Kölbig, K. S.. (1996). "The polygamma function ψ(k)(x) for x=14 and x=34". Journal of Computational and Applied Mathematics.
- Blagouchine, Iaroslav V.. (2014-10-01). "Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results". The Ramanujan Journal.
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