I know that no one has yet been able to prove that the zeta function is irrational at every point $N$. but my question is why is it so difficult to prove the irrationality of the zeta function for $N$?
For example i find a good argument why the zeta function should be irrational (not a proof i know):
Assume that the zeta function at every point is rational you could write it like that
$$\frac{p_n}{q_n}= \zeta(n)= 1+\frac{1}{2^n}+\frac{1}{3^n}+\frac{1}{4^n}+\frac{1}{5^n}+\ldots$$
Now divide the sum on the left into $2$ sums with the proviso that one side shares a factor with $q_n$ and the other does not.
\begin{align*}\frac{p_n}{q_n} &= (\text{not Factor with }q_n) + \left[\frac{1}{q_n^n}+\frac{1}{(2q_n)^n}+\frac{1}{(3q_n)^n}+\frac{1}{(4q_n)^n}+\ldots\right] \\ &= (\text{not Factor with }q_n) + \frac{1}{q_n^n}\cdot\left[1+\frac{1}{2^n}+\frac{1}{3^n}+\frac{1}{4^n}+\frac{1}{5^n}+\ldots\right] \\ &= (\text{not Factor with }q_n)+ \frac{1}{q_n^n}\cdot\zeta(n) \\ &= (\text{not Factor with }q_n) + \frac{p_n}{q_n^{n+1}} \end{align*}
Then reshaping creates the requirement that both sides have a factor with $q_n$, which is a contradiction!
What are your considerations for this "proof"? And does anyone have an idea how to fully prove it with this method?
It's always hard to say when an idea that hasn't produced a proof could lie on the road to a proof. That said, in my mind, your argument doesn't get us any closer.
The key issue: the term you call "(not Factor with $q_n$)". You don't give any suggestion of how we can prove anything about it. Specifically, how do we know it isn't the rational number $\frac{p_n}{q_n}-\frac{p_n}{q_n^{n+1}}$? Proving that looks just as hard as proving that $\zeta(n)$ is irrational.