Why can we not establish the irrationality of Catalan's constant the same way as $zeta(3)$?
$begingroup$
One of the main ingredients in Apéry's proof of the irrationality of $zeta(3)$ is the existence of the fast-converging series:
$$
{displaystyle {begin{aligned}zeta (3)&={frac {5}{2}}sum _{k=1}^{infty }{frac {(-1)^{k-1}}{{binom {2k}{k}}k^{3}}}end{aligned}}}.
$$
Despite numerous attempts, no similar expressions were found for other values of the Riemann $zeta$-function at positive odd integers.
For Catalan's constant, however, we do have such an expression, namely:
$${displaystyle G={frac {pi }{8}}log left(2+{sqrt {3}}right)+{frac {3}{8}}sum _{n=0}^{infty }{frac {1}{(2n+1)^{2}{binom {2n}{n}}}}.}$$
Why is this not sufficient for applying an Apéry-like method for proving its irrationality?
irrational-numbers riemann-zeta irrationality-measure
edited Jan 21 at 17:12
Tito Piezas III
27.3k366174
asked Oct 8 '18 at 12:37
KlangenKlangen
1,75811334
$endgroup$
add a comment |
$begingroup$
One of the main ingredients in Apéry's proof of the irrationality of $zeta(3)$ is the existence of the fast-converging series:
$$
{displaystyle {begin{aligned}zeta (3)&={frac {5}{2}}sum _{k=1}^{infty }{frac {(-1)^{k-1}}{{binom {2k}{k}}k^{3}}}end{aligned}}}.
$$
Despite numerous attempts, no similar expressions were found for other values of the Riemann $zeta$-function at positive odd integers.
For Catalan's constant, however, we do have such an expression, namely:
$${displaystyle G={frac {pi }{8}}log left(2+{sqrt {3}}right)+{frac {3}{8}}sum _{n=0}^{infty }{frac {1}{(2n+1)^{2}{binom {2n}{n}}}}.}$$
Why is this not sufficient for applying an Apéry-like method for proving its irrationality?
irrational-numbers riemann-zeta irrationality-measure
edited Jan 21 at 17:12
Tito Piezas III
27.3k366174
asked Oct 8 '18 at 12:37
KlangenKlangen
1,75811334
$endgroup$
2
$begingroup$
Having such an expression is by no means the same as irrationality. It is not clear how to adapt Apery's proof.
$endgroup$
– Dietrich Burde
Oct 8 '18 at 12:42
add a comment |
$begingroup$
One of the main ingredients in Apéry's proof of the irrationality of $zeta(3)$ is the existence of the fast-converging series:
$$
{displaystyle {begin{aligned}zeta (3)&={frac {5}{2}}sum _{k=1}^{infty }{frac {(-1)^{k-1}}{{binom {2k}{k}}k^{3}}}end{aligned}}}.
$$
Despite numerous attempts, no similar expressions were found for other values of the Riemann $zeta$-function at positive odd integers.
For Catalan's constant, however, we do have such an expression, namely:
$${displaystyle G={frac {pi }{8}}log left(2+{sqrt {3}}right)+{frac {3}{8}}sum _{n=0}^{infty }{frac {1}{(2n+1)^{2}{binom {2n}{n}}}}.}$$
Why is this not sufficient for applying an Apéry-like method for proving its irrationality?
irrational-numbers riemann-zeta irrationality-measure
edited Jan 21 at 17:12
Tito Piezas III
27.3k366174
asked Oct 8 '18 at 12:37
KlangenKlangen
1,75811334
$endgroup$
One of the main ingredients in Apéry's proof of the irrationality of $zeta(3)$ is the existence of the fast-converging series:
$$
{displaystyle {begin{aligned}zeta (3)&={frac {5}{2}}sum _{k=1}^{infty }{frac {(-1)^{k-1}}{{binom {2k}{k}}k^{3}}}end{aligned}}}.
$$
Despite numerous attempts, no similar expressions were found for other values of the Riemann $zeta$-function at positive odd integers.
For Catalan's constant, however, we do have such an expression, namely:
$${displaystyle G={frac {pi }{8}}log left(2+{sqrt {3}}right)+{frac {3}{8}}sum _{n=0}^{infty }{frac {1}{(2n+1)^{2}{binom {2n}{n}}}}.}$$
Why is this not sufficient for applying an Apéry-like method for proving its irrationality?
irrational-numbers riemann-zeta irrationality-measure
irrational-numbers riemann-zeta irrationality-measure
edited Jan 21 at 17:12
Tito Piezas III
27.3k366174
asked Oct 8 '18 at 12:37
KlangenKlangen
1,75811334
edited Jan 21 at 17:12
Tito Piezas III
27.3k366174
asked Oct 8 '18 at 12:37
KlangenKlangen
1,75811334
edited Jan 21 at 17:12
Tito Piezas III
27.3k366174
edited Jan 21 at 17:12
Tito Piezas III
27.3k366174
edited Jan 21 at 17:12
Tito Piezas III
27.3k366174
27.3k366174
asked Oct 8 '18 at 12:37
KlangenKlangen
1,75811334
asked Oct 8 '18 at 12:37
KlangenKlangen
1,75811334
asked Oct 8 '18 at 12:37
KlangenKlangen
1,75811334
1,75811334
2
$begingroup$
Having such an expression is by no means the same as irrationality. It is not clear how to adapt Apery's proof.
$endgroup$
– Dietrich Burde
Oct 8 '18 at 12:42
add a comment |
2
$begingroup$
Having such an expression is by no means the same as irrationality. It is not clear how to adapt Apery's proof.
$endgroup$
– Dietrich Burde
Oct 8 '18 at 12:42
2
2
$begingroup$
Having such an expression is by no means the same as irrationality. It is not clear how to adapt Apery's proof.
$endgroup$
– Dietrich Burde
Oct 8 '18 at 12:42
$begingroup$
Having such an expression is by no means the same as irrationality. It is not clear how to adapt Apery's proof.
$endgroup$
– Dietrich Burde
Oct 8 '18 at 12:42
add a comment |
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$begingroup$
Having such an expression is by no means the same as irrationality. It is not clear how to adapt Apery's proof.
$endgroup$
– Dietrich Burde
Oct 8 '18 at 12:42