My question is twofold:
Can the Waring problem be expressed with the Jacobi theta function or some analog (as is the case for $k=2$) for general $k$? Say for $k=4$ or $k=6$, are these able to be understood with some properties of the theta function?
w. Can the Hardy-Littlewood circle method (used for $k=4$) be used for asymptotic bounds of general $k$, or at least small $k$ (say $k=6$)?