Many people have tried and failed to extend Apery's Irrationality proof of $\zeta(3)$ to Catalan's constant, by looking for a fast converging series for Catalan's constant analogous to the one for $\zeta(3)$ that Apery utilized:

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

see this question: Why can we not establish the irrationality of Catalan's constant the same way as $\zeta(3)$?

which unfortunately didn't receive an answer.

My question is different and specifically relates to Beukers-like irrationality proof for $\zeta(2)$/$\zeta(3)$ as most clearly articulated recently by F. M. S. Lima in Beukers-like irrationality proofs for $\zeta(2)$ and $\zeta(3)$

For analogous Beukers-like irrationality proof applied in the case Catalan's Constant, does a proof fail in the initial lemma's or in evaluating and applying the unit square integral?

Lets rewrite the Lemma's applying to $\zeta(2)$ given by F. M. S. Lima and apply them to an analogous unit square integral for Catalan's Constant, $G$, that is "Lemma N" in the paper becomes "Lemma N_G" here:

Lemma 1G A unit square integral for Catalan's Constant $$\int_0^1 \int_0^1 \frac{1}{1+(x y)^2} \,dx\,dy= G$$

Lemma 2G ($I_{2r,2r}$) For all odd integers $r>0$ $$\int_0^1 \int_0^1 \frac{x^{2r}y^{2r}}{1+(x y)^2} \,dx\,dy= G-\sum_{m=1}^{2r}\frac{(-1)^{m-1}}{(2m-1)^2}$$

Lemma 3G ($I_{2r,2s}$) Let r and s be non-negative odd integers, $r\ne s$. Then $$\int_0^1 \int_0^1 \frac{x^{2r}y^{2s}}{1+(x y)^2} \,dx\,dy=\frac{\widetilde{h_s}-\widetilde{h_r}}{2(r-s)}$$ where $\widetilde{h_n}=\sum_{m=1}^n \frac{(-1)^{m-1}}{(2m-1)}$, an alternating analog of the Harmonic Number.

Lemma 4G ($I_{2r,2r}$ as a linear form). For all odd integers $r>0$ $$I_{2r,2r}=G-\frac{z_{2r}}{(d_{2r})^2} $$ for some $z_{2r} \in {\mathbb{N}}^*$. Where $d_{r}=lcm(1^2,3^2,5^2,...,r^2)$

Lemma 5G ($I_{2r,2s}$ is a positive rational). For all odd $r,s \in {\mathbb{N}},\, r \ne s,$ $$I_{2r,2s}=\frac{z_{2r,2s}}{(d_{2r})^2} $$ for some $z_{2r,2s}\in {\mathbb{N}^*}$

Lemma 6G and Lemma 7G These are written in terms of $n$ and therefore both Lemma's can be written in terms of $2n$

If the analogous lemma's can be all proved correct for odd integer $r,s>0$ then presumably a proof in regards to $G$ must fail in the main part of the proof, what Lima labels Theorem 1, i.e. in the process of evaluating the new unit square integral and applying the result; which for me is the hardest part of this proof.

Added One difficulty is defining the even powered polynomials to multiply together - the resultant terms will have more than one factor of 2 so the lemmas above need to be modified.


Thank you for citing my arXiv paper on the irrationality of zeta(2) and zeta(3). I've also tried to develop a similar proof for Catalan's constant, but the last steps need some magic modifications, so maybe this is not the proper way to deal with the irrationality of this constant. A more promising way is the search for rapidly-converging Apery-like series, as those in my other arXiv paper https://arxiv.org/abs/1207.3139


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.