Apéry's constant

Properties

ζ(3) was named Apéry's constant after the French mathematician Roger Apéry, who proved in 1978 that it is an irrational number. This result is known as Apéry's theorem. The original proof is complex and hard to grasp, and simpler proofs were found later.

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

${\displaystyle \zeta (3)={\int }_0^1{\int }_0^1{\int }_0^1\frac{1}{1-xyz}dxdydz,}$

by the Legendre polynomials. In particular, van der Poorten's article chronicles this approach by noting that

${\displaystyle I_3:=-\frac{1}{2}{\int }_0^1{\int }_0^1\frac{P_n(x)P_n(y)\log (xy)}{1-xy}dxdy=b_n\zeta (3)-a_n,}$

where ${\displaystyle |I|\le \zeta (3)(1-\sqrt{2}{)}^{4n}}$, ${\displaystyle P_n(z)}$ are the Legendre polynomials, and the subsequences ${\displaystyle b_n,2\mathrm{lcm}(1,2,\dots ,n)\cdot a_n\in \mathbb{Z}}$ are integers or almost integers.

Many people have tried to extend Apéry's proof that 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 ζ(2n + 1) are irrational, and that at least one of ζ(5), ζ(7), ζ(9), and ζ(11) must be irrational.

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.

Series representations

Classical

In addition to the fundamental series:

${\displaystyle \zeta (3)=\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k^3},}$

Leonhard Euler gave the series representation:

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

in 1772, which was subsequently rediscovered several times.

Fast convergence

Since the 19th century, a number of mathematicians have found convergence acceleration series for calculating decimal places of 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, rediscovered by Hjortnaes in 1953, and rediscovered once more and widely advertised by Apéry in 1979:

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

The following series representation gives (asymptotically) 1.43 new correct decimal places per term:

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

The following series representation gives (asymptotically) 3.01 new correct decimal places per term:

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

The following series representation gives (asymptotically) 3.92 new correct decimal places per term:

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

The following series representation gives (asymptotically) 5.04 new correct decimal places per term:

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

It has been used to calculate Apéry's constant with several million correct decimal places.

Digit by digit

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.

Others

The following series representation was found by Ramanujan:

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

The following series representation was found by Simon Plouffe in 1998:

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

A generalization of these can be proven with a new technique that relies on products of series for ${\displaystyle a\ne 0}$:

${\displaystyle \zeta (3)=\frac{1+5a^2+a^4}{90a(1+a^2)}{\pi }^3-\frac{2}{1+a^2}\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k^3(e^{2a\pi k}-1)}-\frac{2a^2}{1+a^2}\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k^3(e^{2\pi k/a}-1)}}$

It also admits one based on odd numbers (representing odd numbers as ${\displaystyle N=2k-1}$):

${\displaystyle \zeta (3)=\frac{a{\pi }^3}{14(1+a^2)}+\frac{16}{7(1+a^2)}\sum _{k=1}^{\mathrm{\infty }}\frac{1}{N^3(e^{a\pi N}+1)}+\frac{16a^2}{7(1+a^2)}\sum _{k=1}^{\mathrm{\infty }}\frac{1}{N^3(e^{\pi N/a}+1)}}$

Finally, another compact result:

${\displaystyle \zeta (3)=\frac{19{\pi }^3}{540}-\frac{8}{3}\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k^3(e^{\pi k}-(-1{)}^k)}}$

Srivastava (2000) collected many series that converge to Apéry's constant.

Integral representations

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

Simple formulas

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

${\displaystyle \zeta (3)=\frac{1}{2}{\int }_0^{\mathrm{\infty }}\frac{x^2}{e^x-1}dx}$

More complicated formulas

Other formulas include

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

and

${\displaystyle \zeta (3)=-\frac{1}{2}{\int }_0^1{\int }_0^1\frac{\log (xy)}{1-xy}dxdy=-{\int }_0^1{\int }_0^1\frac{\log (1-xy)}{xy}dxdy.}$

Also,

${\displaystyle \begin{array}{cc}\zeta (3)& =\frac{8{\pi }^2}{7}{\int }_0^1\frac{x(x^4-4x^2+1)\log \log \frac{1}{x}}{(1+x^2{)}^4}dx\\ & =\frac{8{\pi }^2}{7}{\int }_1^{\mathrm{\infty }}\frac{x(x^4-4x^2+1)\log \log x}{(1+x^2{)}^4}dx.\end{array}}$

A connection to the derivatives of the gamma function

${\displaystyle \zeta (3)=-\frac{1}{2}({\mathrm{\Gamma }}^{‴}(1)+{\gamma }^3+\frac{1}{2}{\pi }^2\gamma )=-\frac{1}{2}{\psi }^{(2)}(1)}$

is also very useful for the derivation of various integral representations via the known integral formulas for the gamma and polygamma functions.

Continued fraction

Apéry's constant is related to the following continued fraction:

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

with ${\displaystyle a_n=34n^3+51n^2+27n+5}$ and ${\displaystyle b_n=-n^6}$.

Its simple continued fraction is given by:

${\displaystyle \zeta (3)=1+\frac{1}{4+\frac{1}{1+\frac{1}{18+\frac{1}{1+\frac{1}{1+\frac{1}{\dots }}}}}}}$

Other formulas are given by:

${\displaystyle \zeta (3)=G_0},G_k=1+\frac{1}{4+\frac{1}{1+\frac{(k+1{)}^3}{(k+1)(2k+3)(2k+4)+\frac{(k+2{)}^3}{G_{k+1}}}}}$

${\displaystyle \zeta (3)=\frac{6}{W_0},W_k=4(2k+1{)}^3+\frac{(k+1{)}^3}{1+\frac{(k+1{)}^3}{W_{k+1}}}}$

${\displaystyle 2\zeta (3)=2+\frac{1}{N_0},N_k=2k+2+\frac{(k+1)(k+2)}{2k+4+\frac{(k+1{)}^2}{2k+3+\frac{(k+2{)}^2}{2k+2+\frac{(k+1)(k+2)}{N_{k+1}}}}}}$

${\displaystyle 2\zeta (3)=2+\frac{42}{H_0},H_k=a(k)-\frac{b(k)}{H_{k+1}}}$

with

${\displaystyle a(k)=2(102k^6+612k^5+1462k^4+1768k^3+1143k^2+382k+52)}$

${\displaystyle b(k)=(k+1{)}^3(k+2{)}^3(2k+1)(2k+5)(3k^2+3k+1)(3k^2+15k+19)}$

${\displaystyle 2\zeta (3)=\frac{7}{6}+\frac{73}{12\ast S_0},S_k=a(k)-\frac{b(k)}{S_{k+1}}}$

with

${\displaystyle a(k)=2(204k^6+1173k^5+2668k^4+3065k^3+1905k^2+634k+86)}$

${\displaystyle b(k)=\frac{(k+1{)}^3(k+2{)}^3}{4}(24k^3+30k^2+16k+3)(24k^3+174k^2+424k+347)}$

Known digits

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. This is due both to the increasing performance of computers and to algorithmic improvements.