Apéry's constant

Apéry's constant arises naturally in a number of physical problems, including in the second- and third-order terms of the electron's gyromagnetic ratio using quantum electrodynamics. It also arises in the analysis of random minimum spanning trees{{sfnp|Frieze|1985}} 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.

The reciprocal of {{math|ζ(3)}} (0.8319073725807... {{OEIS|id=A088453}}) is the probability that any three positive integers, chosen at random, will be relatively prime, in the sense that as {{math|N}} approaches infinity, the probability that three positive integers less than {{math|N}} chosen uniformly at random will not share a common prime factor approaches this value. (The probability for n positive integers is {{math|1/ζ(n)}}.{{sfnp|Mollin|2009}}) In the same sense, it is the probability that a positive integer chosen at random will not be evenly divisible by the cube of an integer greater than one. (The probability for not having divisibility by an n-th power is {{math|1/ζ(n)}}.{{sfnp|Mollin|2009}})

{{unsolved|mathematics|Is Apéry's constant transcendental?}}

{{math|ζ(3)}} was named Apéry's constant after the French mathematician Roger Apéry, who proved in 1978 that it is an irrational number.{{sfnp|Apéry|1979}} This result is known as Apéry's theorem. The original proof is complex and hard to grasp,{{sfnp|van der Poorten|1979}} and simpler proofs were found later.{{harvtxt|Beukers|1979}}; {{harvtxt|Zudilin|2002}}.

Beukers's simplified irrationality proof involves approximating the integrand of the known triple integral for {{math|ζ(3)}},

:$\backslash zeta(3)\; =\; \backslash int\_0^1\; \backslash int\_0^1\; \backslash int\_0^1\; \backslash frac\{1\}\{1-xyz\}\backslash ,\; dx\backslash ,\; dy\backslash ,\; dz,$

by the Legendre polynomials.

In particular, van der Poorten's article chronicles this approach by noting that

:$I\_3\; :=\; -\backslash frac\{1\}\{2\}\; \backslash int\_0^1\; \backslash int\_0^1\; \backslash frac\{P\_n(x)\; P\_n(y)\; \backslash log(xy)\}\{1-xy\}\backslash ,\; dx\backslash ,\; dy\; =\; b\_n\; \backslash zeta(3)\; -\; a\_n,$

where $|I|\; \backslash leq\; \backslash zeta(3)\; (1-\backslash sqrt\{2\})^\{4n\}$, $P\_n(z)$ are the Legendre polynomials, and the subsequences $b\_n,\; 2\; \backslash operatorname\{lcm\}(1,2,\backslash ldots,n)\; \backslash cdot\; a\_n\; \backslash in\; \backslash mathbb\{Z\}$ are integers or almost integers.

Many people have tried to extend Apéry's proof that {{math|ζ(3)}} is irrational to other values of the Riemann zeta function with odd arguments. Although this has so far not produced any results on specific numbers, it is known that infinitely many of the odd zeta constants {{math|ζ(2n + 1)}} are irrational.{{sfnp|Rivoal|2000}} In particular at least one of {{math|ζ(5)}}, {{math|ζ(7)}}, {{math|ζ(9)}}, and {{math|ζ(11)}} must be irrational.{{sfnp|Zudilin|2001}}

Apéry's constant has not yet been proved transcendental, but it is known to be an algebraic period. This follows immediately from the form of its triple integral.

In addition to the fundamental series:

: $\backslash zeta(3)\; =\; \backslash sum\_\{k=1\}^\backslash infty\; \backslash frac\{1\}\{k^3\},$

Leonhard Euler gave the series representation:{{sfnp|Euler|1773}}

: $\backslash zeta(3)\; =\; \backslash frac\{\backslash pi^2\}\{7\}\; \backslash left(1\; -\; 4\backslash sum\_\{k=1\}^\backslash infty\; \backslash frac\{\backslash zeta\; (2k)\}\{2^\{2k\}(2k\; +\; 1)(2k\; +\; 2)\}\backslash right)$

in 1772, which was subsequently rediscovered several times.{{sfnp|Srivastava|2000|loc=p. 571 (1.11)}}

Since the 19th century, a number of mathematicians have found convergence acceleration series for calculating decimal places of {{math|ζ(3)}}. Since the 1990s, this search has focused on computationally efficient series with fast convergence rates (see section "Known digits").

