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

${\displaystyle \beta (s)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)^{s}}},}$

or, equivalently,

${\displaystyle \beta (s)={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }{\frac {x^{s-1}e^{-x}}{1+e^{-2x}}}\,dx.}$

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: ^[1]

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

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

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

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:

${\displaystyle \beta (s)={\frac {i}{2}}\left({\text{Li}}_{s}(-i)-{\text{Li}}_{s}(i)\right).}$

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

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

but this formula is only valid at positive integer values of ${\displaystyle s}$.

Euler product formula

It is also the simplest example of a series non-directly related to ${\displaystyle \zeta (s)}$ 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

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

Functional equation

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

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

where Γ(s) is the gamma function. It was conjectured by Euler in 1749 and proved by Malmsten in 1842. ^[2]

Specific values

Positive integers

For every odd positive integer ${\displaystyle 2n+1}$, the following equation holds: ^[3]

${\displaystyle \beta (2n+1)\;=\;{\frac {(-1)^{n}E_{2n}}{2(2n)!}}\left({\frac {\pi }{2}}\right)^{2n+1}}$

where ${\displaystyle E_{n}}$ is the n-th Euler Number. This yields:

${\displaystyle \beta (1)\;=\;{\frac {\pi }{4}},}$

${\displaystyle \beta (3)\;=\;{\frac {\pi ^{3}}{32}},}$

${\displaystyle \beta (5)\;=\;{\frac {5\pi ^{5}}{1536}},}$

${\displaystyle \beta (7)\;=\;{\frac {61\pi ^{7}}{184320}}}$

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 ${\displaystyle \beta (2)=G}$ is known as Catalan's constant.

It has been proven that infinitely many numbers of the form ${\displaystyle \beta (2n)}$ ^[4] and at least one of the numbers ${\displaystyle \beta (2),\beta (4),\beta (6),...,\beta (12)}$ are irrational. ^[5]

The even beta values may be given in terms of the polygamma functions and the Bernoulli numbers: ^[6]

${\displaystyle \beta (2n)={\frac {\psi ^{(2n-1)}(1/4)}{4^{2n-1}(2n)!}}n-{\frac {\pi ^{2n}(2^{2n}-1)|B_{2n}|}{2(2n)!}}}$

We can also express the beta function for positive ${\displaystyle n}$ in terms of the inverse tangent integral:

${\displaystyle \beta (n)={\text{Ti}}_{n}(1)}$

${\displaystyle \beta (1)=\arctan(1)}$

For every positive integer k:^{[ citation needed ]}

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

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

s	approximate value β(s)	OEIS
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	A258814
8	0.9998499902468296563380671	A258815
9	0.9999496841872200898213589	A258816

Negative integers

For negative odd integers, the function is zero:

${\displaystyle \beta (-2n-1)\;=\;0}$

For every negative even integer it holds: ^[3]

${\displaystyle \beta (-2n)\;=\;{\frac {1}{2}}E_{2n}}$.

It further is:

${\displaystyle \beta (0)\;=\;{\frac {1}{2}}}$.

Derivative

We have: ^[3]

${\displaystyle \beta '(-1)={\frac {2G}{\pi }}}$

${\displaystyle \beta '(0)=2\ln \Gamma ({\tfrac {1}{4}})-\ln \pi -{\tfrac {3}{2}}\ln 2}$

${\displaystyle \beta '(1)={\tfrac {\pi }{4}}(\gamma +2\ln 2+3\ln \pi -4\ln \Gamma ({\tfrac {1}{4}}))}$

with ${\displaystyle \gamma }$ being Euler's constant and ${\displaystyle G}$ being Catalan's constant. The last identity was derived by Malmsten in 1842. ^[2]

See also

• Hurwitz zeta function
• Dirichlet eta function
• Polylogarithm

References

1. Dirichlet Beta – Hurwitz zeta relation, Engineering Mathematics
2. Blagouchine, Iaroslav V. (2014-10-01). "Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results". The Ramanujan Journal. 35 (1): 21–110. doi:10.1007/s11139-013-9528-5. ISSN   1572-9303.
3. Weisstein, Eric W. "Dirichlet Beta Function". mathworld.wolfram.com. Retrieved 2024-08-08.
4. Rivoal, T.; Zudilin, W. (2003-08-01). "Diophantine properties of numbers related to Catalan's constant". Mathematische Annalen. 326 (4): 705–721. doi:10.1007/s00208-003-0420-2. hdl: 1959.13/803688 . ISSN   1432-1807.
5. Zudilin, Wadim (2019-05-31). "Arithmetic of Catalan's constant and its relatives". Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. 89 (1): 45–53. arXiv: 1804.09922 . doi: 10.1007/s12188-019-00203-w . ISSN   0025-5858.
6. Kölbig, K. S. (1996-11-12). "The polygamma function ψ(k)(x) for x=14 and x=34". Journal of Computational and Applied Mathematics. 75 (1): 43–46. doi:10.1016/S0377-0427(96)00055-6. ISSN   0377-0427.

• 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.