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

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.