The following series representation was found by A. A. Markov in 1890,{{sfnp|Markov|1890}} rediscovered by Hjortnaes in 1953,{{sfnp|Hjortnaes|1953}} and rediscovered once more and widely advertised by Apéry in 1979:{{sfnp|Apéry|1979}}

: $\backslash zeta(3)\; =\; \backslash frac\{5\}\{2\}\; \backslash sum\_\{k=1\}^\backslash infty\; (-1)^\{k-1\}\; \backslash frac\{k!^2\}\{(2k)!\; k^3\}.$

The following series representation gives (asymptotically) 1.43 new correct decimal places per term:{{sfnp|Amdeberhan|1996}}

: $\backslash zeta(3)\; =\; \backslash frac\{1\}\{4\}\; \backslash sum\_\{k=1\}^\backslash infty\; (-1)^\{k-1\}\; \backslash frac\{(k\; -\; 1)!^3\; (56k^2\; -\; 32k\; +\; 5)\}\{(2k\; -\; 1)^2(3k)!\}.$

The following series representation gives (asymptotically) 3.01 new correct decimal places per term:{{sfnp|Amdeberhan|Zeilberger|1997}}

: $\backslash zeta(3)\; =\; \backslash frac\{1\}\{64\}\; \backslash sum\_\{k=0\}^\backslash infty\; (-1)^k\; \backslash frac\{k!^\{10\}\; (205k^2\; +\; 250k\; +\; 77)\}\{(2k\; +\; 1)!^5\}.$

The following series representation gives (asymptotically) 5.04 new correct decimal places per term:{{harvtxt|Wedeniwski|1998}}; {{harvtxt|Wedeniwski|2001}}. In his message to Simon Plouffe, Sebastian Wedeniwski states that he derived this formula from {{harvtxt|Amdeberhan|Zeilberger|1997}}. The discovery year (1998) is mentioned in [http://plouffe.fr/simon/articles/TableofRecords.pdf Simon Plouffe's Table of Records] (8 April 2001).

: $\backslash zeta(3)\; =\; \backslash frac\{1\}\{24\}\; \backslash sum\_\{k=0\}^\backslash infty\; (-1)^k\; \backslash frac\{(2k\; +\; 1)!^3\; (2k)!^3\; k!^3\; (126392k^5\; +\; 412708k^4\; +\; 531578k^3\; +\; 336367k^2\; +\; 104000k\; +\; 12463)\}\{(3k\; +\; 2)!\; (4k\; +\; 3)!^3\}.$

It has been used to calculate Apéry's constant with several million correct decimal places.{{harvtxt|Wedeniwski|1998}}; {{harvtxt|Wedeniwski|2001}}.

The following series representation gives (asymptotically) 3.92 new correct decimal places per term:{{sfnp|Mohammed|2005}}

: $\backslash zeta(3)\; =\; \backslash frac\{1\}\{2\}\; \backslash sum\_\{k=0\}^\backslash infty\; \backslash frac\{(-1)^k\; (2k)!^3\; (k\; +\; 1)!^6\; (40885k^5\; +\; 124346k^4\; +\; 150160k^3\; +\; 89888k^2\; +\; 26629k\; +\; 3116)\}\{(k\; +\; 1)^2\; (3k\; +\; 3)!^4\}.$

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.{{sfnp|Broadhurst|1998}}

Apéry's constant can be represented in terms of the Thue-Morse sequence $(t\_n)\_\{n\backslash geq0\}$, as follows:

{{citation|last1=Tóth|first1=László|title=Linear Combinations of Dirichlet Series Associated with the Thue-Morse Sequence|journal=Integers|volume=22|year=2022|article-number=98|arxiv=2211.13570 |url=http://math.colgate.edu/~integers/w98/w98.pdf}}

:$\backslash begin\{align\}$

\sum_{n\geq1} \frac{9 t_{n-1} + 7 t_n}{n^3} &= 8 \zeta(3),\end{align}

This is a special case of the following formula (valid for all $s$ with real part greater than $1$):

:$(2^s+1)\; \backslash sum\_\{n\backslash geq1\}\; \backslash frac\{t\_\{n-1\}\}\{n^s\}\; +\; (2^s-1)\; \backslash sum\_\{n\backslash geq1\}\; \backslash frac\{t\_\{n\}\}\{n^s\}\; =\; 2^s\; \backslash zeta(s).$

The following series representation was found by Ramanujan:{{harvtxt|Berndt|1989|loc=chapter 14, formulas 25.1 and 25.3}}.

: $\backslash zeta(3)\; =\; \backslash frac\{7\}\{180\}\; \backslash pi^3\; -\; 2\; \backslash sum\_\{k=1\}^\backslash infty\; \backslash frac\{1\}\{k^3\; (e^\{2\backslash pi\; k\}\; -\; 1)\}.$

The following series representation was found by Simon Plouffe in 1998:{{sfnp|Plouffe|1998}}

: $\backslash zeta(3)\; =\; 14\; \backslash sum\_\{k=1\}^\backslash infty\; \backslash frac\{1\}\{k^3\; \backslash sinh(\backslash pi\; k)\}\; -\; \backslash frac\{11\}\{2\}\; \backslash sum\_\{k=1\}^\backslash infty\; \backslash frac\{1\}\{k^3\; (e^\{2\backslash pi\; k\}\; -\; 1)\}\; -\; \backslash frac\{7\}\{2\}\; \backslash sum\_\{k=1\}^\backslash infty\; \backslash frac\{1\}\{k^3\; (e^\{2\backslash pi\; k\}\; +\; 1)\}.$

{{harvtxt|Srivastava|2000}} collected many series that converge to Apéry's constant.

