Is Apéry's constant a rational multiple of $ pi ^ 3$?
$begingroup$
It is well known that the values of the Riemann zeta function for even positive numbers are of the form:
$$zeta(2k) = rm rational * pi ^{2k},$$
and more specifically $zeta (2k)=(-1)^{{k+1}}{frac {B_{{2k}}(2pi )^{{2k}}}{2(2k)!}}!$. It is not that far-fetched to consider that
$$zeta(2k + 1) = rm rational * pi ^{2k + 1}.$$
Specifically for Apéry's constant (which is $zeta(3)$), did someone prove something like that? The proof should be something like:
$frac{zeta(3)}{pi^3}$ is rational / irrational / transcendental.
EDIT: Even if the question is still open (which I can see it is from the comments), is there any new development on this matter lately? Just curious.
complex-analysis number-theory riemann-zeta
$endgroup$
|
show 1 more comment
$begingroup$
It is well known that the values of the Riemann zeta function for even positive numbers are of the form:
$$zeta(2k) = rm rational * pi ^{2k},$$
and more specifically $zeta (2k)=(-1)^{{k+1}}{frac {B_{{2k}}(2pi )^{{2k}}}{2(2k)!}}!$. It is not that far-fetched to consider that
$$zeta(2k + 1) = rm rational * pi ^{2k + 1}.$$
Specifically for Apéry's constant (which is $zeta(3)$), did someone prove something like that? The proof should be something like:
$frac{zeta(3)}{pi^3}$ is rational / irrational / transcendental.
EDIT: Even if the question is still open (which I can see it is from the comments), is there any new development on this matter lately? Just curious.
complex-analysis number-theory riemann-zeta
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3
$begingroup$
No one has proved anything like this.
$endgroup$
– Lord Shark the Unknown
Sep 16 '17 at 18:02
4
$begingroup$
This is a well-known question, see this MO-question.
$endgroup$
– Dietrich Burde
Sep 16 '17 at 18:04
2
$begingroup$
A significant amount of effort has gone into this, but the nature of Apery's constant is still largely mysterious.
$endgroup$
– George Coote
Sep 16 '17 at 18:07
4
$begingroup$
I am not sure whether the problem is open. But I would be rather surprised if $large frac{zeta(3)}{pi^3}$ turned out to be rational. My guess is that it is even transcendental (of course, only a guess). The continued fraction I calculated with PARI/GP with $20 000$ digits accuracy, contains $19501$ entries not exceeding $134656$. So, if the constant IS rational, numerator and denominator must be very large.
$endgroup$
– Peter
Sep 16 '17 at 18:10
2
$begingroup$
Cross-site duplicate
$endgroup$
– Wojowu
Sep 16 '17 at 18:21
|
show 1 more comment
$begingroup$
It is well known that the values of the Riemann zeta function for even positive numbers are of the form:
$$zeta(2k) = rm rational * pi ^{2k},$$
and more specifically $zeta (2k)=(-1)^{{k+1}}{frac {B_{{2k}}(2pi )^{{2k}}}{2(2k)!}}!$. It is not that far-fetched to consider that
$$zeta(2k + 1) = rm rational * pi ^{2k + 1}.$$
Specifically for Apéry's constant (which is $zeta(3)$), did someone prove something like that? The proof should be something like:
$frac{zeta(3)}{pi^3}$ is rational / irrational / transcendental.
EDIT: Even if the question is still open (which I can see it is from the comments), is there any new development on this matter lately? Just curious.
complex-analysis number-theory riemann-zeta
$endgroup$
It is well known that the values of the Riemann zeta function for even positive numbers are of the form:
$$zeta(2k) = rm rational * pi ^{2k},$$
and more specifically $zeta (2k)=(-1)^{{k+1}}{frac {B_{{2k}}(2pi )^{{2k}}}{2(2k)!}}!$. It is not that far-fetched to consider that
$$zeta(2k + 1) = rm rational * pi ^{2k + 1}.$$
Specifically for Apéry's constant (which is $zeta(3)$), did someone prove something like that? The proof should be something like:
$frac{zeta(3)}{pi^3}$ is rational / irrational / transcendental.
