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!