If I'm right it is easy to write in closed-form the series $$\sum_{n=2}^\infty\frac{S(n)-S(n-1)}{n^3},\tag{1}$$ where for integers $m\geq 1$, the arithmetic function $S(m)$ is defined as $$S(m)=\sum_{k=1}^m m\text{ mod }k,\tag{2}$$ that is, the $S(m)$ is the so-called sum of remainders function.
Example. For $m=5$, one has that $$S(5)=5\text{ mod }1+5\text{ mod }2+5\text{ mod }3+5\text{ mod }4+5\text{ mod }5=0+1+2+1+0=4.\square$$
Then yesterday while I wrote those symbolic calculations I wondered if is it possible to write a sum likes previous $(1)$ as an integral, having such integral representation a simple form.
Question. Write as a simple (definite) integral, any, the quantity $$\sum_{n=2}^\infty\frac{S(n)-S(n-1)}{n^3}.$$ Many thanks.
Isn't required that you calculate the value of this real number since I know it, j, that I am asking is about a simple integral representation for this real number $(1)$. What is your solution? You can use the properties of the integral that you need (Lebesgue, Riemann, Stieltjes).