Theta series of integer squares ($\theta (\tau )={\sum }_{n\in \mathbb{Z}}{e}^{2\pi i{n}^2\tau }$)

This convention for the Jacobi theta function is the usual theta constant evaluated at twice the argument. The Poisson summation formula gives $\theta (-1/(4\tau ))=\sqrt{-2i\tau }\theta (\tau )$, with the square root holomorphic on the half-plane and positive on the imaginary axis after evaluating its positive real argument.

The number of ordered integer eight-tuples with square sum $n$ satisfies the displayed divisor formula for positive $n$, with $r_8(0)=1$ separately. The eighth power of the shifted theta series of integer squares is $(16E_4(2\tau )-E_4(\tau ))/15$. Its coefficient sign is $(-1{)}^n$, because squares and their integer roots have the same parity.