$\sum_{i=1}^n |\{k \in \mathbb{N} \mid k | i\}|$ What is $\sum_{i=1}^n |\{k \in \mathbb{N} \mid k | i\}|$ asymptotically (as a function of $n$)?
(I'm summing, for each of $1,\dotsc,n$, its number of divisors)
Or at least, what's the best upper bound you can find?

You can find a good summary in the following Wikipedia article.
The main term in the estimate for $\sum_{i=1}^n d(i)$ is 
$$n\log n +(2\gamma -1)n,$$
where $\gamma$ is Euler's $\gamma$.  
There is a large literature on the error term. Dirichlet got $O(n^{1/2})$, and it is known that one cannot do better than $O(n^{1/4})$.
Note: For a number-theoretic function, this is a fantastically precise estimate. We have a dominant term, $n\log n$, that gives us the asymptotic behaviour. We have also an explicitly known correction term $(2\gamma-1)n$.  And, by number-theoretic estimate standards, the error term is quite small compared to $(2\gamma-1)n$.  
