Covariance of two independent zeta(2) variables

Let $A$ and $B$ be independent, each with the $\text{Zeta}(2)$ distribution: $\mathbb{P}(A=k)=C/k^2$ for every positive integer $k$, where $C$ is the normalising constant. What is $\operatorname{Cov}(A,B)$? Answer $1$ if it equals $+\infty$, $2$ if it equals $-\infty$, and $3$ if it is $0$ or undefined.

$3$. The mean is $C\sum_k k\cdot\tfrac1{k^2}=C\sum_k\tfrac1k=\infty$, so $A$ and $B$ have no finite expectation. The covariance $\mathbb{E}[AB]-\mathbb{E}[A]\mathbb{E}[B]$ then takes the form $\infty-\infty$ and is undefined. Independence alone does not give $0$ here, because covariance needs finite means.