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My question is twofold:

  1. 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 $k=6$)?

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    $\begingroup$ No, $f(q)=\sum_n q^{n^4}$ is not a theta function. Yes the circle method should give useful estimates because from that the Mellin transform of $\sum_n e^{-n^4 x}$ is $\Gamma(s)\zeta(4s)$ you know a very good asymptotic expansion of $f(q)$ $\endgroup$
    – reuns
    Commented Jan 4, 2020 at 15:24
  • $\begingroup$ @reuns sorry, is the sum you mention the expression for k=4 or have I misunderstood your statement? What would the expression be for k=6? I don’t see, atleast obviously, why there shouldn’t be some expression for general k of the waring problem that atleast resembles a theta function? $\endgroup$
    – alex sharma
    Commented Jan 4, 2020 at 17:15

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