Let $d(n)$ be the number of divisors function, i.e., $d(n)=\sum_{k|n} 1$ of the positive integer $n.$ I know about some of the ''gross'' averages for this function, such as the estimate $$ \sum_{n\leq x} d(n)=x \log x + (2 \gamma -1) x +{\cal O}(\sqrt{x}) $$ as well as its variability, e.g., the lim sup of the fraction $$ \frac{\log d(n)}{\log n/\log \log n} $$ is $\log 2$ while the lim inf of $d(n)$ is $2,$ achieved whenever $n$ is prime.

How much is known about the statistics of $d(n)$? In particular, if we let $N$ grow to infinity, is there any way to bound a sum of the form $$ \left| \sum_{n=1}^N \varepsilon_n d(n) \right| $$ from below for all or almost all $(\varepsilon_1,\cdots,\varepsilon_N)\in \{\pm 1\}^N$?