I attempted to generalise the sum of remanders function $R(n)=\sum_{m=1}^{n}(n\space\text{mod}\space m)$. If we have:
$$P_z(n) := \sum_{m=1}^{\left\lfloor \sqrt[z]{n}\right\rfloor}\left(n\;\text{mod}\;m^z\right),\;\;\; P^k(n) := \sum_{m=1}^{n}\left(n\;\text{mod}\; m\right)^k$$
Do the following hold?
$$\lim_{n\to\infty}\left(\frac{P_z(n)}{n^{1+\frac{1}{z}}}\right)=1-\frac{1}{z+1}\zeta\left(\frac{z+1}{z}\right)$$
$$\lim_{n\to\infty}\left(\frac{P^k(n)}{n^{k+1}}\right)=1-\sum_{m=1}^{k}\frac{\zeta(m+1)}{k+1}$$
$$\lim_{n\to\infty}\left(\sum_{m=1}^{n}\frac{P^k(m)}{n^{k+2}}\right)=\frac{1}{k+2}-\sum_{m=1}^{k}\frac{\zeta(m+1)}{(k+1)(k+2)}$$