Eight-square representation formula ($r_8(n)=16(-1{)}^n{\sum }_{d\mid n}(-1{)}^d{d}^3(n>0)$)

The number $r_8(n)$ of ordered integer eight-tuples whose squares sum to $n$ satisfieswith $r_8(0)=1$ separately. For the theta series of integer squares $\theta (\tau )={\sum }_{m\in \mathbb{Z}}{e}^{2\pi i{m}^2\tau }$, the precise shifted identity iswhere $E_4$ is the normalized Eisenstein series with constant Fourier coefficient one. The coefficient of $q^n$, $q=e^{2\pi i\tau }$, on the left is $(-1{)}^n{r}_8(n)$, because an integer and its square have the same parity. Comparing the Fourier coefficients with the divisor-sum expansion of $E_4$ gives the displayed formula. For example, $r_8(1)=16$, from choosing one of eight coordinates and either sign.