Let $V$ be the set of sequences $a \in\mathbb{R}^\mathbb{N}$ such that $\lim\_{n\to\infty} a\_n = 0$. The set $V$ can be seen as a real vector space, with pointwise addition and scalar multiplication. 


For $a\in V$ we define the "associated $\zeta$-function" to be \begin{eqnarray} \zeta(a) := \sum\_{n=1}^\infty \frac{1}{n^{1+|a\_n|}}
\end{eqnarray}
(similar to post <https://mathoverflow.net/questions/187610/almost-zeta-function>)


Let $D = \{a\in V: \zeta(a) = \infty\}$. Is $D\subseteq V$ a subspace? 



Do the sequences with divergent associated $\zeta$-function form a vector space?
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.