EDIT: Even if the question is still open (which I can see it is from the comments), is there any new development on this matter lately? Just curious.
complex-analysis number-theory riemann-zeta
complex-analysis number-theory riemann-zeta
edited Dec 27 '18 at 11:39
Klangen
1,72811334
1,72811334
asked Sep 16 '17 at 18:01
Sagi ShadurSagi Shadur
340116
340116
3
$begingroup$
No one has proved anything like this.
$endgroup$
– Lord Shark the Unknown
Sep 16 '17 at 18:02
4
$begingroup$
This is a well-known question, see this MO-question.
$endgroup$
– Dietrich Burde
Sep 16 '17 at 18:04
2
$begingroup$
A significant amount of effort has gone into this, but the nature of Apery's constant is still largely mysterious.
$endgroup$
– George Coote
Sep 16 '17 at 18:07
4
$begingroup$
I am not sure whether the problem is open. But I would be rather surprised if $large frac{zeta(3)}{pi^3}$ turned out to be rational. My guess is that it is even transcendental (of course, only a guess). The continued fraction I calculated with PARI/GP with $20 000$ digits accuracy, contains $19501$ entries not exceeding $134656$. So, if the constant IS rational, numerator and denominator must be very large.
$endgroup$
– Peter
Sep 16 '17 at 18:10
2
$begingroup$
Cross-site duplicate
$endgroup$
– Wojowu
Sep 16 '17 at 18:21
|
show 1 more comment
3
$begingroup$
No one has proved anything like this.
$endgroup$
– Lord Shark the Unknown
Sep 16 '17 at 18:02
4
$begingroup$
This is a well-known question, see this MO-question.
$endgroup$
– Dietrich Burde
Sep 16 '17 at 18:04
2
$begingroup$
A significant amount of effort has gone into this, but the nature of Apery's constant is still largely mysterious.
$endgroup$
– George Coote
Sep 16 '17 at 18:07
4
$begingroup$
I am not sure whether the problem is open. But I would be rather surprised if $large frac{zeta(3)}{pi^3}$ turned out to be rational. My guess is that it is even transcendental (of course, only a guess). The continued fraction I calculated with PARI/GP with $20 000$ digits accuracy, contains $19501$ entries not exceeding $134656$. So, if the constant IS rational, numerator and denominator must be very large.
$endgroup$
– Peter
Sep 16 '17 at 18:10
2
$begingroup$
Cross-site duplicate
$endgroup$
– Wojowu
Sep 16 '17 at 18:21
3
3
$begingroup$
No one has proved anything like this.
$endgroup$
– Lord Shark the Unknown
Sep 16 '17 at 18:02
$begingroup$
No one has proved anything like this.
$endgroup$
– Lord Shark the Unknown
Sep 16 '17 at 18:02
4
4
$begingroup$
This is a well-known question, see this MO-question.
$endgroup$
– Dietrich Burde
Sep 16 '17 at 18:04
$begingroup$
This is a well-known question, see this MO-question.
$endgroup$
– Dietrich Burde
Sep 16 '17 at 18:04
2
2
$begingroup$
A significant amount of effort has gone into this, but the nature of Apery's constant is still largely mysterious.
$endgroup$
– George Coote
Sep 16 '17 at 18:07
$begingroup$
A significant amount of effort has gone into this, but the nature of Apery's constant is still largely mysterious.
$endgroup$
– George Coote
Sep 16 '17 at 18:07
4
4
$begingroup$
I am not sure whether the problem is open. But I would be rather surprised if $large frac{zeta(3)}{pi^3}$ turned out to be rational. My guess is that it is even transcendental (of course, only a guess). The continued fraction I calculated with PARI/GP with $20 000$ digits accuracy, contains $19501$ entries not exceeding $134656$. So, if the constant IS rational, numerator and denominator must be very large.
