I am trying exercises from Apostol Modular functions and Dirichlet series in number theory and I am stuck on this problem from Chapter -1 .
Problem image is
Image of theorem 1.18
I am not able to think how to prove this result
Can someone please help.
Your definitions are equivalent to the following versions given by Ramanujan : \begin{align} \sum_{n=1}^{\infty}\tau(n)q^n&=q\prod_{n=1}^{\infty} (1-q^n)^{24}=\eta^{24}(q)\notag\\ Q(q)&=1+240\sum_{n=1}^{\infty}\frac{n^3q^n}{1-q^n}=1+240\sum_{n=1}^{\infty} \sigma_{3}(n)q^n\notag\\ R(q) &=1-504\sum_{n=1}^{\infty} \frac{n^5q^n}{1-q^n}=1-504\sum_{n=1}^{\infty} \sigma_{5}(n)q^n\notag \end{align} It can be proved with some effort that $$Q^{3}(q)-R^{2}(q)=1728\eta^{24}(q)$$ and from this it follows that $$1728\tau(n)=240^3((\sigma_{3}\circ\sigma_{3})\circ\sigma_{3})(n)-504^2(\sigma_{5}\circ\sigma_{5})(n)$$ and dividing by $1728$ we are done.