# nt.number theory – Subsets of special values ​​of \$ zeta (k) \$ containing irrational numbers

We consider all the elements $$zeta (2), zeta & # 39; (3), zeta (4), zeta & # 39; (5), ldots$$ or $$zeta (s)$$ is the zeta function of Riemann and $$zeta (s) = frac {d} {ds} zeta (s)$$ its derivative. We therefore consider this set and we designate it by
$$D = { zeta (k) text {for integers} 2 leq k }.$$
After considering the power set $$mathcal {P} (D)$$ and we take an element $$s in mathcal {P} (D)$$ of cardinality $$| s | geq 2$$.

I would like to know if it is possible to state an assertion similar to that of Zudilin's theorem ((1)), look at the second paragraph of the section Whole positive odds from the Wikipedia Particular values ​​of the Riemann zeta function.

Let $$l geq 1$$ to be an integer and $$s in mathcal {P} (D)$$ a subset with cardinality $$| s | geq 2$$ elements of $$D$$. Then we say that this subset $$s$$ has the property $$W_l$$ if and only if at least $$l$$ elements of $$s$$ are irrational numbers.

Question. I would like to know what work can be done to show examples of subsets $$s in mathcal {P} (D)$$ of cardinality $$| s | geq 2$$ for which the property $$W_l$$ is true for an integer $$l geq 1$$. Thank you very much.

It is well known that the closed form for $$zeta ( 2)$$ in terms of certain constants, and it is possible to obtain formulas for $$lim_ {s to 2k} frac {d} {ds} zeta (s),$$ but i do not know if it is possible to get from these an example for my question.

I hope my question has good mathematical content.

## References:

(1) Wadim Zudilin, One of the numbers $$zeta (5), zeta (7), zeta (9), zeta (11)$$ Is irrational, Russian mathematical surveys. 56(4): 774-776 (2001).