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).