This is a question related to the one I posted here, but I have found some more interesting and general results and thought here might be a better place to ask.
Let $R_{k,N}$ denote the remainder of dividing $N$ by $k$. By some random experiments, I discover the following asymptotic relation: $$\lim_{N\rightarrow\infty} \frac{\sum_{k=1}^K \frac{R_{k,N}^L}{k^M}}{\sum_{k=1}^K\frac{k^{L-M}}{L+1}}\approx 1, $$ for some $K$ growing at a sub-linear rate (e.g. $K\sim\mathcal{O}(N^{2/3})$) and $0\leq M\leq L$. For example, when $M=0$ and $L=1$, we have the partial sum of remainders function: $$ \sum_{k=1}^KR_{k,N}\approx \sum_{k=1}^K\frac{k}{2}=\frac{K(K+1)}{2}. $$$$ \sum_{k=1}^KR_{k,N}\approx \sum_{k=1}^K\frac{k}{2}=\frac{K(K+1)}{4}. $$ When $M=L=1$, we have a linear relation: $$ \sum_{k=1}^K\frac{R_{k,N}}{k}\approx \sum_{k=1}^K\frac{1}{2}=\frac{K}{2}. $$ Heuristically, I can think of $R_{k,N}$ as a sequence of quasi random variable distributed uniformly on the interval $[0,k]$ with moments $E[R_{k,N}^L]=\frac{k^{L}}{L+1}$, and therefore the approximation follows since $$\sum_{k=1}^K \frac{E[R_{k,N}^L]}{k^M}=\sum_{k=1}^K\frac{k^{L-M}}{L+1}.$$
However, is there a formal result on the asymptotical behaviour of $R_{k,N}$ that can justify my argument here?