We can determine the asymptotics of partial sums involving the divisor function accurately by means of, for example, the hyperbola method: $$\sum_{n\leq N}\frac{d(n)}{n}=\frac{1}{2}(\log(N))^{2}+2\gamma\log (N)+\gamma^{2}-2\gamma_{1}+O\left(N^{-1/2}\right). \qquad \qquad (*)$$ A derivation of the result above by means of this method can be found in Eric Naslund's answer over <a href="https://math.stackexchange.com/questions/471326/the-asymptotic-expansion-for-the-weighted-sum-of-divisors-sum-n-leq-x-frac">here</a>. The expression can also be obtained through the use of the Mellin-Perron summation formula. 

I wonder whether the result may also be found through the use of generating functions. Recall that: $$ \sum_{n=0}^{\infty} d(n)q^{n} = \sum_{n=1}^{\infty} \frac{q^{n}}{1-q^{n}} .$$ If we slightly adapt this result to our purposes by means of <a href="https://math.stackexchange.com/questions/273275/generating-function-for-the-divisor-function">this</a> answer, we find that $$\sum_{n=1}^{N} d(n)q^{n} =  \sum_{n=1}^{\infty} \frac{q^{n(N+1)}}{q^{n}-1} - \frac{q^{n}}{q^{n}-1} .$$

So if we now divide by $q$, integrate from $q=0$ to $q=z$, switch integral and summation operators, and let $z \to 1$, we obtain an expression for the titular sum. Let's integrate <a href="https://www.wolframalpha.com/input/?i=integrate+q%5E%28n-1%29%2F%281-q%5E%28n%29%29+dq+">term</a> by <a href="https://www.wolframalpha.com/input/?i=integrate+q%5E%28n*%28N%2B1%29-1%29%2F%28q%5E%28n%29-1%29+dq+">term</a>: 

$$\int \frac{q^{n-1}}{q^{n}-1} dq = \frac{\log(1-q^{n})}{n} + C_{1} ,$$ and $$\int \frac{q^{n(N+1)-1}}{q^{n}-1} dq = - q^{n(N+1)} \frac{\ {_2F_1}\left(1,N+1;N+2;q^{n}\right)}{n(N+1)} + C_{2} .$$ Here, the later equality involves a hypergeometric series. 

So these expressions need to be evaluated in the aforementioned integration limits and summed over $n \in \mathbb{Z}_{\geq 1}$. Now, I'm not sure how to continue. 

**Question**: can an asymptotic expression akin to the one in $(*)$ - i.e., one that includes the constant terms - be obtained by pursuing this generating functions method further?