Eight-square representation formula
= Eight-square representation formula
{title2=$r_8(n)=16(-1)^n\sum_{d\mid n}(-1)^dd^3\quad(n>0)$}
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.