$endgroup$
– Peter
Sep 16 '17 at 18:10
$begingroup$
I am not sure whether the problem is open. But I would be rather surprised if $large frac{zeta(3)}{pi^3}$ turned out to be rational. My guess is that it is even transcendental (of course, only a guess). The continued fraction I calculated with PARI/GP with $20 000$ digits accuracy, contains $19501$ entries not exceeding $134656$. So, if the constant IS rational, numerator and denominator must be very large.
$endgroup$
– Peter
Sep 16 '17 at 18:10
2
2
$begingroup$
Cross-site duplicate
$endgroup$
– Wojowu
Sep 16 '17 at 18:21
$begingroup$
Cross-site duplicate
$endgroup$
– Wojowu
Sep 16 '17 at 18:21
|
show 1 more comment
1 Answer
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$begingroup$
While Apéry proved in 1978 that $zeta(3)$ is irrational, the irrationality of $frac{zeta(3)}{pi^3}$ is still an open problem. There are some formulas expressing odd zeta values in terms of powers of $pi$, the most known ones are due to Plouffe and Borwein & Bradley. Here are some examples:
$$
begin{aligned}
zeta(3)&=frac{7pi^3}{180}-2sum_{n=1}^infty frac{1}{n^3(e^{2pi n}-1)},\
sum_{n=1}^infty frac{1}{n^3,binom {2n}n} &= -frac{4}{3},zeta(3)+frac{pisqrt{3}}{2cdot 3^2},left(zeta(2, tfrac{1}{3})-zeta(2,tfrac{2}{3}) right).
end{aligned}
$$
Moreover, in this Math.SE post we have:
$$
frac{3}{2},zeta(3) = frac{pi^3}{24}sqrt{2}-2sum_{k=1}^infty frac{1}{k^3(e^{pi ksqrt{2}}-1)}-sum_{k=1}^inftyfrac{1}{k^3(e^{2pi ksqrt{2}}-1)}.
$$
You can also check out this paper by Vepstas, which provides a nice generalization to some of these identities.
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1 Answer
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1 Answer
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$begingroup$
While Apéry proved in 1978 that $zeta(3)$ is irrational, the irrationality of $frac{zeta(3)}{pi^3}$ is still an open problem. There are some formulas expressing odd zeta values in terms of powers of $pi$, the most known ones are due to Plouffe and Borwein & Bradley. Here are some examples:
$$
begin{aligned}
zeta(3)&=frac{7pi^3}{180}-2sum_{n=1}^infty frac{1}{n^3(e^{2pi n}-1)},\
sum_{n=1}^infty frac{1}{n^3,binom {2n}n} &= -frac{4}{3},zeta(3)+frac{pisqrt{3}}{2cdot 3^2},left(zeta(2, tfrac{1}{3})-zeta(2,tfrac{2}{3}) right).
end{aligned}
$$
Moreover, in this Math.SE post we have:
$$
frac{3}{2},zeta(3) = frac{pi^3}{24}sqrt{2}-2sum_{k=1}^infty frac{1}{k^3(e^{pi ksqrt{2}}-1)}-sum_{k=1}^inftyfrac{1}{k^3(e^{2pi ksqrt{2}}-1)}.
$$
You can also check out this paper by Vepstas, which provides a nice generalization to some of these identities.
$endgroup$
add a comment |
$begingroup$
While Apéry proved in 1978 that $zeta(3)$ is irrational, the irrationality of $frac{zeta(3)}{pi^3}$ is still an open problem. There are some formulas expressing odd zeta values in terms of powers of $pi$, the most known ones are due to Plouffe and Borwein & Bradley. Here are some examples:
$$
begin{aligned}
zeta(3)&=frac{7pi^3}{180}-2sum_{n=1}^infty frac{1}{n^3(e^{2pi n}-1)},\
sum_{n=1}^infty frac{1}{n^3,binom {2n}n} &= -frac{4}{3},zeta(3)+frac{pisqrt{3}}{2cdot 3^2},left(zeta(2, tfrac{1}{3})-zeta(2,tfrac{2}{3}) right).
end{aligned}
$$
Moreover, in this Math.SE post we have:
$$
frac{3}{2},zeta(3) = frac{pi^3}{24}sqrt{2}-2sum_{k=1}^infty frac{1}{k^3(e^{pi ksqrt{2}}-1)}-sum_{k=1}^inftyfrac{1}{k^3(e^{2pi ksqrt{2}}-1)}.
$$
You can also check out this paper by Vepstas, which provides a nice generalization to some of these identities.
