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Let $\chi_{s,k}$ be the characteristic function of integers $n$ which are expressed as sum of $s$ positive $k$-th powers i.e $\chi_{s,k}(n)=1$ if and only if $n=a_1^k+\cdots+a_s^k.$ Examples of this type of functions is well-studied specially when $k$ and $s$ are of small values:

$\chi_{2,2}(n)=1 $ if and only if $v_p(n)\equiv 0 [2]$ for all $p\equiv 3 [4].$

$\chi_{3,2}(n)=1 $ if and only if $n\neq 4^m(8k+7).$ etc... Can someone help me by providing some references that make us understand the analytic properties of these functions?

Thanks in advance.

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    $\begingroup$ Well, for $k\geq 3$ basically everything we know about $\chi_{s,k}$ and your earlier $r_{s,k}$ goes under the label of "Waring's problem". Use Google and check out Trevor Wooley's webpage for the latest (and strongest) results. $\endgroup$
    – GH from MO
    Commented Jan 22, 2018 at 23:11
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    $\begingroup$ Dear @GHfromMO. I would rephrase OP's question like this: can one define something like a "Waring's zeta function" which gives (perhaps conjectural) information about this chi's? I've thought about some ad-hoc constructions, but maybe they are useless. I don't know... I took a (brief) look at google and MR, and I found some papers that use Hasse-Weil zeta functions. A survey that mentions this (in page 5) is www.personal.psu.edu/rcv4/Waring.pdf. $\endgroup$
    – efs
    Commented Jan 23, 2018 at 13:35

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