It has an approximate value of ζ(3)
Bilangan ini memiliki nilai yang kira-kira sama dengan ζ(3)
≈ 1.202056903159594285399738161511449990764986292… (sequence A002117 in the OEIS).
1.202056903159594285399738161511449990764986292… (barisan A002117 pada OEIS).
It also arises in the analysis of random minimum spanning trees and in conjunction with the gamma function when solving certain integrals involving exponential functions in a quotient, which appear occasionally in physics, for instance, when evaluating the two-dimensional case of the Debye model and the Stefan–Boltzmann law.
Konstanta ini juga ditemukan dalam analisis pohon rentang minimum acak, serta mempunyai hubungan dengan fungsi gamma ketika menyelesaikan integral tertentu yang melibatkan fungsi eksponensial dalam hasil bagi, yang kadangkala ditemukan dalam fisika, sebagai contoh, ketika mengevaluasi kasus dimensi dua dari model Debye dan hukum Stefan–Boltzmann.
ζ(3) was named Apéry's constant after the French mathematician Roger Apéry, who proved in 1978 that it is an irrational number.
ζ(3) disebut sebagai konstanta Apéry, konstanta yang dinamai dari matematikawan berkebangsaan Prancis, Roger Apéry.
This result is known as Apéry's theorem.
Hasil tersebut dikenal sebagai teorema Apéry.
∑ k1 ∞ 1 k 3 , ^{\infty }{\frac {1}{k^{3}}},} Leonhard Euler gave the series representation: ζ ( 3 )π 2 7 ( 1 − 4 ∑ k1 ∞ ζ ( 2 k ) 2 2 k ( 2 k + 1 ) ( 2 k + 2 ) ) }{7}}\left(1-4\sum _{k=1}^{\infty }{\frac {\zeta (2k)}{2^{2k}(2k+1)(2k+2)}}\right)} in 1772, which was subsequently rediscovered several times.
Selain mempunyai deret ζ ( 3 )∑ k1 ∞ 1 k 3 , ^{\infty }{\frac {1}{k^{3}}},} Leonhard Euler memberikan representasi deret ζ ( 3 )π 2 7 ( 1 − 4 ∑ k1 ∞ ζ ( 2 k ) 2 2 k ( 2 k + 1 ) ( 2 k + 2 ) ) }{7}}\left(1-4\sum _{k=1}^{\infty }{\frac {\zeta (2k)}{2^{2k}(2k+1)(2k+2)}}\right)} pada tahun 1772, yang kemudian ditemukan kembali berulang kali.
Since the 19th century, a number of mathematicians have found convergence acceleration series for calculating decimal places of ζ(3).
Sejak pada abad ke-19, sejumlah matematikawwan telah menemukan deret percepatan konvergensi untuk menghitung letak desimal ζ(3).
{4}}\sum _{k=1}^{\infty }(-1)^{k-1}{\frac {(k-1)!^{3}(56k^{2}-32k+5)}{(2k-1)^{2}(3k)!}}.}
{2}}\,\sum _{k=0}^{\infty }{\frac {(-1)^{k}(2k)!^{3}(k+1)!^{6}(40885k^{5}+124346k^{4}+150160k^{3}+89888k^{2}+26629k+3116)}{(k+1)^{2}(3k+3)!^{4}}}.}
{64}}\sum _{k=0}^{\infty }(-1)^{k}{\frac {k!^{10}(205k^{2}+250k+77)}{(2k+1)!^{5}}}.}
{2}}\,\sum _{k=0}^{\infty }{\frac {(-1)^{k}(2k)!^{3}(k+1)!^{6}(40885k^{5}+124346k^{4}+150160k^{3}+89888k^{2}+26629k+3116)}{(k+1)^{2}(3k+3)!^{4}}}.}
{24}}\sum _{k=0}^{\infty }(-1)^{k}{\frac {(2k+1)!^{3}(2k)!^{3}k!^{3}(126392k^{5}+412708k^{4}+531578k^{3}+336367k^{2}+104000k+12463)}{(3k+2)!(4k+3)!^{3}}}.}
{24}}\sum _{k=0}^{\infty }(-1)^{k}{\frac {(2k+1)!^{3}(2k)!^{3}k!^{3}(126392k^{5}+412708k^{4}+531578k^{3}+336367k^{2}+104000k+12463)}{(3k+2)!(4k+3)!^{3}}}.} Representasi deret ini digunakan oleh Wedeniwski untuk menghitung konstanta Apéry dengan jutaan pembulatan letak desimal.
{2}}\sum _{k=0}^{\infty }{\frac {(-1)^{k}(2k)!^{3}(k+1)!^{6}(40885k^{5}+124346k^{4}+150160k^{3}+89888k^{2}+26629k+3116)}{(k+1)^{2}(3k+3)!^{4}}}.}
{2}}\,\sum _{k=0}^{\infty }{\frac {(-1)^{k}(2k)!^{3}(k+1)!^{6}(40885k^{5}+124346k^{4}+150160k^{3}+89888k^{2}+26629k+3116)}{(k+1)^{2}(3k+3)!^{4}}}.}
In 1998, Broadhurst gave a series representation that allows arbitrary binary digits to be computed, and thus, for the constant to be obtained by a spigot algorithm in nearly linear time and logarithmic space.
