Consider the generating function for integer partitions
$$
P(q)=\frac{1}{\prod_{k=1}^{\infty}(1-q^k)}=\sum_{n=0}^{\infty}
p(n)q^n\ ,
$$
where $p(n)$ is the number of integer partitions of $n$.
I am interested in its behavior when $q\rightarrow 1^{-}$. More precisely, let
$$
f(t)=\ln P(e^{-t})
$$
and consider the full asymptotic expansion as $t\rightarrow 0^{+}$.
When working this out, I got
$$
f(t)\sim \frac{\pi^2}{6t}+\frac{1}{2}\ln t-\frac{1}{2}\ln(2\pi)+\frac{t}{24}+O(t^{\infty})\ ,
$$
where $O(t^{\infty})$ means a function of $t$ which vanishes to infinite order when $t\rightarrow 0^{+}$.

However, in Exercise 3 of Chapter 3 of the book "Asymptotic Methods in Analysis", de Bruijn says
$$
f(t)\sim \frac{\pi^2}{6t}+\frac{1}{2}\ln t-\frac{1}{2}\ln(2\pi)-\frac{t}{24}+O(t^{\infty})\ ,
$$
i.e., with the opposite sign for the term linear in $t$.
I have spent a ridiculous amount of time trying to find a sign error in my computations, and I feel I could use some help from the community of experts on this rather classical topic.

**My question:** Are there references where this de Bruijn exercise is worked out completely in detail? That would help me find, in comparison with my work, where the sign error might be.


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Edit: I should have mentioned the references I already looked at, mostly gleaned from the article "Khinchin Families and Hayman Class" by Cantón et al. I have looked at the book by flajolet and Sedgewick, the article "Hardy-Ramanujan's Asymptotic Formula for
Partitions and the Central Limit Theorem" by Báez-Duarte, as well as Section 3.2 of the original article by Hardy and Ramanujan. If there is no article with the wanted detailed calculation, I kind of hoping maybe someone taught, e.g., a course in analytic number theory and wrote that up as homework solutions for the students.