$endgroup$
add a comment |
$begingroup$
While Apéry proved in 1978 that $zeta(3)$ is irrational, the irrationality of $frac{zeta(3)}{pi^3}$ is still an open problem. There are some formulas expressing odd zeta values in terms of powers of $pi$, the most known ones are due to Plouffe and Borwein & Bradley. Here are some examples:
$$
begin{aligned}
zeta(3)&=frac{7pi^3}{180}-2sum_{n=1}^infty frac{1}{n^3(e^{2pi n}-1)},\
sum_{n=1}^infty frac{1}{n^3,binom {2n}n} &= -frac{4}{3},zeta(3)+frac{pisqrt{3}}{2cdot 3^2},left(zeta(2, tfrac{1}{3})-zeta(2,tfrac{2}{3}) right).
end{aligned}
$$
Moreover, in this Math.SE post we have:
$$
frac{3}{2},zeta(3) = frac{pi^3}{24}sqrt{2}-2sum_{k=1}^infty frac{1}{k^3(e^{pi ksqrt{2}}-1)}-sum_{k=1}^inftyfrac{1}{k^3(e^{2pi ksqrt{2}}-1)}.
$$
You can also check out this paper by Vepstas, which provides a nice generalization to some of these identities.
$endgroup$
While Apéry proved in 1978 that $zeta(3)$ is irrational, the irrationality of $frac{zeta(3)}{pi^3}$ is still an open problem. There are some formulas expressing odd zeta values in terms of powers of $pi$, the most known ones are due to Plouffe and Borwein & Bradley. Here are some examples:
$$
begin{aligned}
zeta(3)&=frac{7pi^3}{180}-2sum_{n=1}^infty frac{1}{n^3(e^{2pi n}-1)},\
sum_{n=1}^infty frac{1}{n^3,binom {2n}n} &= -frac{4}{3},zeta(3)+frac{pisqrt{3}}{2cdot 3^2},left(zeta(2, tfrac{1}{3})-zeta(2,tfrac{2}{3}) right).
end{aligned}
$$
Moreover, in this Math.SE post we have:
$$
frac{3}{2},zeta(3) = frac{pi^3}{24}sqrt{2}-2sum_{k=1}^infty frac{1}{k^3(e^{pi ksqrt{2}}-1)}-sum_{k=1}^inftyfrac{1}{k^3(e^{2pi ksqrt{2}}-1)}.
$$
You can also check out this paper by Vepstas, which provides a nice generalization to some of these identities.
answered Dec 10 '18 at 10:38
KlangenKlangen
1,72811334
1,72811334
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3
$begingroup$
No one has proved anything like this.
$endgroup$
– Lord Shark the Unknown
Sep 16 '17 at 18:02
4
$begingroup$
This is a well-known question, see this MO-question.
$endgroup$
– Dietrich Burde
Sep 16 '17 at 18:04
2
$begingroup$
A significant amount of effort has gone into this, but the nature of Apery's constant is still largely mysterious.
$endgroup$
– George Coote
Sep 16 '17 at 18:07
4
$begingroup$
I am not sure whether the problem is open. But I would be rather surprised if $large frac{zeta(3)}{pi^3}$ turned out to be rational. My guess is that it is even transcendental (of course, only a guess). The continued fraction I calculated with PARI/GP with $20 000$ digits accuracy, contains $19501$ entries not exceeding $134656$. So, if the constant IS rational, numerator and denominator must be very large.
$endgroup$
– Peter
Sep 16 '17 at 18:10
2
$begingroup$
Cross-site duplicate
$endgroup$
– Wojowu
Sep 16 '17 at 18:21