Following the definition of Kontsevich and Zagier, is it known if conjecturally $1/\zeta(m)$, with $m\geq 2$ an integer, should be a period? 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).
 A: Here is a heuristic to suggest $(2\pi i)^{-1}$ is not a period in the sense of Kontsevich-Zagier. This would imply that $\zeta(2n)^{-1}$ is not period. The heuristic comes from computing a conjectural action of the absolute Galois group of $\mathbb{Q}$ on the ring of periods. Something similar is probably possible for the odd zeta values.
One source of periods are as matrix coefficients for the comparison between algebraic de Rham cohomology and Betti cohomology of a variety defined over $\mathbb{Q}$. Given a smooth variety $X$, a homology class $\gamma\in H_n(X(\mathbb{C}),\mathbb{Q})$, and an algebraic de Rham cohomology class $\omega\in H^n_{dR}(X)$, the number
$$
\langle\gamma,\omega\rangle:=\int_\gamma\omega\in\mathbb{C}
$$
will be a period.
Let $\mathcal{P}$ denote the ring of periods. Let $\bar{\mathbb{Q}}$ be the algebraic closure of $\mathbb{Q}$ inside $\mathbb{C}$, and write $G=\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$. For each prime $\ell$, there is a conjectural action of $G$ on $\mathcal{P}\otimes_\mathbb{Q}\mathbb{Q}_\ell$ by algebra automorphisms, which is computed on an element $\alpha=\langle\gamma,\omega\rangle\in\mathcal{P}$ as follows. The $\ell$-adic etale cohomology $H^n_{et}(X,\mathbb{Q}_\ell)$ is canonically isomorphic to $H^n(X(\mathbb{C}),\mathbb{Q})\otimes\mathbb{Q}_\ell$, so the action of $\sigma$ on $H^n_{et}(X,\mathbb{Q}_\ell)$ induces an action on $H_n(X(\mathbb{C}),\mathbb{Q})\otimes\mathbb{Q}_\ell$. We may write
$$
\sigma(\gamma)=\sum_i \gamma_i\otimes q_i\in H_n(X(\mathbb{C}),\mathbb{Q})\otimes\mathbb{Q}_\ell,
$$
and we define
$$
\sigma(\alpha):=\sum_i\langle \gamma_i,\omega\rangle\otimes q_i\in\mathcal{P}\otimes\mathbb{Q}_\ell.
$$
A priori the result might depend on the representation of $\alpha$ as $\langle\gamma,\omega\rangle$, but a form of the Grothendieck period conjecture would imply the result is independent of the choice of representation. From now on I will assume this is true.
For all but finitely-many primes $p$, the Galois representation $H^n_{et}(X,\mathbb{Q}_\ell)$ is unramified at $p$, and for such $p$, there is a well-defined action of $\phi_p$ (the Frobenius at $p$) on $H^n_{et}(X,\mathbb{Q}_\ell)$ and the characteristic polynomial of $\phi_p$ has integer coefficients. It follows that if we write $\mathcal{P}(X)$ for finite-dimensional vector space of periods coming from the de Rham-Betti comparison on $X$, then the characteristic polynomial of $\phi_p$ acting on $\mathcal{P}(X)\otimes\mathbb{Q}_\ell$ must also have integer coefficients.
Now, consider the variety $X=\mathbb{A}^1\backslash\{0\}$, which has $2\pi i$ as a period of its first cohomology. The action of $\phi_p$ on $H^1_{et}(X,\mathbb{Q}_\ell)$ is multiplication by $p$, so we have
$$
\phi_p(2\pi i)=2\pi i p.
$$
If $(2\pi i)^{-1}$ were a period, we would have
$$
\phi_p\big((2\pi i)^{-1}\big)=\frac{1}{p}(2\pi i)^{-1},
$$
contradicting the fact that the characteristic polynomial of $\phi_p$ has integer coefficients.
