Affiliation:
1. Department of Mathematics , [ 26467]South China University of Technology, Guangzhou , Guangdong 510640 , P. R. China
2. Department of Mathematics Education , [ 34985]Kyungnam University, Changwon , Gyeongnam 51767 , Republic of Korea
Abstract
Abstract
In 1769, Euler proved the following result:
∫
0
π
2
log
(
sin
θ
)
𝑑
θ
=
-
π
2
log
2
.
\int_{0}^{\frac{\pi}{2}}\log(\sin\theta)\,d\theta=-\frac{\pi}{2}\log 2.
In this paper, as a generalization, we evaluate the definite integrals
∫
0
x
θ
r
-
2
log
(
cos
θ
2
)
𝑑
θ
\int_{0}^{x}\theta^{r-2}\log\biggl{(}\cos\frac{\theta}{2}\biggr{)}\,d\theta
for
r
=
2
,
3
,
4
,
…
r=2,3,4,\dots
.
We show that it can be expressed by the special values of Kurokawa and Koyama’s multiple cosine functions
𝒞
r
(
x
)
{\mathcal{C}_{r}(x)}
or by the special values of alternating zeta
and Dirichlet lambda functions.
In particular, we get the following explicit expression of the zeta value:
ζ
(
3
)
=
4
π
2
21
log
(
e
4
G
π
𝒞
3
(
1
4
)
16
2
)
,
\zeta(3)=\frac{4\pi^{2}}{21}\log\Biggl{(}\frac{e^{\frac{4G}{\pi}}\mathcal{C}_{%
3}\bigl{(}\frac{1}{4}\bigr{)}^{16}}{\sqrt{2}}\Biggr{)},
where G is Catalan’s constant and
𝒞
3
(
1
4
)
{\mathcal{C}_{3}(\frac{1}{4})}
is the special value of Kurokawa and Koyama’s multiple cosine function
𝒞
3
(
x
)
{\mathcal{C}_{3}(x)}
at
1
4
{\frac{1}{4}}
.
Furthermore, we prove several series representations for the logarithm of
multiple cosine functions
log
𝒞
r
(
x
2
)
{\log\mathcal{C}_{r}(\frac{x}{2})}
by zeta functions, L-functions or polylogarithms.
One of them leads to another expression of
ζ
(
3
)
{\zeta(3)}
:
ζ
(
3
)
=
72
π
2
11
log
(
3
1
72
𝒞
3
(
1
6
)
𝒞
2
(
1
6
)
1
3
)
.
\zeta(3)=\frac{72\pi^{2}}{11}\log\Biggl{(}\frac{3^{\frac{1}{72}}\mathcal{C}_{3%
}\bigl{(}\frac{1}{6}\bigr{)}}{\mathcal{C}_{2}\bigl{(}\frac{1}{6}\bigr{)}^{%
\frac{1}{3}}}\Biggr{)}.
Funder
National Research Foundation of Korea
Reference37 articles.
1. V. S. Adamchik,
On the Barnes function,
Proceedings of the 2001 International Symposium on Symbolic and Algebraic Computation,
ACM, New York (2001), 15–20.
2. V. S. Adamchik,
The multiple gamma function and its application to computation of series,
Ramanujan J. 9 (2005), no. 3, 271–288.
3. J.-P. Allouche,
Hölder and Kurokawa meet Borwein–Dykshoorn and Adamchik,
J. Ramanujan Math. Soc. 38 (2023), no. 3, 265–273.
4. R. Apéry,
Irrationalité de
ζ
2
\zeta 2
et
ζ
3
\zeta 3
,
Journées Arithmétiques de Luminy,
Astérisque 61,
Société Mathématique de France, Paris (1979), 11–13.
5. T. M. Apostol,
Introduction to Analytic Number Theory,
Undergrad. Texts Math.,
Springer, New York, 1976.