  We show for even positive integers $n$ that the quotient of the Riemann zeta
values $\zeta(n+1)$ and $\zeta(n)$ satisfies the equation
$$\frac{\zeta(n+1)}{\zeta(n)} = (1-\frac{1}{n}) (1-\frac{1}{2^{n+1}-1})
\frac{\mathcal{L}^\star(\mathfrak{p}_n)}{\mathfrak{p}_n'(0)},$$ where
$\mathfrak{p}_n \in \mathbb{Z}[x]$ is a certain monic polynomial of degree $n$
and $\mathcal{L}^\star: \mathbb{C}[x] \to \mathbb{C}$ is a linear functional,
which is connected with a special Dirichlet series. There exists the
decomposition $\mathfrak{p}_n(x) = x(x+1) \mathfrak{q}_n(x)$. If $n = p+1$
where $p$ is an odd prime, then $\mathfrak{q}_n$ is an Eisenstein polynomial
and therefore irreducible over $\mathbb{Z}[x]$.
