While I check the proof for entry 10(vii) in Chapter 19 in Ramanujan's notebook, I couldn't understand one equality. It is
\begin{equation}
\prod\_{\zeta}\varphi(q^{1/5}\zeta) = \varphi^6(q)/\varphi(q^5)
\end{equation}
where $\zeta$ is an arbitrary fifth root of unity and
\begin{equation}
\varphi(q):=\sum\_{n=-\infty}^{\infty}q^{n^2}=(q^2;q^2)\_{\infty}(-q;q^2)\_{\infty}^2 \qquad, |q|<1\\
(a;q)\_{\infty}:= \prod\_{n=0}^{\infty}(1-aq^n)\quad.
\end{equation}
Could anyone help me to understand how it holds?



How does $\prod_{\zeta}\varphi(q^{1/5}\zeta) = \varphi^6(q)/\varphi(q^5) $ hold?
First note that $\prod\_{\xi} (1 - \xi^n q^n)$ is equal to $(1- q^n)^5$ if $5 | n$ and to $1 - q^{5n}$ otherwise. Moreover
$$
\varphi(q) = \prod\_{n \geq 1} (1 - q^n)^{e\_n}
$$
where $e\_n = 1,-2,3,$ or $-2$ if $n \equiv 0,1,2$ or $3 \pmod 4$ respectively. In particular $e\_n = e\_{5n}$ $(\*)$
. Thus
$$
\prod\_{\xi} \varphi(\xi q) = \prod\_{n \geq 1} (1 - q^n)^{f\_n}
$$
where $f\_n = 5 e\_n \mathbb{1}\_{5 \mid n} + e\_{\frac{n}{5}} \mathbb{1}\_{5 \mid n} - e\_{\frac{n}{5}} \mathbb{1}\_{5^2 \mid n}$, and thus $f\_n = e\_n (6 \mathbb{1}\_{5 \mid n} - \mathbb{1}\_{5^2 \mid n} )$ by $(\*)$. On the other hand
$$
\varphi(q)^6 / \varphi(q^5) = \prod\_{n \geq 1} (1 - q^n)^{g\_n}
$$
where $g\_n = 6 e\_n - e\_{\frac{n}{5}} \mathbb{1}\_{5 \mid n}$, i.e. $g\_n = e\_n (6 - \mathbb{1}\_{5 \mid n} )$ by $(\*)$.
By noting that $f\_n = \mathbb{1}\_{5 \mid n} g\_{\frac{n}{5}}$, one sees that 
$$
\prod\_{\xi} \varphi(\xi q) = \varphi(q^5)^6 / \varphi(q^{25}).
$$
Note that the the only thing about $e\_n$ we used is its $4$-periodicity (plus the fact that $5 \equiv 1 \pmod 4$).


