= Character theta proof of the eta product
{title2=$\eta(\tau)=\tfrac12\sum_n\chi_{12}(n)q^{n^2/24}$}
Take the even <primitive Dirichlet character> $\chi_{12}$ whose values at $1,5,7,11$ modulo twelve are $1,-1,-1,1$. Its <Gauss sum of a Dirichlet character> is $\sqrt{12}$. The displayed series $A$ has $A(-1/\tau)=\sqrt{-i\tau}A(\tau)$ and $A(\tau+1)=e^{\pi i/12}A(\tau)$. Thus $A^{24}$ is the normalized weight-twelve <modular discriminant>. It has no interior zeros by the <valence formula for the modular group>. Its <logarithmic derivative> equals $E_2/24$, so integration gives the <Dedekind eta function> product. The odd character modulo four similarly produces a series $B$ with $B^8=A^{24}$ and leading term $q^{1/8}$; consequently $B=A^3$.
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