There are numerous integral representations for Apéry's constant. Some of them are simple, others are more complicated.

The following formula follows directly from the integral definition of the zeta function:

: $\backslash zeta(3)\; =\; \backslash frac1\{2\}\backslash int\_0^\backslash infty\; \backslash frac\{x^2\}\{e^x\; -\; 1\}\; \backslash ,dx$

Other formulas include{{sfnp|Jensen|1895}}

: $\backslash zeta(3)\; =\; \backslash pi\; \backslash int\_0^\backslash infty\; \backslash frac\{\backslash cos(2\backslash arctan\; x)\}\{(x^2\; +\; 1)\; \backslash left(\backslash cosh\backslash frac\{1\}\{2\}\backslash pi\; x\backslash right)^2\}\; \backslash ,dx$

and{{sfnp|Beukers|1979}}

: $\backslash zeta(3)\; =\; -\backslash frac\{1\}\{2\}\; \backslash int\_0^1\; \backslash !\backslash !\backslash int\_0^1\; \backslash frac\{\backslash log(xy)\}\{1\; -\; xy\}\; \backslash ,dx\backslash ,dy\; =\; -\backslash int\_0^1\; \backslash !\backslash !\backslash int\_0^1\; \backslash frac\{\backslash log(1\; -\; xy)\}\{xy\}\; \backslash ,dx\backslash ,dy.$

Also,{{sfnp|Blagouchine|2014}}

: $$

\begin{align}

\zeta(3) &= \frac{8\pi^2}{7} \int_0^1 \frac{x(x^4 - 4x^2 + 1) \log\log\frac{1}{x}}{(1 + x^2)^4} \,dx \\

&= \frac{8\pi^2}{7} \int_1^\infty \frac{x(x^4 - 4x^2 + 1) \log\log{x}}{(1 + x^2)^4} \,dx.

\end{align}

A connection to the derivatives of the gamma function{{citation|url=https://scipp.ucsc.edu/~haber/archives/physics116A10/psifun_10.pdf|title=The logarithmic derivative of the Gamma function|work=Physics 116A lecture notes|date=Winter 2010|first=Howard E.|last=Haber|publisher=University of California, Santa Cruz}}

: $\backslash zeta(3)\; =\; -\backslash tfrac\{1\}\{2\}(\backslash Gamma\text{'}\text{'}\text{'}(1)\; +\; \backslash gamma^3+\; \backslash tfrac\{1\}\{2\}\backslash pi^2\backslash gamma)\; =\; -\backslash tfrac\{1\}\{2\}\; \backslash psi^\{(2)\}(1)$

is also very useful for the derivation of various integral representations via the known integral formulas for the gamma and polygamma functions.{{sfnp|Evgrafov|Bezhanov|Sidorov|Fedoriuk|1969|loc=exercise 30.10.1}}

Apéry's constant is related to the following continued fraction:{{Cite web |last=Weisstein |first=Eric W. |title=Apéry's Constant |url=https://mathworld.wolfram.com/AperysConstant.html |access-date=2024-09-21 |website=mathworld.wolfram.com |language=en}}

:$\backslash frac\{6\}\{\backslash zeta(3)\}=5-\backslash cfrac\{1\}\{117-\backslash cfrac\{64\}\{535-\backslash cfrac\{729\}\{1436-\backslash cfrac\{4096\}\{3105-\backslash cfrac\{15625\}\{\backslash dots\}\}\}\}\}$

with $a\_n=34n^3+51n^2+27n+5$ and $b\_n=-n^6$.

Its simple continued fraction is given by:{{Cite web |last=Weisstein |first=Eric W. |title=Apéry's Constant Continued Fraction |url=https://mathworld.wolfram.com/AperysConstantContinuedFraction.html |access-date=2024-09-21 |website=mathworld.wolfram.com |language=en}}

:$\backslash zeta(3)=1+\backslash cfrac\{1\}\{4+\backslash cfrac\{1\}\{1+\backslash cfrac\{1\}\{18+\backslash cfrac\{1\}\{1+\backslash cfrac\{1\}\{1+\backslash cfrac\{1\}\{\backslash dots\}\}\}\}\}\}$

The number of known digits of Apéry's constant {{math|ζ(3)}} has increased dramatically during the last decades, and now stands at more than {{val|2|e=12}}. This is due both to the increasing performance of computers and to algorithmic improvements.

:

• Riemann zeta function
• Basel problem — {{math|ζ(2)}}
• Catalan's constant
• List of sums of reciprocals

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{{PlanetMath attribution|urlname=AperysConstant|title=Apéry's constant}}

{{Irrational number}}

{{DEFAULTSORT:Aperys constant}}

Category:Mathematical constants

Category:Analytic number theory

Category:Irrational numbers

Category:Zeta and L-functions