Zeta(2)

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In 1735, Euler proved that

$\zeta (2)=\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}={\frac {\pi ^{2}}{6}},\,$

where $\scriptstyle \zeta (s)\,$ is the Riemann zeta function.

Since pi is transcendental, zeta(2) is also transcendental.

Decimal expansion of zeta(2)

$\zeta (2)={\frac {\pi ^{2}}{6}}=1.6449340668482264364724151666460251892189499\ldots \,$

giving the sequence of decimal digits (A013661)

{1, 6, 4, 4, 9, 3, 4, 0, 6, 6, 8, 4, 8, 2, 2, 6, 4, 3, 6, 4, 7, 2, 4, 1, 5, 1, 6, 6, 6, 4, 6, 0, 2, 5, 1, 8, 9, 2, 1, 8, 9, 4, 9, 9, 0, 1, 2, 0, 6, 7, 9, 8, 4, 3, 7, 7, 3, 5, 5, 5, 8, 2, 2, 9, 3, 7, 0, 0, 0, 7, 4, 7, 0, 4, 0, 3, 2, ...}

Continued fraction expansion of zeta(2)

The simple continued fraction expansion of zeta(2) is

$\zeta (2)={\frac {\pi ^{2}}{6}}=1~+~{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{4+{\cfrac {1}{2+{\cfrac {1}{4+{\cfrac {1}{7+{\cfrac {1}{1+{\cfrac {1}{\ddots }}}}}}}}}}}}}}}}}}\,$

giving the sequence of partial quotients (A013679)

{1, 1, 1, 1, 4, 2, 4, 7, 1, 4, 2, 3, 4, 10, 1, 2, 1, 1, 1, 15, 1, 3, 6, 1, 1, 2, 1, 1, 1, 2, 2, 3, 1, 3, 1, 1, 5, 1, 2, 2, 1, 1, 6, 27, 20, 3, 97, 105, 1, 1, 1, 1, 1, 45, 2, 8, 19, 1, 4, 1, 1, 3, 1, 2, 1, 1, 1, 5, 1, 1, 2, 3, 6, ...}

1/zeta(2)

1/zeta(2) is

• the density of squarefree numbers (the probability that a randomly chosen integer is squarefree);
• the probability that two randomly chosen integers are coprime.

Decimal expansion of 1/zeta(2)

${\frac {1}{\zeta (2)}}={\frac {6}{\pi ^{2}}}=0.6079271018540266286632767792583658334261526480\ldots \,$

giving the sequence of decimal digits (A059956)

{6, 0, 7, 9, 2, 7, 1, 0, 1, 8, 5, 4, 0, 2, 6, 6, 2, 8, 6, 6, 3, 2, 7, 6, 7, 7, 9, 2, 5, 8, 3, 6, 5, 8, 3, 3, 4, 2, 6, 1, 5, 2, 6, 4, 8, 0, 3, 3, 4, 7, 9, 2, 9, 3, 0, 7, ...}

See also

• Zeta(3) (known as Apéry's constant)