Apéry constant

The Apéry constant is a mathematical constant that is used as the value of the series

${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{3}}}={\frac {1}{1^{3}}}+{\frac {1}{2^{3}}}+{\frac {1}{3^{3}}}+{\frac {1}{4^{3}}}+\dotsb }$

is defined. This is the value of the Riemann ζ function at position 3. ${\displaystyle \zeta (3)}$

Basics

Is an approximation

${\displaystyle \zeta (3)=1{,}20205{\text{ }}69031{\text{ }}59594{\text{ }}28539{\text{ }}97381{\text{ }}61511{\text{ }}44999{\text{ }}07649{\text{ }}86292{\text{ }}34049{\text{ }}\dotso }$(Follow A002117 in OEIS ).

Currently (as of August 2020) 1,200,000,000,100 decimal places are known, they were calculated by Seungmin Kim on July 26, 2020.

The constant was considered by Euler as early as 1735 . It is named after Roger Apéry , who proved in 1979 that it is irrational . It is not yet known whether it is also transcendent , nor whether it is normal or whether it is irrational (with circle number ). Little is known about the values ​​of the zeta function for other odd natural numbers - in contrast to the values ​​for even numbers: An infinite number of the numbers must be irrational, with at least one of and . ${\displaystyle \zeta (3)/\pi ^{3}}$ ${\displaystyle \pi }$ ${\displaystyle \zeta (2n+1),\,n=1,2,3,\dotsc }$ ${\displaystyle \zeta (5),\zeta (7),\zeta (9)}$ ${\displaystyle \zeta (11)}$

For the measure of irrationality , where is the set of positive real numbers for which at most finitely many pairs of positive integers and with exist, the bounds are known, in particular, it is not Liouville . ${\displaystyle \operatorname {r} (\zeta )=\inf R}$ ${\displaystyle R}$ ${\displaystyle \rho }$ ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle \textstyle 0<\mid \zeta -{\frac {p}{q}}\mid <{\frac {1}{q^{\rho }}}}$ ${\displaystyle 2\leq r(\zeta (3))<5{,}513891}$ ${\displaystyle \zeta (3)}$

The reciprocal (sequence A088453 in OEIS ) is the asymptotic probability that three integers coprime are, and also the asymptotic probability that an integer cubic free (not a cube number is divisible greater than 1). These are special cases where integers with asymptotic probability do not have a -th power greater than 1 as a common factor. ${\displaystyle {\tfrac {1}{\zeta (3)}}=0{,}83190737258070746868\dotso }$ ${\displaystyle n}$ ${\displaystyle {\tfrac {1}{\zeta (nk)}}}$ ${\displaystyle k}$

Series representations

Apéry used the formula

${\displaystyle \zeta (3)={\frac {5}{2}}\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}}{n^{3}{\binom {2n}{n}}}}.}$

A result already known to Euler is

${\displaystyle \zeta (3)={\frac {1}{2}}\sum _{n=1}^{\infty }{\frac {H_{n}}{n^{2}}}}$

with the harmonic numbers . Numerous related formulas such as ${\displaystyle H_{n}}$

${\displaystyle \zeta (3)={\frac {1}{2}}\sum _{i=1}^{\infty }\sum _{j=1}^{\infty }{\frac {1}{ij(i+j)}}}$

also lead to the Apéry constant. From the Dirichlet λ- and η function is obtained ${\displaystyle \zeta (z)/2^{z}=\lambda (z)/(2^{z}-1)=\eta (z)/(2^{z}-2)}$

${\displaystyle \zeta (3)={\frac {8}{7}}\sum _{n=0}^{\infty }{\frac {1}{(2n+1)^{3}}}={\frac {4}{3}}\sum _{n=1}^{\infty }{\frac {(-1)^{n-1}}{n^{3}}}.}$

A rapidly converging series comes from Tewodros Amdeberhan and Doron Zeilberger (1997):

${\displaystyle \zeta (3)={\frac {1}{24}}\sum _{n=0}^{\infty }(-1)^{n}{\frac {A(n)\cdot (2n+1)!^{3}\cdot (2n)!^{3}\cdot n!^{3}}{(3n+2)!\cdot (4n+3)!^{3}}}}$

with . ${\displaystyle A(n)=126392n^{5}+412708n^{4}+531578n^{3}+336367n^{2}+104000n+12463}$

According to Matyáš Lerch (1900):

${\displaystyle \zeta (3)={\frac {7\pi ^{3}}{180}}-2\sum _{n=1}^{\infty }{\frac {1}{n^{3}(e^{2\pi n}-1)}}}$

Simon Plouffe developed this expression further:

${\displaystyle \zeta (3)={\frac {\pi ^{3}}{28}}+{\frac {16}{7}}\sum _{n=1}^{\infty }{\frac {1}{n^{3}(e^{\pi n}+1)}}-{\frac {2}{7}}\sum _{n=1}^{\infty }{\frac {1}{n^{3}(e^{2\pi n}+1)}}}$

${\displaystyle \zeta (3)=28\sum _{n=1}^{\infty }{\frac {1}{n^{3}(e^{\pi n}-1)}}-37\sum _{n=1}^{\infty }{\frac {1}{n^{3}(e^{2\pi n}-1)}}+7\sum _{n=1}^{\infty }{\frac {1}{n^{3}(e^{4\pi n}-1)}}}$

More formulas

A connection to the prime numbers is

${\displaystyle \zeta (3)=\prod _{p\ \mathrm {prim} }{\frac {1}{1-p^{-3}}}}$

as a special case of the Euler product (Euler 1737).

