Let $\sigma_m (n)=\sum_{d|n}d^m$, and $d(n)=\sigma_0(n)$ as stated in the title.

My question is: Does the q-series $f(q)=\sum_{n\ge 1} d(n)q^n$ gives something like a modular form (i.e. with some kind of identity under the action of $SL_2$) when $q=e^{2\pi i z}$?

When $d(n)$ is replaced by $\sigma_{2m-1} (n)$ the q-series gives out Eisenstein series, which are exactly modular forms. One way to obtain such a relation is to use Hecke's converse theorem for the L-function $\zeta(s) \zeta(s-2m+1)$. So I feel natural to think about $\zeta(s)^2$ for properties of $\sum_{n\ge 1} d(n)q^n$.

I tried to modify the argument of Hecke's converse theorem but it does not really give me a neat relation.

Any help will be appreciated. Thanks in advance.