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.
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 or book with the wanted detailed calculation, I am kind of hoping that maybe someone taught, e.g., a course in analytic number theory and wrote that up as homework solutions for the students.