There are also some integral representations, for example:

${\displaystyle \zeta (3)=\int \limits _{0}^{1}\int \limits _{0}^{1}\int \limits _{0}^{1}{\frac {\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z}{1-xyz}}}$

${\displaystyle \zeta (3)={\frac {1}{2}}\int \limits _{0}^{\infty }{\frac {x^{2}}{\mathrm {e} ^{x}-1}}\mathrm {d} x}$

${\displaystyle \zeta (3)={\frac {2}{3}}\pi ^{3}\int \limits _{0}^{1}x(x-{\frac {1}{2}})(x-1)\cot(\pi x)\mathrm {d} x}$

It also appears as a special case of the second polygamma function , namely:

${\displaystyle \zeta (3)=-{\frac {1}{2}}\psi _{2}(1)}$

literature

• Frits Beukers : A note on the irrationality of and . ${\displaystyle \zeta (2)}$ ${\displaystyle \zeta (3)}$Bulletin of the London Mathematical Society 11, October 1979, pp. 268-272.
• Alfred van der Poorten : A proof that Euler missed… Apéry's proof of the irrationality of ζ (3). An informal report. The Mathematical Intelligencer 1, December 1979, pp. 195-203 (English: Alf's reprints. Paper 45, PDF; 205 kB).
• Steven R. Finch: Apéry's constant. Chapter 1.6 in Mathematical constants. Cambridge University Press, Cambridge 2003, ISBN 0-521-81805-2 , pp. 40-53 (English).

Web links

Wikibooks: Proof of the irrationality of Zeta (3) according to Frits Beukers  - learning and teaching materials

• Eric W. Weisstein : Apéry's Constant . In: MathWorld (English).
• Sequence A013631 in OEIS ( continued fraction expansion of ζ (3))
• Episode A053980 in OEIS (Engel development of ζ (3))

Individual evidence

1. ^ Records set by y-cruncher. Retrieved August 12, 2019 . 
2. ↑ Leonhard Euler : Inventio summae cuiusque seriei ex dato termino generali. October 13, 1735, Commentarii academiae scientiarum imperialis Petropolitanae 8, 1741, pp. 9-22 (Latin; "1.202056903159594" on p. 21 ).
3. ↑ Roger Apéry : Irrationalité de et . ${\displaystyle \zeta (2)}$ ${\displaystyle \zeta (3)}$Astérisque 61, 1979, pp. 11-13 (French).
4. ^ David H. Bailey , Richard E. Crandall : Random Generators and Normal Numbers. ( Memento of October 13, 2003 in the Internet Archive ). (PDF; 399 kB), Experimental Mathematics 11, 2002, pp. 527-546 (English).
5. ↑ Finch: Apéry's constant. 2003, p. 41 (English).
6. ↑ Tanguy Rivoal: La Fonction zêta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs. Comptes rendus de l'Académie des sciences Série I 331, 2000, pp. 267-270 (French; arxiv : math / 0008051v1 ).
7. ↑ WW Zudilin: One of the numbers ζ (5), ζ (7), ζ (9), ζ (11) is irrational . Russian Mathematical Surveys 56, 2001, pp. 774-776 (English).
8. ↑ Georges Rhin, Carlo Viola: The group structure for ζ (3). Acta Arithmetica 97, 2001, pp. 269-293 (English).
9. ↑ M. Beeler, RW Gosper , R. Schroeppel : HAKMEM. MIT AI Memo 239, February 29, 1972 (English), ITEM 53 (Salamin).
10. ↑ Walther Janous: Around apery's constant. Journal of inequalities in pure and applied mathematics 7, 2006, Article 35 (English).
11. ↑ Tewodros Amdeberhan, Doron Zeilberger : Hypergeometric series acceleration via the WZ method. ( Memento of April 30, 2011 in the Internet Archive ). The Electronic Journal of Combinatorics 4 (2), 1997 (English). arxiv : math / 9804121v1
12. ^ The Value of Zeta (3) to 1,000,000 places. In Project Gutenberg (English).
13. ^ Matyáš Lerch : Sur la fonction ζ (s) pour les valeurs impaires de l'argument. Jornal de sciencias mathematicas e astronomicas 14, 1900, pp. 65-69 (French; yearbook summary ).
14. ^ Simon Plouffe : Identities inspired by Ramanujan Notebooks (part 2). April 2006 (English).
15. ^ Leonhard Euler : Variae observationes circa series infinitas. April 25, 1737, Commentarii academiae scientiarum imperialis Petropolitanae 9, 1744, pp. 160-188 (Latin; Euler product as “Theorema 8” on p. 174 f. ).
16. ↑ Abramowitz-Stegun : Riemann Zeta Function and Other Sums of Reciprocal Powers. P. 807, formula 2/23/17.