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Let $r_{s,k}(l)$ be the number of representation of a natural number $l$ as a sum of $s$ positive $k$-th powers. Then the circle method of Hardy and Littlewood gives the asymptotic formula

\begin{equation} r_{s,k}(l) = \frac{\Gamma(1+1/k)^s}{\Gamma(s/k)} \mathfrak{S}_{s,k}(l) l^{s/k-1} + o(l^{s/k-1}), \end{equation} where

$\mathfrak{S}_{s,k}(l)$ is the singular series, which is bounded below and above by positive constants which do not depend in $l$, provided that $s$ is sufficiently large with respect to $k$.

Question. For $s>k$, it would be certainly impossible to expect a better upper bound, namely that $r_{s,k}(l) \ll l^{s/k-1-\delta}$ for a positive $\delta$. It seems that this impossibility is well-known to number theorists and there should be an elementary proof of this impossibility. It would be appreciated if you let me know any reference or argument/proof.

Thank you!

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  • $\begingroup$ Could you help me by providing some references about the properties of the number $r_{s,k}(l)$? I would be very grateful to you! $\endgroup$ Jan 21, 2018 at 17:43

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There are $O(x^{1/k})$ $k$-th powers of size at most $x$, so there are $O(x^{s/k})$ sums of $s$ $k$-th powers of size at most $x$ but only $x$ integers of size at most $x$ so some integer at most $x$ needs to be represented at least $\gg x^{s/k -1}$ times.

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  • $\begingroup$ It's a nice simple answer. You mean, if my understanding is correct, since $x^{s/k} \ll \sum_{l=1}^x r_{s,k}(l)$, we cannot have $r_{s,k}(l) \ll l^{s/k-1-\delta}$ for some positive $\delta$. Thank you very much. $\endgroup$
    – user27855
    Feb 28, 2013 at 19:40

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