# Finding an English version of Hjortnaes' paper on the series representation of $\zeta(3)$

Hjortnaes published a paper in 1953 establishing the following series representation for $\zeta(3)$, where $\zeta$ is the Riemann Zeta function:

{\begin{aligned}\zeta (3)&={\frac {5}{2}}\sum _{k=1}^{\infty }(-1)^{k-1}{\frac {k!^{2}}{(2k)!k^{3}}}\\&={\frac {5}{2}}\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{{\binom {2k}{k}}k^{3}}}\end{aligned}}}

The paper is entitled "Overføring av rekken $\sum _{k=1}^{\infty }\left({\frac {1}{k^{3}}}\right)$ til et bestemt integral", and appeared in the Scandinavian Mathematical Society's Proc. 12th Scandinavian Mathematical Congress. Unfortunately the paper is in Norwegian, which I do not understand. A (quick) Google search did not yield any English translations for this paper. Does anyone know where I can find one?

• I am pretty sure that there is no English version of that paper. Just in case you are more interested in the formula itself, you can look for Roger Apery's paper instead. – polfosol Jun 5 '18 at 17:35
• Wikipedia Apéry's constant writes that Markov found the proof before Hjortnæs. This is their reference: Markov, A. A. (1890), "Mémoire sur la transformation des séries peu convergentes en séries très convergentes", Mém. De l'Acad. Imp. Sci. De St. Pétersbourg, t. XXXVII, No. 9: 18pp. – Jeppe Stig Nielsen Mar 2 at 14:27
• @JeppeStigNielsen That is highly interesting, I have not heard of that before. Do you have a link? I could not find it anywhere. – Klangen Mar 2 at 20:42
• I think this link works: https://www.biodiversitylibrary.org/page/46874381 – Jeppe Stig Nielsen Mar 2 at 21:27
• @JeppeStigNielsen Amazing, thank you – Klangen Mar 3 at 7:59

Unfortunately there is no English translation of Hjortnaes' paper. Moreover, after hours of searching I was not able to find Apéry's original paper proving the irrationality of $\zeta(3)$.