In [1], Kontsevich and Zagier provide us a definition of a period, (page 3) and examples of complex numbers being periods and complex numbers that aren't periods (pages 3-5).
Question. Let $\zeta(s)$ the Riemann's Zeta function. (Since I believe that it is an unsolved problem and very difficult) I am asking about if you can provide us an heuristic to deduce conjecturally if $$\frac{1}{\zeta(s)}$$ for integers $s\geq 2$, are periods? Since I don't know if this problem was in the literature, add the reference, if you know it, as an answer. Many thanks.
I believe that find such integral representation for $1/\zeta(m)$ with $m\geq 2$ an integer should be very difficult or unknown, then I am asking about an heuristic with the purpose to answer the question. What are you saying?
References:
[1] Kontsevich and Zagier, Periods, Institut des Hautes Études Scientifiques (2001).
[2] A different reference, in spanish, is page 555 of Waldschmidt, Una introducción elemental a valores zeta múltiples, La Gaceta de la RSME, Volumen 17, número 3 (2014).