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I'm young, and have been studying this number for quite some time. Possible suspects for a closed form i have personally encountered through ghetto makeshift studyies are:

  • Euler-Mascheroni Constant
  • Glaisher Constant
  • Cube root of two, i.e $\sqrt[3]2$
  • $\displaystyle\frac{\pi\tanh[\pi\sqrt{3}]}{\sqrt{3}}$
  • Random values of Inverse Tangent, Inverse Hyperbolic tangent.

The cube root of two and Euler's Constant are especially likely suspects, but I'm confident that the cube root of two is a coefficient for the true closed form. They appear frequently when I'm trying different methods to evaluate $\zeta(3)$.

I would like to hear your opinions, if you have any, about the relationship between known constants and $\zeta(3)$. I know many people believe that odd values of/for $\zeta(3)$ are unique in a sense where they are unrelated to other known constants, but I am hoping this is not true.

Also, I was wondering if someone could help me find the closed form for the real part of a complex valued Digamma function, or if this series is related to $\zeta(3)$ at all.

RE60K
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Dan Budris
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  • Welcome to math.se! What is your question exactly? – Travis Willse Apr 10 '15 at 07:22
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    The series you mention can be expressed in terms of $~\dfrac{\sinh\big(\pi\sqrt i~\big)}{\sin\big(\pi\sqrt i~\big)}$ – Lucian Apr 10 '15 at 07:31
  • A lot of work has been done, trying to relate $\zeta(3)$ to other constants. Type "integer relations" into the internet, and see what comes back at you. – Gerry Myerson May 01 '15 at 12:28
  • If you don't mind, could you share your work on Apery's Constant with me? – zerosofthezeta Aug 18 '15 at 21:55

2 Answers2

5

This is not an answer, but it is too long for a comment.

If you look at sequence $\rm A002117$ at $\rm OEIS$, you will find a very nice approximation of Apéry's constant . It is given by $$\zeta(3) \approx\frac{236 }{197}\log ^3(2)-\frac{283\pi}{394} \log ^2(2)+\frac{11\pi ^2}{394} \log (2)+\frac{209}{394} \log ^3\left(1+\sqrt{2}\right)+\frac{93 \pi C}{197}-\frac{5}{197}$$ and the first $22$ digits are correct.

RE60K
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Claude Leibovici
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  • Seemingly remarkable mathematical coincidences are easy to generate... For all those constants sitting within that expression, 22 significant digits are a meager prize... – Klangen Jan 04 '19 at 11:29
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No closed-form expression is known for $\zeta(3)$. The closest you can get to an expresison involing known constants are due to Plouffe and Borwein & Bradley:

$$ \begin{aligned} \zeta(3)&=\frac{7\pi^3}{180}-2\sum_{n=1}^\infty \frac{1}{n^3(e^{2\pi n}-1)},\\ \sum_{n=1}^\infty \frac{1}{n^3\,\binom {2n}n} &= -\frac{4}{3}\,\zeta(3)+\frac{\pi\sqrt{3}}{2\cdot 3^2}\,\left(\zeta(2, \tfrac{1}{3})-\zeta(2,\tfrac{2}{3}) \right). \end{aligned} $$

Moreover, in this Math.SE post we have:

$$ \frac{3}{2}\,\zeta(3) = \frac{\pi^3}{24}\sqrt{2}-2\sum_{k=1}^\infty \frac{1}{k^3(e^{\pi k\sqrt{2}}-1)}-\sum_{k=1}^\infty\frac{1}{k^3(e^{2\pi k\sqrt{2}}-1)}. $$

You can also check out this paper by Vepstas, which provides a nice generalization to some of these identities.

Klangen
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