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!



Upper bounds on the number of representation of a natural number as a sum of $s$ positive $k$-th powers.
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.