Pada tahun 1998, Broadhurst memberikan representasi deret yang memungkinkan menghitung digit biner sembarang, dan untuk konstanta yang akan diperoleh dalam waktu linier dekat, dan ruang logaritma.
^{\infty }{\frac {1}{k^{3}\sinh(\pi k)}}-{\frac {11}{2}}\sum _{k=1}^{\infty }{\frac {1}{k^{3}(e^{2\pi k}-1)}}-{\frac {7}{2}}\sum _{k=1}^{\infty }{\frac {1}{k^{3}(e^{2\pi k}+1)}}.} Srivastava (2000) collected many series that converge to Apéry's constant.
^{\infty }{\frac {1}{k^{3}\sinh(\pi k)}}-{\frac {11}{2}}\sum _{k=1}^{\infty }{\frac {1}{k^{3}(e^{2\pi k}-1)}}-{\frac {7}{2}}\sum _{k=1}^{\infty }{\frac {1}{k^{3}(e^{2\pi k}+1)}}.} (Srivastava 2000) mengumpulkan banyak deret yang konvergen menuju ke konstanta Apéry.
There are numerous integral representations for Apéry's constant.
Ada banyak representasi integral untuk konstanta Apéry.
π ∫ 0 ∞ cos ⁡ ( 2 arctan ⁡ x ) ( x 2 + 1 ) ( cosh ⁡ 1 2 π x ) 2 d x ^{\infty }{\frac {\cos(2\arctan x)}{(x^{2}+1)\left(\cosh {\frac {1}{2}}\pi x\right)^{2}}}\,dx} and ζ ( 3 )− 1 2 ∫ 0 1 ∫ 0 1 log ⁡ ( x y ) 1 − x y d x d y− ∫ 0 1 ∫ 0 1 log ⁡ ( 1 − x y )
π ∫ 0 ∞ cos ⁡ ( 2 arctan ⁡ x ) ( x 2 + 1 ) ( cosh ⁡ 1 2 π x ) 2 d x , ^{\infty }\!{\frac {\cos(2\arctan {x})}{\left(x^{2}+1\right)\left(\cosh {\frac {1}{2}}\pi x\right)^{2}}}\,dx,} dan ζ ( 3 )− 1 2 ∫ 0 1 ∫ 0 1 log ⁡ ( x y ) 1 − x y d x d y− ∫ 0 1 ∫ 0 1 log ⁡ ( 1 − x y )
{2}}\int _{0}^{1}\!\!\int _{0}^{1}{\frac {\log(xy)}{1-xy}}\,dx\,dy=-\int _{0}^{1}\!\!\int _{0}^{1}{\frac {\log(1-xy)}{xy}}\,dx\,dy.}
{2}}\int _{0}^{1}\!\!\int _{0}^{1}{\frac {\log(xy)}{\,1-xy\,}}\,dx\,dy=-\int _{0}^{1}\!\!\int _{0}^{1}{\frac {\log(1-xy)}{\,xy\,}}\,dx\,dy.}
2 ( 102 k 6 + 612 k 5 + 1462 k 4 + 1768 k 3 + 1143 k 2 + 382 k + 52 ) +612k^{5}+1462k^{4}+1768k^{3}+1143k^{2}+382k+52)} b ( k )
1 24 ∑ k0 ∞ ( − 1 ) k ( 2 k + 1 ) ! 3 ( 2 k ) ! 3 k ! 3 ( 126392 k 5 + 412708 k 4 + 531578 k 3 + 336367 k 2 + 104000 k + 12463 ) ( 3 k + 2 ) ! ( 4 k + 3 ) ! 3 .
( k + 1 ) 3 ( k + 2 ) 3 4 ( 24 k 3 + 30 k 2 + 16 k + 3 ) ( 24 k 3 + 174 k 2 + 424 k + 347 ) (k+2)^{3}}{4}}(24k^{3}+30k^{2}+16k+3)(24k^{3}+174k^{2}+424k+347)} The number of known digits of Apéry's constant ζ(3) has increased dramatically during the last decades, and now stands at more than 2×1012.
1 24 ∑ k0 ∞ ( − 1 ) k ( 2 k + 1 ) ! 3 ( 2 k ) ! 3 k ! 3 ( 126392 k 5 + 412708 k 4 + 531578 k 3 + 336367 k 2 + 104000 k + 12463 ) ( 3 k + 2 ) ! ( 4 k + 3 ) ! 3 .
This is due both to the increasing performance of computers and to algorithmic improvements.
Hal ini disebabkan karena peningkatan kinerja komputer dan algoritme yang berkembang.