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
β
(
s
)
=
∑
n
=
0
∞
(
−
1
)
n
(
2
n
+
1
)
s
,
{\displaystyle \beta (s)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)^{s}}},}
or, equivalently,
β
(
s
)
=
1
Γ
(
s
)
∫
0
∞
x
s
−
1
e
−
x
1
+
e
−
2
x
d
x
.
{\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:
β
(
s
)
=
4
−
s
(
ζ
(
s
,
1
4
)
−
ζ
(
s
,
3
4
)
)
.
{\displaystyle \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:
β
(
s
)
=
2
−
s
Φ
(
−
1
,
s
,
1
2
)
,
{\displaystyle \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
β
(
s
)
=
1
2
s
∑
n
=
0
∞
(
−
1
)
n
(
n
+
1
2
)
s
=
1
(
−
2
)
2
s
(
s
−
1
)
!
[
ψ
(
s
−
1
)
(
1
4
)
−
ψ
(
s
−
1
)
(
3
4
)
]
.
{\displaystyle \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
ζ
(
s
)
{\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:
β
(
s
)
=
∏
p
≡
1
m
o
d
4
1
1
−
p
−
s
∏
p
≡
3
m
o
d
4
1
1
+
p
−
s
{\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
β
(
s
)
=
∏
p
>
2
p
prime
1
1
−
(
−
1
)
p
−
1
2
p
−
s
.
{\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
β
(
1
−
s
)
=
(
π
2
)
−
s
sin
(
π
2
s
)
Γ
(
s
)
β
(
s
)
{\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 .
Special values
Some special values include:
β
(
0
)
=
1
2
,
{\displaystyle \beta (0)={\frac {1}{2}},}
β
(
1
)
=
arctan
(
1
)
=
π
4
,
{\displaystyle \beta (1)\;=\;\arctan(1)\;=\;{\frac {\pi }{4}},}
β
(
2
)
=
G
,
{\displaystyle \beta (2)\;=\;G,}
where G represents Catalan's constant , and
β
(
3
)
=
π
3
32
,
{\displaystyle \beta (3)\;=\;{\frac {\pi ^{3}}{32}},}
β
(
4
)
=
1
768
(
ψ
3
(
1
4
)
−
8
π
4
)
,
{\displaystyle \beta (4)\;=\;{\frac {1}{768}}\left(\psi _{3}\left({\frac {1}{4}}\right)-8\pi ^{4}\right),}
β
(
5
)
=
5
π
5
1536
,
{\displaystyle \beta (5)\;=\;{\frac {5\pi ^{5}}{1536}},}
β
(
7
)
=
61
π
7
184320
,
{\displaystyle \beta (7)\;=\;{\frac {61\pi ^{7}}{184320}},}
where
ψ
3
(
1
/
4
)
{\displaystyle \psi _{3}(1/4)}
in the above is an example of the polygamma function . More generally, for any positive integer k :
β
(
2
k
+
1
)
=
(
−
1
)
k
E
2
k
π
2
k
+
1
4
k
+
1
(
2
k
)
!
,
{\displaystyle \beta (2k+1)={{({-1})^{k}}{E_{2k}}{\pi ^{2k+1}} \over {4^{k+1}}(2k)!},}
where
E
n
{\displaystyle \!\ E_{n}}
represent the Euler numbers . For integer k ≥ 0, this extends to:
β
(
−
k
)
=
E
k
2
.
{\displaystyle \beta (-k)={{E_{k}} \over {2}}.}
Hence, the function vanishes for all odd negative integral values of the argument.
For every positive integer k :
β
(
2
k
)
=
1
2
(
2
k
−
1
)
!
∑
m
=
0
∞
(
(
∑
l
=
0
k
−
1
(
2
k
−
1
2
l
)
(
−
1
)
l
A
2
k
−
2
l
−
1
2
l
+
2
m
+
1
)
−
(
−
1
)
k
−
1
2
m
+
2
k
)
A
2
m
(
2
m
)
!
(
π
2
)
2
m
+
2
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},}
[citation needed ]
where
A
k
{\displaystyle A_{k}}
is the Euler zigzag number .
Also it was derived by Malmsten in 1842 that
β
′
(
1
)
=
∑
n
=
1
∞
(
−
1
)
n
+
1
ln
(
2
n
+
1
)
2
n
+
1
=
π
4
(
γ
−
ln
π
)
+
π
ln
Γ
(
3
4
)
{\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)}
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