Define 
$$
a_n=\begin{cases}\frac{2\log\log n}{\log n}&:&\text{$n$ is odd,}\\0&:&\text{$n$ is even,}\end{cases}\hspace{1cm}b_n=\begin{cases}0&:&\text{$n$ is odd,}\\\frac{2\log\log n}{\log n}&:&\text{$n$ is even.}\end{cases}
$$
It's not hard to check that $\zeta(a)=\zeta(b)=\infty$, but $\zeta(a+b)<\infty$, so $D$ is not closed under addition.