It's a follow up of Show that : $\frac{1}{\zeta(3)}<2C-1$
I showed by hand that :
$$\frac{1}{\zeta(3)}<2C-1$$
Using classical continued fraction .
Now I want to go further and a conjecture :
Conjecture:
$\exists a_i,a_i>3 \exists b_i, b_i \in [1,2]$ all integers then it seems we have :
$$\zeta(3)(2C-1)=1+\sum_{i=1}^{\infty}(-1)^{b_i}(\zeta(a_i)-1)$$
For example we have :
$$0<\sum_{n=1}^{100}\frac{1}{n^{15}}-\frac{1}{2}\int_{0}^{\infty}\frac{x^{2}}{e^{x}-1}dx\int_{0}^{\infty}\left(\frac{x}{\cosh\left(x\right)}-xe^{-x}\right)dx-\sum_{n=2}^{100}\frac{1}{n^{19}}-\sum_{n=2}^{100}\frac{1}{n^{24}}+\sum_{n=2}^{100}\frac{1}{n^{27}}+\sum_{n=2}^{100}\frac{1}{n^{33}}<7*10^{-11}$$
Question :
As I have no idea to show or evaluate the difficulty of the problem :
How to show the existence of $a_i,b_i$ ?
Any helps is awesomely appreciated .