Let $r_{s,k}(n)$ be the number of representation of a naturel number $l$ as a sum of $s$ positive $k$-th powers. Joung Min Song introduced some conditions to study asymtotic behavior of some positive and multiplicative functions $f$ satisfying these conditions:
(i)$$\sum_{p\leq z} f(p) \log(p)=\kappa z+O(z/(\log{z})^{\delta}) \quad (z>1), $$ (ii) $$\sum_{p} \sum_{\nu \geq 2} \frac{f(p^{\nu})}{p^{(1-\eta)\nu}}\leq A$$ for some suitable constants $\kappa,$ $\delta>0,$ $0<\eta<1/2$ and $A>0.$
My question is: can the function $r_{s,k}$ satisfy these two conditions and for which values of variables $\kappa,$ $\delta,$ $\eta$ and $A$?
Many thanks in advance.
