Is Apéry's constant a rational multiple of $ pi ^ 3$?












2












$begingroup$


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







share|cite|improve this question











$endgroup$








  • 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


















2












$begingroup$


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







share|cite|improve this question











$endgroup$








  • 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
















2












2








2





$begingroup$


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







share|cite|improve this question











$endgroup$




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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








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
















  • 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












1 Answer
1






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0












$begingroup$

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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    active

    oldest

    votes









    0












    $begingroup$

    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.






    share|cite|improve this answer









    $endgroup$


















      0












      $begingroup$

      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.






      share|cite|improve this answer









      $endgroup$
















        0












        0








        0





        $begingroup$

        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.






        share|cite|improve this answer









        $endgroup$



        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.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 10 '18 at 10:38









        KlangenKlangen

        1,72811334




        1,72811334






























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